F- 7 - . J . . ' ' :: , . ?. , I , . , . - - ----:. . . - - > . , , x . . rrr- , *.

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STUDENTS TEXT uNiT No. ,l

! !

MATHEMATICS FOR JUNIOR HIGH SCHOOL VOLUME 1

PART II

SCHOOL MATHEMATICS STUDY GROUP

YALE UNIVERSIlY PRESS

Scllool Mathe~natics Studv Group

Mathc~natics for Junior High C Scl~ool, Volume I

Mathe~natics for Junior High School, ~ o l u ~ n e I

S~rrdt~ri t'r Test, Pr7l.t I 1

Prepared undrr rhc supcrvis io~~ of thc

Yancl on Sevcntll JUJ F.1~11th Gmdes

of the Schcv)l M:lrhc*~n.~tirs St id? C;rt>up:

Ii. 1:). Andcrsol-I

J. A. Bro~vn

L c r l r j ~ - ~ ? J i ) I l l ~

H. W. J< l t~cs

1'. 5 . J O I I ~

1. It. Mayor

l,oui\iana Statc Urllvcrrit?-

U~i;versitv of Dclaw*arc

University of' Cll~cago

Uliivcr<ir\- r > f Colc~radr~

Univcrcit? r>Kh.l tcl l~g~~

Arnerlcall A \ v - . ~ ~ i ~ t i c ~ n for thc

Advaricr!ucrir ilf Scicncc

Un~ i . c~ - \ i t v ~t hliri~icsota

Supcryisor of M n t l ~ ~ ~ n a t i c s ,

Washington. D.C.

C o ~ v r i ~ h t . L FJ 1960, 1961 by Yale Utlivcr~ity. Printed in thc Unitcd Sratzs of Amcrica.

All righn resencd. This book may not bc rcpsoduced, in whole or in part, in ntiv form, without wt-ftccn permission from the publishcr~.

Financial support for thc School Machematicr Stud). Group has bccn provided by rhe National Scicncc Foundation.

The preliminary edition of this volume was prepared at a writing session h e l d a t t h e University or Michigan during the summer of 1959, baaed, in par:, on material prepared at the first 3MSG wrkt lne session, held a t Yale Unlvers l ty in t h e summer of 1958. This re- vision was prepared at Stanford Univers i ty i n the summe:. oP 1960, taklng i n t o account t h e classroom experience with the preliminary edition during the academic year 1959-60.

The following is a list of all those who have participated in the preparation of this volume.

R . D . Anderson, Louisiana State University

B.H. Arnold, Oregon S t a t e College

J.A. Brawn, University of Delaware

Kenneth E. Brown, U.S. Office of Education

Mildred B. Cole, K.D. Waldo Junior High School, Aurors, Illinois

3.H. Colrin, Raein~ Scientific Research Laboratories

J.A. Cooley, Unlverslty o f Tennessee

Richard Dean, California Institute or Technology

H.M. Gehman, University of Buffalo

L. Roland Genise, Brentwood Junios High School, Brentwood, N t w York

E. Glenadine Gibb, Iowa S t a t e Teachers College

Richard Good, University of Maryland

Alice Hach, Racine PuEllc Schoola, Racine, Wisconsin

S.B. Jackson, University of Maryland

Lenore John, University High School, University of Chicago

3.W. Jones, University of Colorado

P.5. Jones, University of Michigan

Houston Karnes, Loulslana State University

Mildred Keif fe r , C inc inna t i Public Schools, Cincinnati, Ohio

Nick Lovdjieff, Anthony Yunlor Hi& School, Ylnneapolis, Ninnesota

J . R . Mayor, American Association Sor the Advancement of Science

Sheldon Meyers , Educatf onal Testing Service

Muriel Mills, Hill Junior High School, Denver, Colorado

P.C. Rosenbloorn, University of Minnesota

Elizabeth Roudebush, Seattle Public Schools, Sea t t l e , Washington

V e r y l Schult , Washington Public Schools, Washington, D . C .

George Schaefer, Alexis I. DuPont High School, Wilrnington, Delaware

Allen Shields, University of Michigan

Rothwell Stephena, Knox College

John Wagner, School Mathematics Study Group, Yew Raven, Connecticut

Ray Walch, Westport Public Schoola, Westport, Connecticut

C.C. Webber, University of Delaware

A.B. Willcox, Amherst College

CONTENTS

Chapter

9 . RATIOS. PERCENTS. AND DECIMALS . . . . . . . . . . 347 9- 1 . Ratios . . . . . . . . . . . . . . . . 347 9- 2 . Percent . . . . . . . . . . . . . . . . 354 9- 3 . DecimalNotation . . . . . . . . . . . 362 9- 4 . operations with Decimals . . . . . . . 365 9- 5 . DeclmalExpansion . . . . . . . . . . . 371 9- 6 . Rounding . . . . . . . . . . . . . . . 375 9- 7 . Percents and Decimals . . . . . . . . . 377 9- 8 . Applications of Percent . . . . . . . . 382

10 . PARALLELS. PARALLELOGRAMS. TRIANGLES. AND R I M i T PRISMS . . . . . . . . . . . . . . . . . . 397 10- 1 . T w o L i n e s in a Plane . . . . . . . . . 397 10- 2 . Three Lines in a Plane . . . . . . . . 404 10- 3 . Parallel Lines and Corresponding Angles 409 10- 4 . Converse ( ~ u r n i n g a Statement round) . 412 10- 5 . Triangles . . . . . . . . . . . . . . . 416 10- 6 . Angles o f a T r i a n g l e . . . . . . . . . 424 10- 7 . Parallelograms . . . . . . . . . . . . 431 10- 8 . Areas of Parallelograms and Triangles . 438 10- 9 . Right Pplsrns . . . . . . . . . . . . . 448 10-10 . Surmnary . . * . . . . . . . . . . * . . 458 10.11 . Historical Note . . . . . . . . . . . . 461

11 . CIRCLES . 11- 1 . 11- 2 . 11- 3 . 11- 4 . 11- 5 . 11- 6 . 11- 7 . 11- 8 . 11- 9 . 11-10 .

. . . . . . . . . . . . . . . . . . . . 463 Circles and the compass . . . . . . . . 463 I n t e r i o r s and Intersections . . . . . . 469 Diameters and Tangents . . . . . . . . 475 Arcs . . . . . . . . . . . . . . . . . 481 Central Angles . . . . . . . . . . . . 486 Circumference of Circles . . . . . . . 492 AreaofaCirc le . . . . . . . . . . . 500 Cylindrlcal Solids-- Volume . . . . . . %9 Cylindrical Sol lds - - Surface Area . . . 514 Optional: Review of Chapters 10 and 11 519

12 . MATHEMATICAL SYSTEMS . . . . . . . . . . . . . . . 12- 1 . A New Kind of Addition . . . . . . . . 12- 2 . A W e w K i n d o f Efultlpllcation . . . . . 12- 3 . What is an Operation? . . . . . . . . . 12- 4 . Closure . . . . . . . . . . . . . . . . 12- 5 . I d e n t i t y Element; Inverse of an

Element . . . . . . . . . . . . . . . . . . . 12- 6 . What is aMathematica1 System? 12- 7 . Mathematical Systems Without

Numbers . . . . . . . . . . . . . . . 12- 8 . The Counting Numbers and the Whole

Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 12- 9 Modular Arithmetic 12-10 . Sumnary and Review . . . . . . . . . .

Chapter

. . . . . . . . . . . . . . 13. STATISTICS AND GRAPHS 579 . . . . . . . . . . . . . 13- 1 Gathering Data 579 . . . . . . . . . . 13- 2 The Broken Llne Graph 582 . 13- 3 Bar a ~ a p h s . . . . . . . . . . . . . . 588 . 13.4 CircleGraphs . . . . . . . . . . . . . 591 . . . . . . . . . . . . 13- 5 Summarizing Data 594 . . . , . . . . . . . . . . . * 13- 6 Sampling 604 . 13- 7 Summary . . . . . . . . . . . . . . . . 607

. . . . . . . . . . 14. MATHEMATICS AT WORK IN SCIENCE 609 . . , . . . . . . . 14- 1 The S c i e n t i f i c Seesaw 609 . . . . . . . . . 14- 2 A Laboratory Experiment 610 14- 3 . Caution: Inductive Reasonlng at Work . 613 . . . . . . . . 14- 4 Graphical In terpreta t ion 615 . . . . . . . . . 14- 5. Other Kinds of Levers 619 14- 6 . The Role of Mathematics in

~cientific ~xper iment . . . . . . . . 619 INDEX . . . . . . . . . . . . . . . . . . following page . 623 (

Chapter 9

RATIOS, PERCENTS, AND DECIMALS

One sunny day a boy measured the length of the shadow cast

by each member of h i s family. He also measured the length of t h e

shadow cast by a b i g t r e e in their yard. He found that his father, who I s 72 inches t a l l , c a s t a shadow 48 inches long. His mother,

who is 63 inches t a l l , c a s t a shadow 42 inches long. His little

bro ther , who is only 30 inches high, caa t a shadow 20 inches long. He didn't know how tall t h e t r e e was, but i t s shadow was 40 feet

long.

L e t us arrange t h i s i n f o r m t i o n In a table.

Fat h e r

Mother

Bro ther

Tree

Shadow length H e i g h t

48 inches 72 inches 42 inches 63 inches

20 inches 30 inches

40 feet ?

We see that the t a l l e r people have longer shadows. But let

! us examine this more c loae ly . Suppoae we d i v i d e the number of I inches in the shadow l ength of the l i t t l e b r o t h e r by the number

20 2 of inches in h l s heigh t . Me get or 3. Suppose we try t h e

same t h ing for the father, It w i l l be easier to measure the father's height and shadow in feet, The father I s 6 feet t a l l and his shadow is 4 feet long. If we divide t h e measure of the shadow length in feet by the measure of his he igh t In feet, we

4 2 get 6 or 3. D i v i d e t h e number of inches in the shadow length of t he

mother by t h e number of inches i n h e r he igh t . Do you agafn 2 g e t 3?

L e t us assume that t h l a principle h o l d s f o r a l l objects (we must meaaure the shadows at the same time and' place; the

shadow changes during the day as the position of t h e sun changes).

Then t h e t r e e must be 60 feet tall in o r d e r that the measure of 2 its shadow length d i v i d e d by t h e measure of its he igh t be 3.

Thus we can discover how t a l l the tree Is without actually

measuring it ! In Chapter 6 we used the term "rat io" and studied some

applications of r a t l o . Def in i t ion . The r a t i o of a number a t o a number b

( b # 0), is the quotient $. (~ometimea t h i a r a t i o is written a:b.)

Ve have formed t h e r a t i o of the measure of shadow length t o t h e measure of height , and we discovered t h a t t h i a ratlo. was

t h e same f o r a l l the people whom we measured. Using this we were a b l e t o dfscover that t h e tree was 60 feet t a l l .

Suppose t h a t the boy's uncle l a 66 inches t a l l ( 5 f e e t 6 inches ) . How long would h i s shadow be if It were measured at the same t i m e and place as t h e o the r people?

To answer t h i s question, we let e be the number of inches

in his shadow l eng th . Then we must have:

The number $ can be expressed as a f r a c t i o n with denominator 66.

s = 44 The uncle's shadow would be 44 inches long.

You have seen t ha t a r a t i o is a comparison of two numbers

by division. The numbers may be measures of physical quantities.

The word, "per," indicates division. We use it t o express the r a t i o of the measures of two physical quantities, such as miles - and - hours. Store prices provide a d d i t i o n a l examples. Pricea relate value to amount such as $1.00 pep pound, or $ .59 per dozen. In each case, the per indicates a ra t io , generally between two different k l n d s of quantities, This l a aometlmes called a rate. Notice further that the second quantity in each caae

represents the s t a n d a r d of comparison. A store charges " 10 6 per comb" ; one comb I s t h e standard of comparison, and t h e y want 10 # In t h e cash r eg i s t e r fo r each comb aa ld . Of course, this standard

does n o t always represent a quanllty of one. For example, cents p e r dozen, dollars per pair (for shoes), dollars per 1000

(for br icks ) . Taking another look a t our example and a t the d e f i n i t i o n of

r a t i o , we see that a r a t i o compares just two numbers. In our - example we had two sets of numbers, one s e t from the lengths of t h e shadows, and one set from the heights, I n each case we formed

the r a t i o of the f i r a t number ( shadow length) t o t h e second number

( h e i g h t ) . In o u r example these r a t i o s were all equal to one another . I n such a situation, w e say t h a t the phys i ca l quantities

measured by the numbers are p r o p o r t i o n a l to one anather . A

proportion I s a statement of equality of two r a t l o s .

Thus In ou r example on t he first page of t h i s chapter, the

shadow length Is proport ional t o t h e height. The rat10 of each corresponding pair of measures, i n that example, is 3, lfe w i l l

use t h i s r a t i o aga in to f i nd t h e height of a water tower if the

shadow length o f a water tower is 2 1 f ee t . Let w be the number

o f feet in the heigh t of t h e tower .

Then t h e statement

21 2 7 = ' 5

is a proportion, Let us now consider statements of thls form,

and then a little l a t e r we w i l l r e t u r n to t h l s statement and show

a method f o r finding a number to substitute for w which w i l l

make the statement tme, We know from Chapter 6 t h a t

2 6 , ' ( c ) 3 =%; ( d ) 4 =a ( a ) 3 = 5 ; (b) T i = - ,

2 6 In ( a ) J = q, we see t h a t 2 9 I 18 and 3 6 18; thus

3 24 2 . 9 = 3 . 6 . In (b) ==, we see that 3. 32 - 96 and

4 . 24 = 96; thus 3 32 = 4 - 24. Simi lar ly , in ( d ) 3 = 6% B

property f o r ( c ) 2 = gm [sec . 9-13

T r y other examples of pairs of f r ac t ions which name t h e same r a t i o n a l number, Are products of the numerator o f one f r a c t i o n and t h e denomimtor of the o t h e r equal? Let us see if t h l s prop-

erty is true f o r a l l pairs of fractions, which are numerals f o r t h e same r a t i o n a l number.

and b # 0 and d f 0, then If 5=7j

a d b ; ; * a = $ a . 6 a 1s multiplied by ,, d whlch equals

c b 1, and - i s mul t ip l ied by .T. d

These products are obtained so that a c the two r a t i o n a l numbers 6 and 7

will be expressed a s f r a c t i o n s with a common denominator .

Thus ad = bc. If two f r a c t i o n s w i t h t h e same denominator are names f o r t h e same ra tLondl number,

t h e i r numerators are equal .

We have shown:

Proper ty - 1. a c

If 5 = 7 and b # 0 and d # 0, then a d = bc.

T h i s p roper ty m y be useful to you in aolvlng problems

Involving proportion. It is also true t h a t if a d = bc, then

C + T provided b. # 0 and d # 0, We s h a l l leave the proof of

t h i s p r o p e r t y as a problem. ( see Problem *18,)

'nre can now r e tu rn to t h e problem of t h e he igh t of t h e water tower .

then 21 3 = 2 w Proper ty 1.

63 = 2w

63 - = w 2

Def in i t i on of a r a t l o n a l

1 number. 312 = w

1 The water tower I s 312 f ee t in height.

3 51

Exercises 9-1

1. What is your height i n inches? Wbat would be the length of your shadow if it were measured at the same time and place as

the shadows of the people in our s t o r y were measured? 2. Some other objects were measured at another time and place,

and the data are recorded below. Copy and complete the t a b l e .

Shadow length Height Ratio

3 feet 8 feet - Clothes pole 36 inches d

1 72 feet 20 feet

144 inches 4 1 llT inches 30 inches

3. In a clasa there are 30 students of whom 12 a re gfrla,

(a) What is the r a t i o of t h e number of gir l s to the total number of students in the class?

(b) What I s the r a t i o of the-number of boys to the t o t a l

number of students in the class?

( c ) What l a the r a t l o of the number of girls to t h e number of boy87

4. In another class, the mtio of the number of girls to the 2 number of boys is 3, the same aa the r a t i o for the class in

Problem 3. This clasa has 10 girls . How mny boys does

it have?

5. In Problem *18 you are asked to show that - $, if a d = bc

and b # O and d # 0. Use thfs property to find which of t h e

following pairs of rat ios are equal,

6. In each of the following, find x so that these will be true statements.

7. Joyce has a p i c t u r e 4 inches wide and 5 inchea long. She

wants an enlargement that w i l l be 8 inches wide . How long

w i l l the enlarged p r i n t be?

8, If a water tower casts a shadow 75 feet long and a 6-foot man casts a shadow 4 feet long, how tall is the water tower?

9. Mr. Landry was paid $135 for a job which required 40 hours of work. At t h l a rate, how much would he be paid for a job

t h a t required 60 hours of work?

10. A cookie recipe calla for the following items. 1 cup butter 1 1 7 CUPS flour

2 CUP sugar 1 teaspoon vanilla

2 eggs This recipe w l l l r n a k e 30 cookiea.

3 (a) Rewrite the recipe by enlarging it in the r a t i o T . (b) How mny cookiea w i l l t h e new recipe make now?

( c ) Suppose you wanted t o make 45 cookies, how much would

you need of each o f t h e itema listed above? 11. U s e a proportion t o a o l v e these problema, Check your work

by solving the problems by another method.

( a ) What i a the c o s t of 3 dozen doughnuts at $.55 per dozen?

(b) What ie the cost of 12 candy bars at 4 for 15#? (HOW

could you s t a t e th la price using the word "per"?)

I ( c ) What i a the cost of 8500 bricks at $14 per thousand?

( d ) A road has a grade of 6 %, which means that it rlses

6 feet per 100 feet of road, How much doea it rlse

In a mile? F i n d the answer to the nearest f o o t , 12. Common unita to express apeed are "miles per hour" and

"feet pep second," A motor scooter can go 30 miles per hour. How many feet per second is t h i s ? he danger of of high speed under various conditions often can be real ized better when the speed l a given in feet per second.)

13. The following table list8 Nips of numbers, A and B. In 6

each pair the r a t i o of A t o B is the number 7. Complete the table.

A B Rat l o R a t l a i n airnplest form

( a ) 12 14 YF 4 12

(b) 21

( 4 30

( d l 100

( e ) 100

*14. Sometlrnea i t is natural and desirable to compare more t h a n

j u s t ' t w o quantities, For example, a mixture of nuts calla f o r 5 pounds of peanuts, 2 pounds of cashews, and 1 pound of pecana. Here t he r a t l o of the number of pounds of peanuts to the number of pounds of cashews is 2 or 5 to 2 ,

and the r a t i o of the number of pounds of cashews t o the

number of pounds of pecans is 2 to 1, This may be stated

b r i e f l y as 5 to 2 t o 1. Suppose a grocer wants t o prepare 24 pounds of' th i s mixture. How many pounds of peanuts, cashews and pecans should he use? To determine your answer, f irst answer theae questions:

(I) If 5 pounds of peanuts, 2 pounds of cashews, and 1 pound of pecana are mixed together, how mny pounds

of nuts are there?

( 2 What is t h e r a t i o of the number of pounds of peanuts to the t o t a l number of pounds?

( 3) Since this r a t i o w i l l be t h e same in the mixture

whose t o t a l weight is 24 pounds, how mny pounds of peanuts are required?

( 4 ) Answer questions ( 2 ) and (3) when " peanuts" are replaced by "cashewsl1.

( 5 ) Answer questions (2) and (3) when "peanuts" are replaced by l1 pecans. 'I

+15. Fi f ty - s ix pounds of a nut mixture, with the same ingredients in t h e same r a t i o as in Problem 14 w l l l be prepared. How mny pounds of each kind of nuts will the grocer need?

*16, If a nut mixture with the same Ingredients as In Problem 14, but in the r a t i o 5 to 3 to 2 has a total weight of 100 pounds how many pounds of each kind of nuts are in the mixture?

*I?. A t r l ang le has a i d e s of length 11 inches, 8 inches, and

6 inchea. If ano the r triangle, the measures of whose sides are in t h e r a t i o 11 to 8 to 6 , is t o be dmwn with a

perimeter of 100 Inches, how long w l l l the shortest side be? *18. Prove t h e property,

if ad = bo and b # 0 and d 0, then $ = 3. When you have proved this statement, you can now say

if % = 2, and b # 0 and d # 0, then ad - bc, and only then. - - (~fnt: If ad = bc, then a = - be we can express d '

C as IJ w

2 Percent

Many of you are familiar with the word "percent," and you m y know something about i t s meaning. If your teacher says,

90 pepcent of the answers o r ~ this paper are correct," would you

know what he means? The word "percent" comes from the l a t in phmse

[sec . 9-21

" p e r centurn," which rnedns "by t h e hundred ." If t h e p i p e r with

90 of t h e answers c o r r e c t h a s 100 dnswers, then 90 dnswers ou t

o f t h e 100 are co r r ec t . The m t l o $ could be used i n s t ead of

the phrase " 9 0 percent" t o describe t h e part of' the dnswers w h i c h

dre correct. The word " p p e r c e n t v ~ i uused when d r a t i o is expressed 711th d denominator o f 100.

1 90 pe rcen t = = 90 . F o r convenience the symbol, ??, , la used f o r the word "percen t . "

1 This symbol is a s h o r t way of say ing .

Suppose t ha t t h e paper has 90 co r rec t answers out of t h e 100;

6 Incorrect answers out of t h e 100; k answers on i t t ed o u t o f t h e

100. Since we know t h e total number of problems we can express

t h i s infomtation I n terns of percent.

90 1 LP m = 9 O * m = 9 0 ~ ( gOO/, of the problems were correct. )

6 1 * = G o r n = 6 '70 ( 6% of t h e problems were incorrect. )

4 1 m = 4 o m = " 0 ( 4 %of t h e problems were omitted. )

6 4 100 G + m + m = m ( a l l possible answers, )

go%+ 15% S % = 100% ( d l 1 poss ib le answers. )

Another name f o r t h e number one is 100 %. The number 2 cdn be wrlt ten

1 200 In o t h e r words 2 0 ~ ~ ~ r n e a n s 200 x - = 2 .

A c l d s s of 25 p u p i l s is mdde u p cf 11 girls dr!d 14 boys . The

r d t i o of the number o f g i r l s to the number of p u p i l s in t h e c.lsss

cdn be expressed mny ways. For instance:

I f we wish to i n d i c d t e t h e percent of the c l d s s t h i t t is g i r l s ,

whlch f r d c t l o n g i v e s t he in fo rmdt lon most e a s i l y ? ' , ;hy? The

r a t l a of the number of t o y s to t h e t o t d l number in T h e class m y

be w r i t t e n

Whdt numbers dre represented by t h e l e t t e r s c , d, e , f? 114 I' (,qirls) dnd - ( b o g ? ) . Yhdt is the Notice t h e two r a t i o s -

25 2 5 sum o f the two r d t , i o s ? F i n d t h e sum of the t w o r a t i o s 4 4

d n d Express t h e t w o r a t i o s dnd t h e i r ~ ~ u n a s pe rcen t s

u s i n g t h e s y r n b ~ l , ? ~ . The entire c l a s s is consloered to be 1013 ' r ' , o f the class becduse

d Any number - can be expressed 6 percen t by the b -- - number c s u c h t h a t

ii c - - 1 T;=m c = C % *

Exercises 9-2d

1, IJsing saudred pdper 3rdw d l a r g e squdre whose i n t e r i o r I s

d i v i d e d I n t o 100 smll squares. Write the l e t t e r , in 1G

$ 1 ~ 1 1 squares. Write t h e l e t t e r B i n 2Oy0of t h e squdrcs.

!?/r i te the l e t t e r C in 35Y0 of t h e squdres . ':lrZte t h e l e t t e r

D in 30 of the squares. Q r i t e the l e t t e r X in t h e

remainder of t h e s q u a r e s .

( 3 ) In what f r a c t i o r l a l wrt of t h e squctres is the l e t t e r A?

(b) In whcit percent of the squares is the l e t t e r A?

( c ) In how many squares I s the l e t t e r B?

( d ) In what f r a c t i ona l part of the squares Is t h e letter B?

(e) In how many squares is the l e t t e r C ?

(f) In what f r a c t i o n d l part of the squares is the le t ter C?

( g ) I n what fractional pdrt of the squares is the l e t t e r D?

( h ) In what percent of the squares is t h e l e t t e r D?

( i ) I n what f r d c t i o n a l pdrt of the squarea is t h e l e t t e r X? (j) In hat percent of t h e squares is t h e letter X?

2. What is the sum of t h e numbers i n your answers f o r ( a ) , ( d ) ,

(f), (g), (i), of Problem l?

3. In Problem 1, what l a the sum of the percents of the squares containing t h e letters A, B, C, D, X?

4. write each of the following numbers a s a percent.

1 1 (a) 7 ( d l 5 3

(9) 2 7 ( J ) g

1 5 , Consider t h e following class. Studen t

Robert Joan Mary John Jae B e t t y

Don hrgaret David

Hair Color -- blond brown blond red brown brown b 1 ond brown red brown

(a ) What percent are boys?

( b). What percent are blonds?

( c ) What percent dre redheads?

( d ) I-t percent are not redheads?

( e ) What percent are brown-haired g ir l s?

I f ) What percent are redheaded boys?

3 9

6. Jean h a s a weekly allowance of 50 centa. One week ahe spends 12 centa f o r a penci l , 10 cents f o r an ice cream cone, 15 cents f o r Sunday-school collection, and she pvts the rest in h e r

bdnk.

(a) Write the rat io of each expense and the money banked to

the allowance, and exppess each ratio as a percent.

(b) F i n d the awn of t h e ratios.

( c ) Find the sum of the numbers named by the percents,

( d ) W h a t check on the separate answers above do you observe?

7. The monthly income f o r a f a m i l y i a $400, The family budget

for the month is shown.

Payment on the mortgage for the home $ 80

Taxes 20

Payment on t h e c a r 36 Food 120

Clothing 48 Opera t ing expenses 32 Health, Recreation, etc. 24

Savings, and Insurance 40

( a ) What percent of the Income is assigned to each item of t h e budget?

(b). What I s one check on the accuracy of the 8 answers?

Percent is a convenient tool f o r g i v i n g information involving rd t io s . A t h l e t i c standings sometimes are given i n percent. Two seventh-grade pupi l s discovered the reason f o r this use of percent, The boys were discussing the scores of their baaeball teams. In the L i t t l e k a m e one team won 15 games out of 20 games played. Another team won 18 out of 25 gamea, Which team had a bet ter ~ e c o r d ? The second team had won 3 more games, but the first team played fewer gamea. Look a t the r a t l o s of t h e number OF games won to the number of games played f o r each team. The ratios 3 and

l8 cannot be compared a t a glance. L e t us use percent for the s comparison. -

The first team won of t h e games played.

3 59

3 =a = 75% They won 75Xof the games played.

The second team won la of the games played. = 25

They won 72%or the gamea played.

The first team which won 75%of i t s games had a higher standing t han t h e second team which won 727,of I t s games. We could say t h a t

72% < 757* or 7%> 72% Data about business, school, Scouta -- are sometimes given in

percent. It I s of ten more convenient to refer to the data at some later time if they are given in percent than if they are given otherwise.

A few years ago the director of a camp kept some records for fu ture use, Some I n f o m t l o n was glven in percent, and some was

no t . The records gave t h e following items of infomat ion,

(1) There were 200 boys in camp.

(2) One hundred percent of the boys were hungry for t h e

f i r s t dlnner In camp.

( 3 ) On t h e aecond day in camp 44 boys caught fish,

(4) One boy wanted t o go home the f l r a t n ight .

1 ( 5 ) A neighboring camp d irec tor said "forty percent of t h e

boys in my camp w i l l learn to s w l m this summer. We shall teach 32 boys to s w i m . "

From (1) and ( 2 ) , how many hungry boys came to dinner the f i r s t day?

Of course we should know without computation that 100%'oof

200 is 200.

From (31, we can f ind the percent of boys who caught fish on the second day. The r a t i o of t h e number of boy8 who caught f i s h

44 to the number of boys in camp is . If we call x% t h e percent of the boys who caught fish, then

x = 22

~wenty- two%of the boys caught fish.

From (41, we can find the percent of the boys who were homesick. If this is x?,, then

1 & = m 200x = 100

100 ( Property 1. )

X = m 1 X = - 2

I Of t h e boys, q%were homesick. T h i s m y be read "one-half

percent" or "one half of one percent." You m y prefer t o say

"one half of one percent," since t h i s emphasizes the smallness of

it.

From t h e informat ion in Item 5, the total number of boys in t h e second camp can be found. C a l l this number of boys x. '' of the group, and also refers t o 32 boys. 40% means

-

4 0 32 We wish t o find x such that = y .

( Property 1 + )

Exercises 9-2b

1. A L i t t l e League team won 3 out of t h e first 5 games played.

(a) K U t percent of t h e f i r s t 5 games d i d t h e team win?

( b ) What percent of t h e first 5 games did t h e team lose?

h t e r in the season t h e team had won 8 o u t of t h e 16 gdmes

p l a y e d .

( a ) ':!hat was t h e p e r c e n t of games won at t h i s time?

( b ) Has t h e percent of' games won increased or dec reased?

At t h e end of the seiison, t h e team had won 26 games o u t of 40.

(a) !:.!hat percent. o f t he gdrnes played d i d t h e team win by t h e

end of t h e season?

( b ) How does this percent compare w i t h the o t h e r two?

The re a r e 6 d ~ seventh-grade pupils in a j u n i o r h i g h s c h o o l .

Thq p r i n c i p a l hopes t o d i v i d e t h e pupils i n t o 20 s e c t i o n s o f

equal s i z e . ( How many pup i l s would be in each s e c t i o n ?

( b ) \$hat percen t cf t h e p u p i l s w o u l d be in each sectiatz?

( c ) What number of p u p i l s is 1 %of the number of p u p i l s In

the seventh grade?

( d ) \!'hat number of p u p i l s is lO%of t h e number of p u p i l s in

the seven th grade?

Suppose one section contdins 36 pup i l s . Hhdt p e r c e n t o f t h e

seventh-grade pup i l s a re i n that section?

One hundred fifty seventh-grude pup i l s come to school on the

s c h o o l bus.

(a) What percen t o f t h e seventh-grade p u p i l s come by

s c h c o l bus?

(b) blha t percent uf t h e seven th -g rdde p u p i l s come t o s c h o o l

by some o t h e r mcdns?

In d section of 30 p u p i l s , 3 were t a r d y one ddy.

( idhat f r a c t i o n a l p a r t of t h a t s e c t i o n was t a r d y ?

(b) What percent o f thdt section was t a r d y ?

There were '150 e i g h t h - g r a d e pupfls . The nutnbcr of eighth-

grdde p u p i l s i s what percent of the nunber of seventh-grdde

p u p i l s ?

h e day 3 seventh-grade pupils came to s c h o o l on c r u t c h e s

( t h e y had been s k i i n g ) . ?That pe rcen t of t h e nunber of

seventh-grade pupil 7 : ; c r c o n c r u t c h e s ?

10. One day a seventh-grade p u p i l heard t h e princ ipa l say,

' 'Four percent of the ninth g r a d e r s are absent today." A

list of absentees f o r t h a t day had 22 names of ninth-grade p u p i l s on it. From theae two piece s of i n f o r m t i o n , t h e

seventh-grade pup11 discovered t h e number of ninth-grade

puplls In the ,school. How many ninth-grade pupils are there?

9-3. Decimal Notation

You recall from Chapter 2 t h a t a base ten numeral, such

as 3284, wri t ten as 3( lo3) + 2( lo2) + 8( lo1) + 4 was said to be in expanded form, Written a3 3284 t h e numeral is said to be in p o s i t i o n a l notation. In base t e n t h l a form is also c a l l e d

d e c i m l nota t ion . Each d i g i t represents a cer ta in value ac-

cording to i t s place in the numeral, In the above example, t h e 3 is in thousands place, the 2 is in hundreds place and s o on.

The form for place value in base ten shows that the value of

each place immediately t o t h e l e f t or a given place is t en times t h e value of the given place, Each place value immediately to the right of a gfven place is one tenth of the place value of t h e given place. Looking back to t he number 3284 wri t t en In

expanded form you w i l l notice t h a t reading from left t o right

the exponents of t en decrease. Suppose we want t o wri te 5634.728 in expanded form, We could w r i t e

5(103) + 6(102) + 3(101) + 4 + 7 ( ? ) + 2( ? ) + 8 ( 1 ). he 1

4 ia in unitie (or onets) place. The value of t h i s place is of the value of the place before it. If we extend our numeral

to the r i g h t of one's place and s t i l l keep t h i s pattern, what w i l l be the value of the nex t place? Of the next place af ter t h a t one? To write th is in expanded form to the right of unit's place as well as to the l e f t of unit's place we have:

1 1 5(103) + 6(102) + 3(101) + 4 + 7(-&) + 2(7) + 8(--.5) where

10 10

t h e f l r s t place to the r i g h t o f one1 s place is the one-tenth's place o r simply tenthas place.

The f o l l o w i n g c h a r t shows the place values bo th to the

l e f t and to t h e r l g h t o f the unit's place.

Place Value Chart

We usually re fe r to the places to the right of oriels p l a c e

as t h e decimal p l a c e s , When t h e number 3284.569 is written i n decimal notat ion

the decimal p o i n t l o c a t e s the one 's place. The number is read

" ~ h r e e thousand two hundred e igh ty- four - and five hundred s l x t y -

nine thousandths" or "~hree two e i g h t four -- point -- f i v e six n i n e .

Example 1.

1 Write 5(102) + 7(101) + 3(m) in decimal n o t a t i o n .

Notice t h a t t h e onets place was not written. Could oners place be written O(loO) ? Your answer 570.3 will h e l p you answer t h i s question.

Exercf ses 9-3

1. Write each of the following in decimal notation:

2. Write each of the following In expanded form:

(a) 52.55

(b) 1.213

( c ) O.#

( d ) 3.01

( e ) 0.0102

( f ) 0.10001

(g) 30.03

3. Wrlte each of the following in words: For example, 3.001 is wri t ten three and one thousandth.

(a) 7.236 ( d ) 1.0101

(b) 0.004 ( e ) 909.009

( c ) 360.101 I f ) 3,0044

4. Write in d e c i m l numeral form:

(a) Three hundred and f i f t y - t w o hundredths

( b ) Five hundred seven ten-thousandths

( c ) Fourteen thousandths

d ) S ixty and 7 hundredths

( e ) Thir ty- two hundred-thousandths

( f ) E i g h t and nineteen thousandths

* 5 . Write 0. ltWo i n base t en .

t

1 * 6 . Write p in the duodecimal system.

'7. Write 2 In the duodecimal system.

'8. Change 10.O1ltw, t o base ten.

9-4. Operations - w i t h Decimala

Suppose we wish to a d d two numbers in decimal form; f o r example, 0.73 4- 0.84. We know how t o , a d d : 73 + 84 = 157. How do we handle the d e c i m a l places? We may proceed a s follo

1 1 0.73 = 73 x and 0.84 = 811 x m. T h e r e f ore,

1 0.73 + 0.84 = ( 7 3 x + (84 x A) 1

= (73 + 84) x m 1

= 157 X ~ l ~ f = 1-57.

Notice t h a t we have used the distributive property here,

Suppose now we wish t o find the sum, 0.73 + 0.125, Just

as before, we f i r s t rewrite our numbers as f r a c t i o n s :

1 and 0.125 = 125 X m. 1 0.73 = 7 3 X = 730 X

1 We use t he last forrp of 0-73, since then appears in both producta .

1 - 855 x - = 0.855.

These examples can be handled more conveniently by wr i t i ng

one number below the other as follows:

Notice that we write the d e c a l points direc t ly under one I

another, This is because we want to add the number in the m place in the f i r s t addend to the number in the 6 place In t h e

1 second addend, and the number In the place i n the first 1

addend to the number In the & place in the second, e t c . Thus,

7 3 8 4 0.73 =, +, and 0.84 = + and therefore

Subtraction can be handled In the same wag, For example,

Exercises 9-4a

1, Add the following numbers.

(a) 0.76 + 0.84 ( c ) 1.002 -I- 0.0102

(b) 0.719 + 0.382 ( a ) 1.05 + 0.75 + 21.5

Subtract the following.

( a ) 0.84 - 0.76 ( b ) 0.625 - 0.550 ( c ) 0.500 - 0.125 ( a ) 1.005 - 0.0005

Perform the opera t ions Indicated.

(a) 1.051 - 0.702 + 0.066

(b) 0.407 - 0.32 + 0,076 Four men en t e r a hardware store, and t h e f i r s t wants t o huy

10.1 feet of copper wire, t h e second wanta 15.1 f e e t , t h e

t h i r d wants 8.6 f ee t , and the f o u r t h wanta 16.6 f e e t . h he storekeeper has 50 feet of wire in his store. Can he give

each man what he wants? A storekeeper has 11.5 pounds of sugar. A womn buys 5.6 pounds, Another womn buys 4.8 pounda, Then a d e l i v e r y t ruck brings 25 pounds to the s tore . F l n a l l y , mice eat 0.05 pounds, How much augar is l e f t ? There are 16 ounces in 1 pound. Which is heavier, 7 ounces or 0.45 lb.?

In most of Europe, distances are measured in kilometers (remember that a kilometer is about 5/8 of a mile), The

distance from city A to Paris is 37.5 kilometers and t h e

dis tance from city B t o Parfs is 113.2 kilometers. How

far in kilometers is I t f r o m city A to city B by way of

Paris? Suppoae t h e t h x e c i t i e s , A, 3 and Paris in the previous problem are on the same road and c i t y A is between Parfs and c l t y B. Then how far in kilometers i s It between c i t i e s A and B along this road?

Find the value of the sum, in the decfmal system, of the

numbers 10. O l t w o and 1. Oltw, . F i r s t change i n t o t h e

decimal system and then add,

Suppose we w i s h to multlply two numbers in decimal form. For exdmple, 0.3 x 0.25. We know how t o multiply these numbers:

3 x 25 = 75. Just as before, we wri te

(1) How many dig i t s are there to the r ight of the decimal point in 0.3?

(2) How many d i g i t s are there to the right of t h e decimal

point In 0.25?

( 3 ) What is t h e sum of the answers t o (1) and (2)?

( 4 ) How many digits are there to the right of the dec-1

point III 0.075?

( 5 ) Compare t h e answers to ( 3 ) and ( 4 ) . Now mul t lp ly 0.4 x 0.25, What I s your answer? Anawer the

five questions above, (11, (2), (3), ( 4 ) , ( 5 ) for these nubera . Do the answers to ( 3 ) and ( 4 ) s t i l l agree?

Property 2: To f ind the number of decimal place8 in the product when two numbers are multiplied, add the number of decimal places in t h e two numerals.

For example, suppose we w i s h to multiply 0.732 by 0.25.

The first numeral has three d e c i m a l places and t h e second has two, so there will be flue d e c h a l places in the answer. We find the

product 732 x 25, and then mark o f f five decimal places in the

answer, counting from r ight to l e f t ,

We consider now the problem of d i v i d i n g one number in decimal form by another, for example, 0.125 + 0.5.

The f i r s t step is usually to find a fraction whose denominator

is a whole number and so that the new fmctlon l a also a name

[sec. 9-41

0 12 1 f o r 0.125 + 0.5. In this case, we start with ,

Then, in order to replace the denominator by a whole number, we 1 2 i mult ip ly numerator and denomimtar by 10 to get e. Using

i f r a c t i o n s we could work it out l i k e t h i s :

But a much shorter way is to use the usual form f o r d i v l a l o n .

Then: ( div isor) ( quotient ) ( div idend)

5 x 0.25 - 1.25 -

and we see that in t h e equality the number of decimal places ( two) in the product is t h e sum of the number of d e c a l places in t h e members of the product ( 0 + 2 ) Just as we say I n

Property 2, Whenever the d i v l a o r is a whole number, the d l v i d e n 8

and the quotient have the same number of dec-1 places, By

placing the decimal poin t of the quotient d i r e c t l y above that of the dAvidend, we locate the decimal point of the quotient automatically in the correct place.

It is very easy to nuke a mistake In placing the decinral

point of the answer and hence it is alwaya a good plan to check by estimating the answer. For t h i s example we see that

which is reasonably cloae to our product. Try a more complicated fract ion: . If we m l t i p l i e d

the denominator by 10 we would have 25.3 which still I s not a

whole number. But If we mult ip ly by 100, it becomes 253, which 31 .3 is a whole number. The given f ract ion then I s equal to

and we perform the d i v i s i o n in the usual way:

Then 253 x 2.1 = 531.3 and we see again that the number of I I

decimal places in 253 (which is zero) plus the number of decimal

places in 2.1 (whlch is 1) is equal t o the number of d e c i m a l

places In 531.3 ( whlch I s 1). You ahould check the following:

Of course we cannot expect our divisions to "come out exactly" in a l l cases. Suppose we try to f i n d the decimal expression f o r

If we want the quotient to thme decimal places, we w i l l 7' d i v i d e 2,000 by 7 , If it were to be s ix places, we w o u l d d i v i d e

2.000000 by 7 , and so forth, Suppow we f i n d it to s ix places:

(Assume that the d e c i m l s are exact . )

1. Find the following products.

( a ) 0.009 x 0.09

( b ) 0.0025 x 2.5

( d ) 0.135 X 0,202

Find the following quotients.

( a ) 0.009+ 30

(b) 0.015 - 0.05 ( c ) 0.575 + 0.4

( a ) 2.04 + 0.008

Express the following numbers as decinuals.

John's father has a garden 25.3 f ee t long and 15.7 feet w i d e ,

How many square feet are there in the garden?

A rectangle is 14.2 meters long and 5.7 meters wide. What

i a i t s area in square meters? The dimensions of a box are: 17.3 meters by 8.3 meters by 2.5 meters. What is i t s volume i n cubic meters? bout how many miles is a dlatance of 3 .8 kilometers? (One kilometer is about 3 of a mile.) Find the fol lowing p ~ o d u c t :

9-5. Declml Expansion

L e t us look more cloaely at r a t i ona l numbers in d e c i m l form.

You recall that is the quotient of 1 d i v i d e d by 8.

So and 0.125 are names f o r the same r a t i o n a l number.

Consider the mtioml number3. We a l l b o w the d e c i m l

It i s found by dividing 1 by 3. numeral which names 7. 0.333.. .

3- Without performing the d i v i s i o n do you know what d l ita will appear in each of the next b places? At some stage in the d i v i s i o n do we get a reminder of zero?

The 3 dots indicate that the decimal numeral never ends.

Recall that Look again at the decimal numeral for 8. it was obtained by d i v i d i n g 1 by 8.

8 '20 After the first subtraction the remin- 16 d e r l a 2 ,

T O The second remainder Is 4 . h0 0 The t h i r d reminder i s 0.

0 D The f o u r t h remainder is 0. Can you

predict the f i f t h and sixth rematnders?

We usually stop our d i v i s i o n process at t h e stage where we

obtain a zero reminder, However we could just a a well continue

d i v i d i n g , getting at each new stage a reminder of zero, and a

quotient of zero . It is clear that once we get a reminder of zero every

remainder thereafter w i l l be zero, We could write 1 0.125000.. . just as we wrote g = 0.333.. . , but we seldom do. 8 =

Therefore O.l25OOO... is a repeat ing deciml with 0 repeat ing over and o v e r again. Likewise 0.333,. . I s a repeating decimal with the dlgit 3 repeating,

By decimal expansion we mean that there is a digit for 1 every dacirrual place, Note that the declmal expansion of 3 and

the decimal expansion of $ are both repeating.

Class Discussion Exercf aea

Let us look at the decimal which names %he r s t i o n a l

number ; 1, Can you tell, without performing the d iv i s ion , the d l g i t a

t h a t should appear In each of the next six placsa? 2. Is t h e r e a block of digits which continues to repeat

endlessly? L e t us place a hopizontal bar over the block of digits which repeats. Thus 0.142857142857,,.uses the bar

to mean t h a t the same digits repeat i n the same o r d e r and t h e

3 do ts t o m a n that t h e decimal never ends. 1

3. Name TI by a decimal numeral. 4. Hov! soon can you recognize a pattern? 5. 5 ? i l l there be a zero remainder If you continue d i v i d i n g ?

6 , Does t h i s decimal repeat? How should you i n d i c a t e t h i s ?

7. Observe t h a t t h e decfmal repeats aa t h e remainder repeats. Look at the procedure by which we found a decimal numeral

1

f o r +.

After the f i r s t s u b t r a c t i o n t h e reminder is 3.

The second reminder is 2.

The t h i r d r e m i n d e r is 6 .

The f o u r t h remainder is 4.

The fifth remainder is 5.

The sixth' remainder is 1.

The seventh remainder is 3 , Is t h e seventh d i g i t t o t h e r i g h t of the decimal point t h e same as t h e ffrst?

1 4 T o The e igh th remainder is t h e same as

second. Is the e igh th d l g l t t o the Po r i g h t o f t h e dec imal point t h e same as t h e second?

40

Notice t h a t the d i g i t s in t h i s quo t i en t begin t o repeat whenever a number appears as t h e remainder f o r the second time.

8, Make similar observations when you d f v i d e 1 by 37 to f lnd 1 a dec iml numeral naming r, 7

Hence from these many i l l u s t r a t i o n s you may conclude t h a t

every r a t i o n a l number can be named by a decimal numeral which

#ither repeats a s i n g l e d i g i t o r a block of digits over and

over again.

[see . 9-51

Exercises 9-5 - 1 1. Write a d e c i m l numeral (or decimal) for T3.

( a ) How soon can you recognize a pattern? (b) Does t h i s d e c i m l end?

(c) How should you indicate t ha t It doea not end?

( d ) Is t h e r e a s e t of digits which repeats periodically?

( e ) How should you i n d i c a t e t h i s ?

2. Write decimals f o r :

3. Vrlte the d e c i m l s for:

See how soon you can recognize a pat tern In each case. In performing the division watch the reminders. They my gtve you a clue about when to expect the decimal numeral to begin t o repeat.

4. Study these d e c i r m l numerals and see if you can find a re l a t i ons h ip

( a ) between the decimal naming $ and the d e c i m l

( b ) between the d e e i m l naming 4 and the d e c i m l

5. Can you f i n d a decimal f o r d without d i v i d i n g ?

6 . Is it tme that t h e nwnber 0.63E.,. is seven tlmea the - number 0.0909... ?

7 . Find the decimal numeral for the f i r s t number in each group and calculate the o the r s without dividing.

9-6. Rounding

2 Suppose we wish only an approximate decimal va lue for 7 . We might, f o r instance, want to represent thJs on the number line where each segment represent8 0.1 a s below:

Where would be on t h i s line? We know its decimal form is 7 0.285714.. . . Perhaps you can see immediately from this t h a t 3 7 must l i e between 0.20 and 0.30. If this is not c lear , look a t

it a l i t t l e more closely. First:

0.285714.. . = 0.2 -t 0.08 + 0.005 + ( o t h e r d e c i m l s ) , and

hence it is ce r t a in ly g r e a t e r than 0.2, the f i r s t number ln the

sum. Second, 0.2 I s less t h a n 0.3; 0.28 2s less than 0.30 ;

0.285 is less t h a n 0.300 and so f o r t h . No natter how far t h e

decimal is computed, what we get is always less than 0.3000000.. . A l s o you can see that Is c lose r to 0.30 t h a n to 0.20 since 0.25 7 is halfway between 0.20 and 0.30 and the number 0.2857111.. . is

2 between 0.25 and 0.30 . 53 7 goes qn t h e number Line in abou t

the place I n d i c a t e d by t h e arrow. We could represent sore clasely on the number line if we 7

d i v i d e d each of t h e segments i n t o ten parta. Then we would see

t h a t $ is between 0.28 and 0.29 but a l i t t l e closer to 0.29.

(you ky want to write t h e reasons o u t in mope d e t a i l t h a n we d i d 2 above, In o r d e r to inalce it clearer . ) We would say t h a t 7 is 0.29 I

to t h e nearest h u n d r e d t h . We ca l l t h i s " round ing t o two places." 2 7

What would be, pounded t o t h r e e places?

The answer is: 0.286.

This rounding is use fu l In estimating r e s u l t s . Fnr instance,

suppose we have to find the produc t : 1 .34 x 3.56. This w o u l d be

approximately 1 x I! = 4 or, if we wanted a little c lose r estimate, we c o u l d compute: 1.3 x 3.6 = 4.7 approximately.

Rounding is a l s o usefu l when we a r e considering approximations

in percents. For instance, i f it t u r n e d out t o be true that about

2 out of 7 families have dogs, I t would be f o o l i s h t o carry t h i a

o u t t o many declmal places in o r d e r t o g e t an answer In percent.

We would usually j u s t use two places and say t h a t about 29Toof a l l fami l ies have dogs, or we could round t h i a s t i l l f u r t h e r and

re fe r to 30% . A special problem arises when the number to be rounded occurs

exactly half -way between t h e two approximating numbers. F3r in-

stance, how does one round 3.1215 t o three decimal places? The

two approximating numbers are 3.121 and 3.122 and one is just as

accurate an approximation as the other . Often it makes little

difference which decimal is used. However, if, for several numbers

of a sum, one always rounded to the lower f i g u r e , the answer would probably be too snall. For this reaBon, one often makes the agree-

ment that he w i l l choose the decimal whose last d i g i t 1s even.

Thus In the above case, we could choose 3.122 slnce its las t

d i g i t is even; t h i s number is l a rger than the given number. But

if the given number were 3.1425, we would c h o ~ s e 3.11~2 as t h e approximating number since t h e giT.-en number 1 i e s between 3.1.J!2

and 3.lh3 and it is the d e c i m a l 3.142 whose l a s t d i g i t is even;

here we have chosen t h e smaller oP the two approximating numbers.

This is not espectally important for thls class but you m y find

in y o u r science c las s t h a t this I s what is done.

Exercises 9-6 - 1, Round the following numbers to two places,

( a ) 0.0351

(b) 0.0449 ( c ) 0.0051

( d ) O.Gl93

Round the following numbers to three places.

( a ) 0.1599 (b) 0.0009

( c ) 0.0000g

( d ) 0.3249

Express the following numbers as decimals cor rec t to t h r e e

place a .

Express the following numbers as decimals correct t o one

place.

( a ) A piece of land i a measured and the measurements are rounded t o the nearest tenth of a r o d (in o t h e r words, the measures are rounded t o one decimal place ) . The

length, aftep rounding is 11.1 rods and w i d t h 1s 3.9 rods. Find the area rounded to t h e nearest tenth of a square r o d .

(b) Suppoae t h a t the length is 13.14 rods and t h e w i d t h I s

3.94 rods rounded t o the nearest hundredth of a r o d .

Find t h e area rounded t o the nearest hundredth ~ 3 f a

square rod. What Is t h e difference between t h l a answer and the previous one?

i 9-7. Percents - and Decimals You have learned that the number & can be w r i t t e n as 51%

and also as 0.51. Actually we read both 0.51 and & as ''fifty-one hundredths". We have three different expressions f o r

the same number:

which we would read: " f i f t y - o n e hundredths is equal to f i f t y -one

percent is equal t o f i f t y -one hundmdths.' Similarly we have

50 4 = ma = 0.50 = 50%= 0.5 ,

13 Also 65% = 0.65 = % - ,

and 3 60 7 =

= 0.60 = 6O%= 0.6 .

Class Exerciaes 9-7a

1. Express as percents:

( b ) 0.4 ( d ) 1.2

2. Express as decimals:

( a ) 5 3 %

~t 13 a l i t t l e more d i f f ~ c u l t to express $ as a percent since 8 is not a factor of 100. We know that i t s d e c i m l form 18 0.125. Other ways of w r l t l n g thls number are: or 12.5%;

1 1 also or 1$% . S i m i l a r l y , 0.375 l a the same as 37.5% or m 1 3T2% and may also be written as:

3 Thus 0.375 is equal to both 37+% and 3 .

Class Exercises 9-7b

1. Express as percents:

2. Express as declmla:

1 How do we find the percent equivalent of 3 whose decimal, 0.333 . . . repeats endlessly? If we wish an approximate value we can round the decimal to two places and have:

$ is approximately equal to 0.33 or 33% . An accurate name In percent for one-third can be found as follows:

S O that an accurate expression for 4 I s 34 9. .

Claas Exercises 9-7c - 1. Express approximately a8 percents:

2. Express t h e above accurately as percents.

4 How can 28 %be expressed as a f rac t ion? 7

Class Exerclaea 9-7d - - Express the following a8 fractions:

(a) 66 9% ( 4 2 5 T % 1

(b) 11$% ( d l 125 +%.

Exercises 9-7 - 1. Copy the following chart and fill in the missing mmes of

numbers. The completed chart w i l l be helpful to you in

future lesson8 . Frac t ion

Simplest form

(a ) $ - .

1 (b) g

I c >

( a ) - ( e !

(f)

4 (€3) 7

. -

(h)

( I ) --

( J ) -- - .--

Percent

50 70

-

40 %

- .

Hundred as denominator Deciml --

3 0.50 ---

&

60 m --

-. -- 7 0

m- --

-

0.20

--

... --

0.35, .. -

- . 0.6;. . .

381

Draw a number line and name points on it with t h e percents in

Problem 1.

,3 . Using squared paper, draw a large square containing 100 small

squares. By proper shading indica te the percents in parts

w, ( d ) , (11, (P), (d*

Frac t ion Simplest form

3 (i.1 m

(1)

( 4 1 ' (4 g -

( 0 )

( P I ( ~ 1

( r)

(3)

7 (t) g

(4

(4

(4 2 (XI 7J 1

(Y>

( 2 )

Percent

90 %

- .

150%

100 7%

-

Hundred a a denominator

-..I

300 m

%

1% m

64 m

Decimal

0.10

0.375

0.01

0.005

4 . W h a t f r a c t i o n in simplest form is another name f o r

(a) 3236 (b) 90% ( c ) 120% 3

5. Give the percent mmes f o r t h e fol lowiqg numbers:

9-8, Applications - of Percent

Percent is used to express ratios of numerical quantaties in everyday experience. It is important for you t o understand the notation of percent, and a lso , to be accurate in computing with numbers written as percents. kt US look at some examples in the

use of percent. Example 1. Suppose that a family has an annual income of

44500 (after withholding t a x ) . The faml ly budget Includes an 1 item far food of 33 3 x o f the budget. How much money is allowed

f o r food f o r the year? Can you answer this question without using a pencil and paper? If you can, do so, and check what you find with the result below, We work t h i s example as we do, t o show a

method which we sha l l be using In more difficult examples later. 1 1 We know t h a t 33 -5%, 1s equal to -5 and hence if we let x stand

f o r the number of dollars allowed f o r food we have

To f i nd x we use Property 1 of Section 9-1 which te l l s us that

if = then ad = bc.

For example, thiq means that 6 if m = 9 then 6 15 .; 9 10 .

Then in our example,

X 1 if = J then 3x = 4500 . Hence x - .?- - 1500, and 1500 w i l l be the number of dol lars

used for food.

38 3

Example 2. Suppose the annual income of t h e farnlly in

Example 1 is $4560 and 32% of the budget is allowed f o r food f o r t h e year. Then if x stands for the number of d o l l a r s allowed f o r food, we have

32 ~ = m * Usfng Property 1, we have

100~ = 32(4560) = 145,920 . x = 3459.20

and t h e amount spent f o r food would be $1459.20 which is close t o

the answer in Example 1,

Exanlple 3. The number 900 is what percent of 1500? Can you

find the answer t o this example without use of pencil? T h i s is a

little harder than Example 1 but you m y be able to solve it

mentally, But the same method we used above, works here also.

If x x atands for t h e percent which 900 i s of 1500, we have

&=S. Agaln us ing Property 1, we have

1500~ = loo( 900) = 90000,

go 000 x =*d = 60,

and 900 I s 60% or 1500.

Example 4. Suppose a f a m i l y with an annual income of $4560 renta a houae f o r $ 7 7 , 0 0 per month, What percent of the f a m i l y income w i l l be spent for rent? F i r s t , the rent for the year w i l l be twelve times the monthly rent, that is $94.00. If x % 1s the percent that 924 I s of 4560, we have, as before

and, using Property 1 again, we have 4 5 6 0 ~ = 100(924)

4 5 6 0 ~ = 92,400

Hence xx is about 20% ; that is, about 20% of t he family income Is spent for rent.

Example 5 , An advertisement said t h a t a b i cyc l e could be purchased I ' on time" by nuking a down gayment of $14 -70. The merchdnt s t a t e d f u r t h e r t h a t t h i s payment was 25%of the prlce, l d k t was t h e price of t h e b i c y c l e ? If t he pr lce of the bicyc le is x d o l l a r s , we wish to find x such that

Before going on to finding x accurately, you should with your p e n c i l and paper, estimate what x should be, Then, t o find the

accura te v a l u e of x, we use the same method as before to get:

l O O ( 1 4 . 7 0 ) = 25X

t h a t is,

So the price must have been $58.80.

Commissions and Discounts

People who work as salesmen often are paid a commlsslon

i n s t e a d of a salary. A book salesman is paid a commlasion of 2%

of t h e s e l l i n g p r i c e of the s e t of books. If he sells a s e t of books f o r $60.00, h i s commission would be 2 9 of 8a.00 or $15.00. Sometimes the percent of the se l l ing price which gives the commission is c a l l e d the - rate of the commission.

D e f l n l t i o n . Commission I s the payment, often based on a

percent of s e l l i n g pr i ce , tha t is paid to a aalesrnan for h i s

services . Merchants sometimes sell articles at a discount. During a

sale, an advertisement s t a t e d " ~ l l c o a t a w i l l be sold at a dis- count of 30$. " A coat marked $10.,00 then a e l l a at a discount of 30s of $70.00 or $21.00. The sale price (sometimes ca l l ed t h e

n e t prlce) is $70.00 - $21.00 o r #49,00. -- D e f i n i t i o n . Discount is the amount subtracted from the

marked price.

D e f i n i t f o n . Sale prlce or net price I s the marked price less the discount.

Exercises 9 - h

In t h e follovring problems It may be necessary t o round some dnswers. Round t h e money answers t o t h e nearest cent , and round percent

anzwers t o t h e nearest whole percen t .

1. On an exarnlnation there was a total of 40 problems, The

teacher considered a l l of t h e problems of equa l value, and

dss igned grades by percen t . How many correct answers a r e in-

d i c a t e d by t h e follawing gmdes?

r l a ] 100% ( b ) 80'?0 C C ) 50% ( as 65% 2. In grad ing t h e papers referred to in Problem 1 what percent

would t h e tedcher assign f o r the following papers? ( a ) A l l problems worked, b u t 10 ansrt!ers a r e wrong.

(b) 26 problems worked, a l l answers cor rec t . ( c ) 20 problems worked, 2 wrong anzwers.

( d ) One problem not answered, and 1 wrong answer.

3. If t h e sa les t a x In a c e r t a i n s t a t e is 11% of t h e purchase

pr i ce , what t ax would be collected on the following purchases?

( a ) A dres s selling f o r $17.50

(b) A b i c y c l e sell in^, f o r $119.50

4. A real e s t a t e q e n t r e ce lve s a cornmission of 5Tofor any sale thdt h e make?. :-.'hat would be h2s com~ission on the sale of

a house f o r $17,500? 5. The r e a l estdte agent referred to in Problem 4 wishes to earn

an annual income from commissions of at least $9000. To earn

t h i s Inco3e, what would hks yearly sales need to total?

6 A za lesmn who se l l s vacuum cleaners earns a commission of

$25.50 on each one s o l d . If t h e selling price of t h e cleaner is $85.00 what i s t h e ra te o f commission f o r the salesman?

7 . Sometimes t h e rate of commission is very s m l l . Salesmen for heavy machinery o r t e n rece ive a commission of 1%. If, in one year, such a salesman s e l l s a machine t o an industrial pldnt

f o r $658,000 and ano the r machine f o r $482,000, has he earned

a good income f o r t h e year?

8. A spo r t s s t o r e a d v e r t i s e d d sa le o f football equipment. The

discount was to be 279, .

(a) What would be the sale pr ice of a foo tba l l whose mrked

price is $5.98?

( b ) What would be the sale price of a helmet whose m r k e d

price is $3.40?

9. In a junior high school there are 380 seventh-grade pupils, 385 eighth-grade pupi l s , and 352 ninth-grade pupils ,

(a) What is t h e total enrollment of the school?

(b) What percent of the enrollment is in the seventh grade?

( c ) W h a t percent of the enrollment is in the e igh th grade?

( d ) What percent of the enrollment l a in t h e ninth grade?

( e ) What is the sum of the numbers represented by the

answers to (b), ( c ) , and ( d ) ?

10. Mr, Martin keeps a record of the arnounta of money h i s family pays i n sales tax, A t the end of one year he found t h a t the t o t a l was $96,00 for the year. If t h e salea tax rate Is 4%, what was the t o t a l amount of taxable purchases made by the

FvZartin fami ly during the year?

11, Mr. Brawn asked h i s bank f o r a loan of $1000 and promised to

pay it back in a year . For this t h e bank charged h i m 6 % of the $1000. This is called the "rate of interest ." W h a t was

t h e amount he had t o pay in Interest? If he repaid the amount of the loan a t the end of the year together with the interest, what d i d he pay the bank?

12. Mr. Jacobs arranged a loan from his bank of $2000. At the

end of a year he paid to the bank $2140. !4hat was the rate

of in te res t? What percent is $2140 of 12000?

*13. A government bond which costs $18.7~ is w10rth $z5.00 t e n years a f t e r it is purchased, By what percent d i d it increaae in

value over the t en yeara? What was the average percent of increase per year?

14. The real e s t a t e tax is $12 on a house valued at $1000, $24 on a house valued at $2000, $36 on a house valued a t $3000. What

percent of the value of the houses is the tax in each case?

If the percent l a the same, what would be t h e tax on a house

valued at $25,000 ?

387

A c e r t a i n store gives a l0%dis~0unt for cash and a 5Xdis - count for purchases made on Mondays. T h a t is, if a customer purchased an a r t i c l e priced a t $100 and paid cash it would

cost him $100 - $10 = $90 . Then I f t h e day of h i s purchase

were Monday he would g e t a f u r t h e r 5%discount, which would

make t h e net prlce $85.50 since 5% of 90 I s 4.50 and

90 - 4.50 = 85.50 . Suppose t he 5% discount on $100 had been computed f i r s t and t h e l ~ % s e c o n d , w ~ u l d t h e f i n a l n e t p r i c e be t h e same?

Would these t w o ways of computing the f i n a l net p r i ce give

t h e same resu l t fo r an article priced at $200? Vhy?

In a cer ta in store each customer p a y s a sa les tax o f 2% and

is given a 1 0 ~ d i s c o u n t f o r cash. That is, if a customer purchased an a r t i c l e priced a t $100 and paid cash it would

c o s t him $90 plus t h e sales tax or$91.80, since 2% of $90

is $1.80 , Suppose the sales tax were computed on $100 and

then the 10% discount al lowed, would the resulting net c o s t

be the same? t h y or why not?

A cuatomer in the store of Problem 15 added the discounts and

thought that s ince he was paying cash f o r an a r t i c l e on

Monday, he should receive 159, discount. If t h l s were the case he would have paid $85. f o r t h e article priced at $100,

inatead of the $85.50 . How should the shopkeeper have

worded h i s notice of discounts to make it clear that he had

In mind the calculation given in Problem *15?

I i

: Percents -- Used for Comparison

In Problem 1 of Exercises 9-7, some of the percents were not 1 1 whole percents. The number 3 written as a percent is 12 9.. You

1 l

have learned that $% can be read1'$ of 1%. Decimals are of ten used in percents, f o r example, 0.7%

1 O m 7 = 0.007 0.79. (0.7 of 1%) = 0.7 X = =

Theae are all names for the same number. If we wfsh t o find 0.7%

of $300, we w i s h t o f ind x such that

O.flo is l e s s than 1% . Since 1% of $300 I s $3.00, the answer $2.10 Is reasonable.

Suppose t h a t we wish to find 2,3%of $500.

2. 3 23 0.023. Find the number such that 2.3% = m =m =

Baseball batting averages are found f o r each player by dividing the number of h i t s he has made by the number of times he

has been at bat . The d i v i s i o n is usually carried to the nearest

thousdndth. So the bat t ing avemge can be conaidered a s a percent

expressed to the neareat tenth of a percent. If a player has 23

hits out of 71 t lmes at bat, h i s battlng average is or ,324.

Sometimes answers are c a l l e d for to the nearest t en th of a

percent. A teacher lsauea 163 grades, 35 of them R. He might

be asked what percent of hls grades are B. He wishes t o find x such that a = .

In section 9-6 you learned how to round dec lmls . If the

percent f s ca l led for to the nearest tenth of a percent, 21.47.. . would round to 21.5% . Percents of Increase and Decreaae

Percent l a used to indicate an increase o r a decrease in some quantity. Suppose t h a t Cent ra l City had a population of 32,000

(rounded to the neamat thousand) in 1950. If the population

increased to 40,000 by 1960, what was t h e percent of increase?

8 ooo . ~ g m = & - 32,000

8,000 (actual increase ) 800 ooo " - 3 g m

x = 25 There was an increase of 2 5 % .

N o t l c e that the percent of increase is found by corngaring the actual increase with the ea r l i e r population figure.

The 40,000 was made up of t h e 32,000 (low plus the increase of 8,000 (2573. So t he population of 40,000 in 1960 was 125Xof the populat ion of 32,000 in 1950.

Suppose t h a t H l l l City had a population o f 15,000 in 1950. If the population in 1960 was 12,000, what was the percent of decrease? If x mpresents the percent of decrease, then

15,000 3 000

- 12,000 Zm-Ttfa 3,000 actual decrease x = 100

x 3 20

There was a decrease of 20% . ! Notice that the population decrease a lao is found by comparing the

a c t u a l decrease with the earlier populat ion figure.

The 12,000 i s t h e d i f f e r e n c e between 15,000 ( 10Wd and the d.ecrease of 3,000 (2a). So the popula t ion i n 1960 of 12,000

was 80% of t h e population of 15,000 in 1950.

If the rents in an apartment house are increased 5 $ , each

tenant can compute h i a new rent, Suppose t h a t a tenant is paying

$80 f o r rent, what will he pay in rent a f t e r the increase? If x represents the increase i n number of d o l l a r a , then

x = 4 (The increase was $4.00.) The new rent will be $80 c $4 = $84.

Exerclses 9-8b - 1. A Junior high schoo l mthematics teacher h a d 176 pupils in

his 5 classes, The s e m e s t e r grades of t h e pupils were

A, 20; B, 37; C, 65; 11, bO; E, 14.

( a ) What percent of the grades were A? ( t o nearest t e n t h

percent )

( b ) What percent of the grades were B?

( c ) What percent of t h e grades were C?

( d ) What percent of the gradea were D?

( e ) What percent of the grades were E? ( f ) What i s the sum of the answers i n part8 (a), (b), ( c ) ,

( d ) , ( e ) ? Does t h f s sum help check the answers?

2. Bob's weight increased during the school year from 65 pounds

to 78 pounds. What was t h e percent of increase?

3 . Durlng t h e same year, Bobts mother reduced he r weight from 160 pounds to 144 pounds. >hat waa the percent of decrease?

4. The enrollment in a jun io r high school waa 1240 in 1954. By 1960 t h e enrollment had increased 25 9,. k h t was the

enro l lmen t in 1960? 5 . Jean earned $14.00 during August, In September she earned

only $9.50, Vmat was t h e percent of decrease in her earnings?

A salesmn of heavy mchinery earned a commission of $4850 on t h e sale of a mchine f o r $970,000.

( a ) At what rate is his commission paid?

(b) W h a t w i l l be h i s commission for the sale of another machine for $847,500?

James was 5 ft. t a l l in September. In June h i s heigh t is 5 f 5 i n Both heights were measured to the nearest inch, I ~ a t

i a the percent of increase in height? Do you know what your height was a t the beginning of the

school year? Now? Do you know what your weight was a t the

beginning of the school year? Now?

(a) W h a t I s the percent of increase in your height since

last September?

(b) What is t he percent of increase in your welght since last September?

A basebal l player named Jonea mde 25 hits out of 83 times at bat. Another player named Smith made 42 hits out of 143 t i m e at bat.

( a ) What is the ba t t ing average of each player?

( b ) Which player has t h e bet ter record? An elementary school had an enrollment of 790 pupils in September, 1955. In September, 1959, the enrollment was 1012

W h a t was the percent of increase in enrollment?

On the first of September Bob's mother weighed 130 pounds.

During that month she decreased her weight by 15 %. However, during the month of October her weight increased by 15%. What d i d she weigh on the first of November? Are you surprised at your answer? Why or why not? Suppose In the previous problem the weight of Bob's mother

Increased by 15%durIng the month of September and decreased by 15%during the month of October. Would the result be the same as in the previous problem? Can you see why or why not?

Then check your guess by working it ou t .

[sec . 9-81

Two Methods f o r Solvin~ Problems of Percent of Increase and Decrease - - - - In t h e s o l u t l o n of problems i n v o l v i n g percent of increase or

percent of decrease, two approaches can be used.

If the cost of b u t t e r increases from 80g a pound to 92$ a

pound, what is the percent of increase? The method you have been

using 1s

92$ - 806 = 12$ (increase). ~f 12 is x percent of 80, t h e n

x = 15 There was a 15% increase.

A second method finds what percent 92p is of 80$. Since 92

Is l a rger than 80, then 92 is more than 100%of 80. If 92 I s x

peroent of 80, then

x - 115 This means that 925 Is 115Xof 80g. If 100%t,is subtracted from

115%, t he percent of increase (l57d resu l t s .

In a cer ta in c i t y t h e fire department extinguished 160 fires during 1958, During 1959, the number of fires extinguished dropped to 120. What was the percent of decrease? \le w i l l show two

ways to so lve t h i s problem, In one method we f lnd what percent the difference ( 160 - 120) is of 160. In the 0 t h ~ ~ method we find

what percent the numbep of f i r es in the la ter year is of the num- ber of flres in the earlier year. T h i s percent is then compared

with 100% . If 120 is y percent of 160, then

160 120 m a = & - 120 - 120 y = 100

4 0 (fewer f i r e s )

Y = 75

100% - 75% = 25%

if 1s x percent of 160, The later nwnber of fires was

then 75% of the earlier number. There

g=& was a decrease of 25T0for t h e num-

ber of f i r e s in 1959 when compared 40 x = l O O = m w i t h t h e number in 1958.

There was a decrease of 25% in x = 25

t h e number of f i r e s i n 1959 when compared wlth the number in 1958.

Of course, the answers should be t h e same for t h e two methods of solution.

Exercises 9-8c

In each problem, 1 through 5 , compute t h e percent of increase o r . decrease by both methods. If necessary, round percents t o t h e

nearest tenth of a percent. 1, In a junior high school the lists of seventh-grade absentees

for a week numbered 29, 31, 32, 28, 30, The next week the five l i s t s numbered 22, 26, 24, 25, 23.

( a ) What was the total number of pupil-days of absence f o r

the f irst week?

( b ) What was the total f o r the second week?

{ c ) Compute the percent 'of increase or decrease In t h e

number of pupil-days of absence.

1 2. On the f i rs t day of school a junior high school h a d an enroll-

ment of 1050 pupils. One month later the enrollment was 1200. What was the percent of Increase?

3. One week the school lunchroom took in $450. The following week the amount was $425. What was the percent of decrease?

4. Fron the weight at b i r t h , a babyts weight usually Increases 100% i n s i x months.

( a ) What shou ld a baby weigh at s ix months, If I t s weight at birth is 7 lb. g oz.?

(b) Suppose t h a t t h e baby in ( a ) weighs 17 lb, at the age of

s ix montha, What is the pekefi t of increaae?

5. During 1958 a family spent $1490 on food, In 1959 the aame f a m i l y spent $1950 on food. m a t was t h e percent of increase

in t h e money spent for f o o d ?

* 6 . During 1958 t h e owner of a business found that sales were below normal. The owner announced to h i s employees that a l l wages f o r 1959 would be cu t 20%. By t h e end of 1959 the owner noted t h a t aa l e s had returned to t h e 1957 l eve l s . The owner then announced to t h e employees that the 1960 wages would be in- creased 20% over those of 1959,

( a ) blhich of t h e following statements I s true?

(1) The 1960 wages are the same a s the 1955 wages.

( 2 ) The 1960 wages are less than t h e 1958 wages.

( 3 ) The 1960 wages are more than t h e 1958 wages. ( b ) If y o u r answer t o part ( a ) is ( I ) , j u s t i f y your answer.

If your answer to part (a) is ( 2 ) or ( 3 ) express the

1960 wages a s a percent of 1958 wages.

*7. In an automobile factory t h e number of cars coming of f the

assembly line in one day is supposed t o be 500, One week the

plant operated normally on Monday. On Tuesday there was a

breakdown which decreased the number of completed cars t o 425 f o r t h e d a y . On Wednesday operat ions were back t o n o m l .

(a) What was the percent of decrease in production on Tuesday

compared w i t h Monday?

(b) What was the percent of increase i n production on

'dednesday compared w i t h Tuesday?

8. ( ) A salesman getn a a commission on an a r t i c l e which he

s e l l s f o r $1000. How much commission does he get?

( b ) A bank g e t s 6% interest per year on a loan of $1000. How much i n t e r e s t does t h e bank ge t?

395

(G) The tax on some jewelry I s 6% . How much tax would one

have t o pay on a pearl necklace worth $1000.?

( d ) Can you rind a n y r e l a t i o n s h i p among ( a ) , (b), and ( c ) ?

9. (a ) An a r t i c l e 13 sold at a 5% discount . ff the s t a t e d

pr i ce I s $510, what did it sell for?

( b ) A small town had a populat ion of 510 people on January first, 1958, The population decreased 5% during the following twelve months, What was i t s population on January f frst, 1959?

( c ) On a l o a n of $510, instead of charging interest, a bank loaned It on a discount of 57; that i a , gave the customer $510 minus 5%of $510 at the beginning of the

year w i t h the understanding t h a t the amount of $510 would be paid back at t,he end of the year. How much d i d the

customer receive at the beg inn ing of the year?

( d ) Can you f ind any re la t ionship among (a) , (b), and ( c ) ?

10. What was the Interest rate in p~oblem 9 ( c ) above? 11, ( a ) Mr. Brown paid $210 In gasoline taxea d u r i n g a year.

If t h e tax on gasoline is 31%, how much d i d he spend on gasol ine?

(b) Mr. Smlth made a down payment of $210 on a washing machine. If t h i s was 31% of t h e total cost, what d i d

the washing machine cost?

( c ) In a cer ta in town 31Xof the populat ion was chi ldren. If there were 210 chi ldren, what was the population of the town?

( d ) On an a r t i c l e priced at $810, a merchant made a p r o f i t

of 31%. What was the amount of h i s p r o f i t in d o l l a r s ?

( e ) Can you find any r e l a t i o n s h i p among (a) , (b), ( c ) ,

and ( d ) ?

12. The income tax c o l l e c t o r looked at the Income tax r e t u r n of Mr. Brown mentioned In Problem 11 (a). He fnquired t o find

t nd t M r . Brown d r o v e a Volkswagen which would go about 30

miles on a gal lon of gasol ine. He a l so found t h a t Mr. Brown

could walk to work.

[sec . 9-81

396

(a) If gasoline c a s t 9.30 a gallon (including tax), how many

gallon8 d i d Mr. Brown buy? (use in fomat ion from Problem 11 (dl .)

(b) How far could he drive w i t h t h i s amount of gasoline?

( c ) What would be the average per day?

( d ) Why d i d t h e tax col lector question Mr. Brown's return?

Chapter 10

PARALLELS, PARALLELOGRAMS, TRIANGLES, AND RIGHT PRISMS

10-1. Two Lines In a P l a n e

A s early as 4000 B.C. geometry had an in f luence on the way I n which man l i v e d . Clay t a b l e t s made in Babylon some 6000 years

ago show tha t Babylonians knew how t o f i n d the area of a

r ec t angu la r f i e l d . They used the same relation you learned in

Chapter 8 . The g rea t pyramid at Giza in Egypt was built about 2900 B,C .

The cons t ruc t i ons of t h i s and other pyramids show that t h e

Egyp t i ans t o o must have kno~im much about geometry. About

1850 B.C. , Egyptian manuscripts were being wrftten which show

t h a t t h e Egypt ians were developing geometric rules. One of these

was the general r u l e fo r f i n d i n g the area of a t r i a n g l e , which

w i l l be discussed l a t e r in thls chap te r ,

tlhen a mathematician begins an investigation, he u s u a l l y

s t a r t s with a very simple case. After he feels t h a t he under-

[ stands this case, he may then proceed to more complicated

1 s i t u a t i o n s . In order to get a r ee l ing for space r e l a t i o n s h i p s ,

let us begin by studying more about figures formed by two lines. t Figure 10-1-a shows that the intersection of lines 1

4 a n d l :, is p o i n t A . Rags on A2 are AB and . (~ecall

-+ that A 3 means the ray with A as endpoint and containing -+ + po in t B). Rays on el are AE and AD.

Ff gure 10-1-a

Observe t h a t there are four angles formed by the rays in

Figure 10-1-a: angle BAE, angle BAD, angle CAD, and angle

CAE. We s h a l l speak of these angles as angles formed b y y l

and&*. In t h i s chapter, then, we shall speak of two i n t e r -

s ec t ing llnes as determining four angles.

Look at angles CAD and DAB. These two angles have a

common ray, AD, and a common vertex, poin t A , Any two angles

whlch have a common ray, a common vertex, and whose i n t e r i o r s have no point i n common are ca l l ed adjacent angles. ( ~ d jacent

means "neighboring" . ) Thus, angles CAD and DAB are adjacent angles. Are there any more pairs of adjacent angles I n the

f igu re? Yes: DAB and BAE, BAE and EAC, EAC and CAD,

are a l l pairs of adjacent angles,

Are angles BAD and EAC adjacent ang les? No, they are not! But they a re both formed by rays on the two l i n e s & 1

and xg. In this chapter, when we speak of a line we vlll always mean a straight line, extending wlthout end in both d i r e c t i o n s .

Men t w o lines intersect , the two palrs of non-adjacent angles

formed by these lines are given a spec ia l name, v e r t i c a l angles. Note t h a t here "ve r t i ca l " is not associated with "horizontal .It

Thus, angles BAE and CAD are a p a i r of v e r t i c a l angles. In Figure 10-14, consider the line %? conta in ing the rays

4 ----* AB and A C . Recall that in Chapter 7 you learned tha t if we + place a p r o t r a c t o r with vertex at A so that AC is t h e zero

4 ray, then AB corresponds to the number 180 on a protractor.

A line divides a plane i n t o two hal f -p lanes . In Figure

10-1-b, one half-plane determined by 1 is shown by the shaded

area a b o v e e l . The other half-plane determined b y l l is

shown as the non-shaded area be low/ l .

In Figure 10-1-c, the two half-planes determined by/

B P ~ shown by the shaded area below l2 and the non-shaded area

Figure 10-1 -b

f Figure 10-1 -c

i : Thus, two intersecting linea separate a plane Into f ou r

i regions. Each of these f o u r regions I s the interior of an angle. i ! The i n t e r i o r of each angle 1s the intersection of two half-planes. i i The intersection of the shaded half-plane in Figure 10-1-b and r

the shaded half-plane in Figure 10-1-c is shown in Figure 10-1-d. i I i The i n t e r s e c t i o n of these two half-planes is the interior of angle 6 / BAE. The i n t e r f o r of angle CAD is the intersection of the two I remaining unshaded half-planes. These two angles, angle BAE r i and angle CAD, are v e r t i c a l angles. Can you see the t w o half-

i planes which inc lude the Interior of angle CAD? Can you see

' the two half-planes which include angle BAD?

[aec . 10-1)

Figure 10-1-d

If the sum of the measures in degrees of two anglea is 180,

the angles are c a l l e d supplementary angles. In Figure 10-I-d, the shaded angles BAE and BAD are supplementary, Are any o ther pa i r s o f angles in t h i s f igure supplementary? A r e L CAD

and 1 BAD supplementary angles? hhat about L CAE and 1 CAD?

In Figure 10-1 -a the supplementary anglea a re a1 so adjacent angles. Angles BAE and BAD are adjacent anglea, as are angles CAD

and BAD. In Figure 10-l-e, if the measure of angle M is 40

and t h e measure of angle N I s 140, then angles M and N

a r e supplementary angles.

Figure 10-1-e

You learned in Chapter 7 how to measure angles. We shall

f requent ly t a l k about the "measure of an anglet1 In this chapter . Therefore, i t is convenient to have a symbol f o r t h i s statement.

To i n d i c a t e the number of units in an angle, we will use t he symbol It I t m fallowed by t h e name of the angle enclosed in parentheses. For example, m ( L ABC) means the number of units in angle ABC.

As you learned, any angle can be used as a unit of measure,

but in thfs chapter we w i l l use the degree as the standard unit.

Thus when we write r n ( ~ ABC) = 40, we will understand angle ABC

I s a 40 degree (40') angle. Note that slnce m(L ABC) is

a number, we write only rn(L ABC) = 40, not l'rn(b~~) = 4 0 ~ ~ "

Exercises 10-1

Use Figure 10-1-f in answering 1 through 3 .

Figure 10-1-f

I. ( a ) Name t he anglea adjacent t o L JKM. ( b ) Name t h e angles adjacent to L LKN. ( c ) Name the angles adjacent to L JKL.

2. ( a ) Name the angle which with L JKM completes a pa i r of v e r t i c a l angles.

(b) Name another p a i r of v e r t i c a l angles in t h e f i g u r e .

( c ) Ifhen two l i n e s in te r sec t in a point , how many pairs of

v e r t i c a l angles are formed?

3 . ( a ) Use a p r o t r a c t o r to find the measures of the v e r t i c a l

angles, L JKM and L LKN . (b) k h a t a re t h e measures of angles NKN and JKL?

( c ) Vhat appears t o be true concerning the measures of a

pa i r of v e r t i c a l angles?

( d ) Draw s e t s of t w o intersecting llnes as in Figure 10-1-f

Vary the size of the angles between t h e lines. h i t h a

pro t rac to r find t h e measures of each p a i r of v e r t i c a l

angles . Do they appear equal ?

4. Study your answers to Problem 3 . Now s t a t e your r e s u l t s in

t h e form of a general proper ty by copyLng and completing

t h e following sentence:

P rope r ty 1: khen two lines Intersect, the two angles in --- -- each pair of ? angles which are formed have ?

measure, OF are congruent . *5. You have found by experiment a c e r t a i n relation to be true

in a number of cases. Let us see why this relation must be t r u e i n a l l cases.

(a ) In Figure 10-1-f what is t he sum of the measures of L Jm and L JKM?

(b) t h a t is the sum of the measures of L IXM and L JKM? ( c ) If the measure of JKM is 60, then what is the

measureof LJXL? Of LNKM? (d) If the measure of angle JKM is 70, then what is the

measure of Jrct? Of L NKM? ( e ) fihat can you say about the measures of angles JKL

and NKM when the measure of angle J K M changes?

Explain why angles JKL and NKM in Ff @re 10-1-f

must have the same measure.

403

The following f igure is simflar to Figure 10-1-f. L e t

x, y, and z represent t h e angles LKJ, 3Ki4, and NKM r e s p e c t i v e l y . The angles are indicated In this way in the

f igure .

Copy and complete t h e following statements:

(a) m(L X I + m(L YI = ? . ( b ) m ( L . z ) + m ( L y ) = ? . ( c ) If m{L y) is known, how can you find m(L x)? How

can you find m ( L z ) ?

( d ) \!rite your answer f o r part ( c ) in the form of a

number sentence as is done in parts ( a ) and (b)

by copying and completing the following:

m(L X) = ? - ?

m ( L 2 ) = ? - ?

( e ) Write a number sentence to show the r e l a t i on between rn(L X ) and m ( L 2).

Imagine two lfnes In s p a c e .

(a) Is there any poss ib le r e l a t i o n between two lines I n

space which would not occur between two lfnes in a plane?

(b) Find an i l l u s t r a t i o n in your classroom to explain your

answer .

10-2. Three Lines in a Plane

The Greeks were the first t o study geometry as a field of

knowledge. Thales (640 B .C . - 546 8 .C . ) studied in Egypt and

introduced the study of geometry i n t o Greece. The discovery

of the property dealing with pairs of vertical angles whlch we studied in the previous sec t ion is credfted to Thales. As you

study geometry you will l e a r n about properties that he and other Greek mathernat l c l a n s discovered.

In t he prevlous section we primarily studied f igures formed

by two lines in a plane. In this section we will study figures

formed by t h ree l l n e s in a plane .

D r a w a f igure similar t o 10-1-a. Can you draw another line,

jj, through point A? Can 1 ines 1 1 *, and jj have a

comiion point of i n t e r s e c t i o n ? Your drawing should look something l i k e this:

Figure 10-2-a

Will three lines drawn on a plane always have a common point

of intersection? Look a t Figure 10-2-b where line t Intersects lines 1 and 4 *. In the language of s e t s we would say

Ant is no t t h e empty set , (and Jgn t is not the empty s e t ) .

A line whlch intersects two or more lines in d i s t i n c t po ints is

ca l l ed a transversal of those llnes. Since line t intersects ,el and d2, t is a transversal of l ines al and kg.

Figure 10-2-b

Exercises 10-2

Draw a figure similar to the

one a t the right. Lines

t2 and t2 do not intersect.

(tl n t, Is the empty s e t ) .

Line t Intersects tl. Call 3 the poin t of intersection A .

Line tj ln te r sec t s t2. Call

the po in t of intersectlan 3.

(a) How many p a i r s of ver t ica l angles are there in the f i g u r e you have drawn?

(b) How many pairs of adjacent angles are there in the f igure you have arawn?

( c ) Is l i n e t a transversal of llnes tl and t2? 3

( d ) Is tl a t r a n s v e r s a l of tg and t3? Explain.

2. Draw two lines, ml and m2, which I n t e r s e c t in point Q.

Draw a t h i r d line, m j , which in te rsec t s rnl and me,

but not in p o l n t Q. Call the intersection of ml and 5 point S , and call the in tersect ion of m2 and 5 p o i n t R .

( a ) What is the name of the s e t of points made up - - of segments SQ, QR,

and RS in t he f igure you have drawn? (see

Chapter 4 if you don't know, )

(b) How many p a i r s of v e r t i c a l angles are there in the

figure you have drawn?

( c ) How many pairs of adjacent angles are there in the

figure you have drawn?

3 . Use Figure 10-2-c I n answering t t

the following questions:

(a) Name the transversal

shown in the drawing I and tell what l i n e s it I intersects . I

( b ) In this drawing, how I many anglea are formed I by the three lines? "1'

( c ) How many pairs of ver t i ca l angles are t h e r e in this drawing?

( d ) Vhat do you know about the measures of each of a p a i r

of ve r t l c a l angles?

( e ) Does m(L h) = m ( L f)? HOW can you tell?

( f ) Is L c congruent to L d? How can you t e l l ?

(a ) How many pairs of adjacent angles are there in Figure

10-2-c ?

(b) tibat do you know about the sum of t h e measures of any

pair of adjacent angles In Figure 10-2-c?

(a) Are angles c and d in Figure 10-2-c supplementary?

(b) Are angles a and d in Figure 10-24 supplementary?

( c ) Are the angles in each p a i r of adjacent angles in Figure

10-2-c supplementary?

t ( d ) If the measure of L h is 80, what is t he measure

of L g? Of L e ? Of L f ?

I ( e ) IS the measure of L h is 90, are angles h and f

supplementary?

(f) If m ( L h ) = m(L g ) , are angles e and g

supplementary?

.6. Look a t t h e two angles, angle b l and angle f , in Flgures 10-24

a t the r i g h t . From the vertex of angle f there is a ray which

I extends upward on line t . This

1 F ray contains a ray of angle b . Also, the i n t e r i o r s of angle b

I

and angle f are on the same < e s ide of the transversal t . Angles

h f > S

placed in this way are cal led v corresponding a n ~ l e s . Figure 10-2-d

( a ) Name another pair of corresponding angles on the same

s i d e of the transversal as angles b and f .

( b ) Are angles a and e corresponding angles? ( ~ o t e

t h a t the r a y on line t which forms angle a is

only a p a r t of the ray on line t which forms angle e; however, n e i t h e r of t h e rays forming angle

e are parts of the rays forming angle a. )

(c ) Are L c and i g corresponding angles?

( d ) How many pairs of corresponding angles are in

Figure 10-2-c?

( e ) If t h e measure of L b is 80, can you t e l l what

t h e measure of L f is?

(f) If the measures of L a and L e are 90, what can you say about the measures of all the angles i n

Figure 10-2-c ?

(g) If t h e measures of L a and L e are 90, are

angles h and b supplementary angles?

( a ) How are t h e figures In Problems 1, 2 , and 3 alike?

(b) How are t h e f i g u r e s different?

( c ) Can you think of another way t o draw a set of t h r e e

lines i n a plane? (DO not u s e the i n t e r s e c t i o n of

t h ree l i n e s in a poin t . )

( d ) Copy and complete t h e fo l lowing statement:

"The In t e r sec t ions of th ree d i f f e r e n t l i n e s in t h e same plane may consist of -9 ? -' 9 - ? ,

o r ? points ,"

*a, If you look around you, you can find illustrations of sets

t of three lines like those i n the f lgures you drew in these I exercises. Now t r y t o irnaglne three planes in space.

1 I Describe a situation which represents three planes in space i I having t h e following intersections. I i t (a) A poin t . ! (b) Two parallel lines.

( c ) Three parallel lines.

( d ) The empty set.

10-3. Parallel -- Lines and Corresponding Angles We have two l i n e s rl and r2, c u t by the t ransversa l t .

Lines r and r2 do not I n t e r s e c t . In the language of sets 1 we now have the following:

Figure 10-3a

rl n t is no t the empty set,

r2 n t 3s not t h e empty se t ,

rl n r, is the empty s e t .

That I s , rl and r2 are parallel. Nelther rl nor r2 is

parallel to t.

[sec. 10-31

Class Exercise and Discussion

1 You are to make a drawing similar t o Figure lo-3-a. Draw

a l i n e t, and loca te points A and E about 1; inches

a p a r t . At p o i n t A , draw an angle w i t h measure 70 f o r

L a . A t p o i n t B, use a measure of 40 f o r angle i b. Do the l i n e s Intersect If extended? If so, on which side of t, the Left or t h e right?

3 . Draw anc",hcr figure making 30 t h e measure of L a and

40 t h e measure of L b. Do t h e lincs intersect I S extended?

If so, on whfch side of t, t he l e f t o r the r i g h t ?

3 . Make at l e a s t six experiments of this kind w i t h various

measures for angles a and b. In at l e a s t two cases use

t h e same measure f o r L a t h a t you use f o r 1 b . Record

your r e s u l t s l i k e this:

4. Cqpy t h e following t ab l e . Pred lc t whether rl and r2

i n t e r s e c t and, if so, where. Make a drawing t o check your

prediction. (YOU may extend the table and choose your own measures of' angles i f you wish. )

Measure of a

in degrees

70

30

40

Measure of L b in degrees

40

4 0

4 0

Measure of L a

In degrees

50

50

50

Intersection of r and r2 1

Left of t

right of t

?

Measure of i b in degrees

80

50

40

Intersection of r1 and r2

After c lass discussion of Problems 1 th rough-4 , copy and

complete the following statement:

Proper ty 2 , When,in the same plane, a transversal ---- - in te r sec t s two l i n e s and %pair of corresponding angles --- - have dif ferent ? , then t h e l i n e s ? . - ---- Consider t h e case where the corresponding angles are

congruent bave e q u a l rneasure4. !.:.rite a statement for thr; s

c a s e . Copy and complete:

Property 2 a . When, in the same plane, a transversal A--

intersects two lines and a pair of corresponding angles ---- - are conaruent. then t h e lines are ? .

1. Make a f i g u r e l i k e t h e one below. M a t ang le forms with

L x a pair of corresponding angles? Label it i p. If angles x and p have t h e followlog measurements, do

A1 a n d l n i n t e r s e c t above t , below t, o r a re t hey

parallel?

41 2

Copy and complete:

2. L i s t several examples of parallel l ines tha t you f i n d In

the classroom.

Are Intersect Intersect

* 3 . Lay a yards t i ck or a r u l e r across some of the parallel l i n e s you find in Problem 2 so as to form a transversal. Do this

so that t h e intersections w i t h the parallel l i n e s are a t a

minimum d i s t ance a p a r t . Measure this distance in each case.

What is the measure of the angle formed by t he transversal

and each line when the minimum distance is found?

(b 1 1-20

( c > 120

( a ) 90

(4 90 1

I f ) 90

(g) 40

(h 1 40

(1 40

10-4. Converse ( ~ u r n l n q - a Statement round) Me have seen that if c e r t a i n t h lngs are t r u e , then cer tafn

other things are t r u e . For example:

120

90

1-20

90

70

90

40

20

I Suppose we make a new statement by in te rchanging t h e "if"

part and the "then1' part . This is t h e new statement:

It (b) - If two angles have t h e same measure, then t h e angles

I are vertical angles .I1 A statement obtained by such an

interchange is called a converse statement. In the

example above, (b) is called the converse of ( a ) .

Since such an interchange in (b) brings us back to

( a ) , we a l s o c a l l ( a ) the converse of (b) .

I If you "turn a t r u e statement around," will t h e new statement alvays b e t r u e ? L e t us look at two statements and t h e i r converses and d e c i d e .

1 . lll;f Mary and Sue a r e s i s ters , then Mary and Sue are

g i r l s . I t

Converse o f 1 : "If - Mary and Sue are girls, then k r y and Sue are sisters." Is t h e o r i g i n a l statement t r u e ? Is the

converse a l s o t r u e ?

1 N o w c o n s i d e r t h e n e x t s t a t e m e n t :

I 2 . "If L i e f is t h e son of Eric, then E r i c is t h e father

of Lief ." Converse of 2: "If E r i c is the f a t h e r of L i e f , then L i e f

is t h e son of E r i c ." Ts t h e o r i g i n a l statement t rue? Is the converse true?

I We can see from these two Illustrations t h a t , if a statement

is true, a converse obtained by interchanging t h e "ift' part and

.the "then" part , may be t r u e o r may be f a l s e .

3 . Is statement (a) above, dealing with v e r t f c a l angles, true? Is the converse statement, (b), true? We cannot accept a converse statement as always being true. Sometimes when a

t r ue statement is t t t u r n e d around," a converse I s true. Sometimes

when a true statement I s "turned around," a converse is f a l s e .

414

Exercises 10-4

1. Make a drawing f o r which a converse of statement (a) in Sect ion 10-4 is not t r u e . Must any two angles whlch

have the same measure always be v e r t i c a l angles?

2 . For each of the following statements wri te "true" If the statement is always true; "false" if the statement is sometimes false.

( a ) I f Slackie I s a dog, then Blackie is a cocker spaniel.

( b ) If it Is J u l y 4th, then it is a holiday i n the United States.

( c ) If Robert is the tallest boy in his school , Robert is the t a l l e s t boy i n his class.

( d ) If an animal is a horse, t h e animal has four lega ,

( e ) If an animal 1s a bear, t h e animal has thick f u r .

( f ) If Susan is Mark's sister, then Mark is Susan's brother

3 . I1:rite a converse f o r each statement in Problem 2 and tell whether the converse is true or f a l s e .

4. Read the following statements. Wrfte "true" ff the statemen{ is always t r u e ; "false1' if the statement is sometimes fa l se .

(a) If a figure I s a circle, then the figure is a slmple,

closed curve .

( b ) I f a f igure is a simple closed curve composed of three line segnents, then the f igure Is a triangle.

( c ) If two angles are congruent, they are right angles,

( d ) If two lines ape para l l e l , then the l i n e s have no point in common.

( e ) If two angles are supplementary, they are adjacent.

( f ) If two adjacent angles are both right angles, they are

supplementary.

[ s e c . 10-41

( a ) Write a converse statement f o r the following property.

Property 2a: If, In the same plane, a

transversal i n t e r s e c t s two l i n e s , and a pair of

corresponding angles are congruent, then the lines

are parallel.

( b ) To t e s t the possible t r u t h

of the converse, draw rnl and m2, and trans-

< {' >

versa1 t as in the figure . \ Are corresponding angles

congruent? Now draw any

other t ransversal of mL

and m2. Call it tl.

Measure the angles in each

pair of corresponding angles

along tl . Are they congruent?

Compare your results with those of your classmates.

( c ) On the basis of this work, do you think the converse

of Property 2a stated in ( a ) above I s t r u e o r fa1 se?

Xrlte a converse statement f o r Property 2: If, in t he

same plane , a transversal in tersects two lines, and a pair

of corresponding angles have different measures, then t he

l i n e s i n t e r sec t . J%es the statement seem to be t r u e or false? You can test your results by making drawings as

In Problem 5 .

Can a converse for a false statement be true? If so, can

you give an example?

10-5. Triangles

You have been discovering angle relations in a figure

composed of t h ree l i n e s , two parallel lines and a transversal.

Suppose the lines are arranged so t h a t each' l i n e is a t ransversa l f o r the other two lines. Would auch a f igure appear like this

? f igure?

In the drawing above, / is a transversal f o r 4 and

jj; R 2 is a transversal f o r d 1 a n d a j : A, is a transversal - f o r 1, and e2. A , B, and C are three polnts and AB, - - BC, and CA are segments joining them in pairs. In Chapter 4 on-~etrlc ~eornetry) the union of , three p o i n t s not on the same

l i n e and the segments Joining them in p a i r s was c a l l e d a t r i a n ~ l e . Our sketch, according to this deflnltlon, contains triangle ABC.

The points A , B, and C are called the v e r t f c e s of the - - t r f ang le and the segments AB, BC, and CA are c a l l e d the sides of the triangle .

lklich of the figures below are triangles? If a f igu re is

not a t r i a n g l e , tell which requirement of t he definition is

l ack ing .

The triangles in the group below are members of a p a r t i c u l a r s e t of triangles because they have a common proper ty . \%at do

you notice about the lengths of the sides of the t r l a n g l e s ? (The symbol 3" means three inches . The symbol 3 means

three feet. )

Note that each triangle has at least two sides which are equal in length. The triangles of this set, and a l l o the r

triangles t h a t have this property, are called isosceles triangles. The triangles in the group below are members of another

particular s e t of t r2angles. \!hat is the common property i n

this set of triangles?

Note t h a t each triangle h a s t h r e e sldes which are equal in length.

Another way of saying this is that a l l sides In the same t r l ang le have equal measures. All triangles of thfs set, and all other

triangles that have this proper ty , are ca l l ed equilateral t r i a n g l e s . The word "equi la tera l1 ' comes from the words, "equi"

meaning equal and " la te ra l t t meaning side . The triangles in the following'group are members of still

another set of t r i ang l e s . Do the triangles in t h i s set appear

t o have a proper ty in common?

Do two sides in any of the above triangles have the same

measure? Do any of the triangles have three sides w i t h equal

measures? No two sides in any of the triangles above have the

same m e a s u r e . The triangles in this p a r t i c u l a r se t , and all o the r triangles h a v i n g t h e same proper ty , a re called scalene

triangles.

Here is a p i c t u r e of a soda straw and here It is creased at two p o i n t s n b

ready to be fo lded so that t h e ends n v v I come together like this:

khat kinds of triangles could be represented In t h i s way by

folding the following straws:

Now you are to draw some t r i a n g l e s . D r a w an angle , and

name the ver tex A . Locate a poln t B on one r a y of t h e angle , - and a poin t C on t he o the r r a y . Draw segment BC Your drawing

shou ld look something l i k e this.

- - - The union of AB, BC, and AC 1s called " t r i a n g l e AEC ." Notice that in your drawing, points C and B are the only points - shared o y angle A and s ide BC . Angle A and side BC are sald to be opposite each other because their intersection consists - only of t he endpoints of BC. Is L I3 opposite E? Why? Is L C opposite E? \,by? hha t o the r angle and side are opposite

t o each o the r in your drawing? It'hy? [sec . 10-51

4 21

Draw another t r l a n g l e as before, bu t this time very c a r e f u l l y

loca te po in t B and point C so that AB and AC are congruent

ae shown below:

A

Khat kind of a t r iangle is A ABC? Why? Make a copy of angle B

and try to fit the copy upon angle C. What does this procedure

suggest about the measure of angles B and C? Make several

more copies of the isosceles triangles on the second page of t h i s section. What do you observe? Copy and complete the

following statement :

Property 2. If two a ides of a triangle are ? in length, ----- --- then the andea -- ? these sides have ? measures. ----

Now make a drawing l i k e the one below. Draw AB about four Inches in length . Draw 40 degree angles at A and B as

shown. Locate points C and D at some convenient place on t he

rays of these angles.

Notice tha t we have drawn two angles whose measures are the same.

Extend AC and E. Name t h e i r point of i n t e r s e c t i o n E. Thus

triangle A B E is formed.

What do you observe about the sides opposlte L A and L 3 in this triangle? Make several other sketches accordfng to the d i r e c t i o n s given. Keep angles A and B acute and equal In

measure, b u t vary the measure f o r di f f erent triangles. What do

you observe about the sides I n each f i g u ~ e ? To what special set of triangles do your figures appear to belong?

Copy and complete the following statement:

If two a n ~ l e s o f g t r i a n g l e are ? in measure, then the -- --- -- sides ? these angles are ? in length. --- ---

Have you not iced that this statement resembles the statement

completed as Property 3 previously? What* new word learned by

studying Sect lon 10-4 could be used to describe the relationship between these two statements? We will c a l l this statement the converse of Property 3.

Class Discussfon Problems

1 , Could t h e three sets of triangles we have discussed

(isosceles, equilateral, and scalene) be determined by some common property per ta in ing to angle measures instead of s ide measures? What advantages or disadvantages would this method of classification have?

2. How could the converse of Proper ty 3 be used to show that

a triangle having a l l three of its angles equal in measure must necessarily be an equi la tera l triangle?

Exercises 10-5

1. Draw an isosceles triangle,

2. Draw an equilateral triangle.

3. Draw a scalene t r i a n g l e .

4. If a t r langle i a isosceles is it also equilateral? Explain

your answer.

5 . If a t r i a n g l e is equilateral, is It a l s o I sosce les? Explain

your answer.

6 . Draw an equilateral t r i a n g l e and letter the vertices P, Q,

and R. Match each angle w i t h I t s opposite s i d e by listing

them in p a i r s .

7 . ( a ) Could a t r i a n g l e be represented by folding t h e soda

straw shown In this ffgure? Explain your answer.

( b ) Could a triangle be represented ,by fo ld ing the soda

straw shown here? Explain your answer.

( c ) S t a t e a proper ty about the l engths of the ;ides of a t r i a n g l e as suggested by your observations In parts

(a) and (b),

8. OPTIONAL: Copy the following pat terns . Fold on the dotted

l i n e s a f t e r cutting along the edges, and paste the f l a p s ,

Note the equilateral triangles. Try to enlarge these

patterns f o r b e t t e r r e s u l t s .

t e t r a h e d r o n octahedron

( f o u r faces) (e ight faces)

10-6. Angles of a Triangle

In t h e previous s ec t ion we studied special properties for

c e r t a i n triangles. In t h i s section we come to a property which

is t r u e f o r a l l t r i ang les .

Class Exercises and Discussion

Draw a triangle, making each side about two o r three I n c h e s

i n length . Cut ou t the t r i angu la r region by cutting along the s ides of the triangle. Tear off two of the corners of this region and mount t h e whole f igu re on cardboard o r a

sheet of paper as shown in the figure below. Note: Corners

B and C are placed around the vertex A . (The corners

may be pasted or stapled In place , )

( a ) Find the measures of t h e three angles with ver t ices

at A . Find the sum of these three measures and compare your results with those of your classmates. In t h e

+ . --+ drawing above, does it appear that AD and AC a re on the same l i n e ? Does this appear to be t r u e on t he

I f i g u r e you made?

I ,(b) t ihat do you observe about angle I , angle 2, and

i angle BAG in this new arrangement?

D r a w a triangle making the 2 * longest side about 4 o -8 inches long, one of the inch= remaining two sides about

3 inches long, and the thlrd side about 2 inches u long. Cut out the i n t e r i o r / \ of the trlangle.

t a b o u t 4 inches

(b) Label t h e v e r t i c e s A ,

B, and C in the interior

as shown in t h e drawing a t

the right. Mark off t he - midpoint of AB. Label t h e

midpoint D. (The midpoint

is halfway from one endpolnt

of a l i n e segment t o the o t h e r endpoin t . ) Find the - midpoint of BC. Label the

midpo in t E. AD and

w i l l have the same length .

BE and EC will have the same 1 ength .

( c ) Draw a l l n e segment joining D and E. Fold downward

the p o r t i o n of t he t r i ang l e

containing the vertex B - along the line segment DE

so that the vertex B f a l l s

on E , Label the po in t where 3 falls on AC as

G. The fold is along the - segment DE ,

(d) Fold the portion containing

A t o t h e right so tha t the

vertex A falls on the po in t G , Fold the portton containing

C to t he l e f t so t h a t the

ve r t ex C a l s o f a l l s on pa in t

G . The resulting f i g u r e will % be a rec tangle .

[sec . 10-61

( e ) bha t appears to be t r u e about t h e sum of t h e measures

of t he angles A , B, and C.

(f) Does this experiment work with o t h e r t r i a n g l e s ?

Check your r e s u l t s with those of your classmates.

( g ) Does t h e property you found in Problem 1 agree with these observations?

+ --* Consider the t r i ang le ABC and the rays AP and BA

shown below. Line RS I s drawn through point C so t h a t

t he measure of L y and the measure of L y f are equal Here we are using a new notation, yl. y f is read

"y prime." (1n this problem we are using this notation

In naming angles. )

Uze a p ~ o p e r t y to explah "why" for each of the following

whenever you can: - - (a) I s R S parallel to AB? !Jhy?

( b ) \ h a t kind of angles are the pair of angles

marked x and x t ? Is m(L x ) = rn(L x i ) ? khy?

( c ) t h a t klnd of angles are the pair of angles

marked z and z l ? Is rn(L 2) = rn(L z t ) ? khy?

I d ) ~ ( L Y ) = ~ ( L Y ' ) why?

( f ) m(L X ) + m(L y ) + m(L z ) is t h e sum of the

measures of the angles of t h e triangle. Why?

(1) We conclude therefore that t h e sum of t h e

measures of the angles of the t r i a n g l e is 180.

khy?

This is a proof of Proper ty 4.

Property 4 . The sum of the measures in degrees of ---- - - the angles of any triangle is 180. - -

Ie w i l l not study many proofs t h i s year, but in some exercises you may be asked to try to discover a proo f . As you

s t u d y more geometry I n l a t e r years you should become quite able

t o discover proofs . N o t i c e t h a t in this proof we drew j u s t any t r i ang l e . Does

t h e proof apply to - all triangles? If you are In doubt about this,

you might draw some o the r triangles quite d i f f e r e n t in shape from

t h e one in this section, label points , angles, segments, rays, and lines In the same way. Then, t r y the proof above f o r the f igure

you have drawn.

Exercises 10-6

1. that Is t h e measure of each angle of an e q u i i a t e r a l t r i ang le?

[sec. 10-61

2 . \ h a t is the measure of the third angle of the t r i a n g l e s If

two of the angles of the triangle have the following

measures?

( a ) 4 0 and 80.

{ b ) 100 and 50.

( c ) 70 and 105 .

( d ) 80 and 8 0 .

3 . Suppose one angle of an i s o s c e l e s triangle has a measure of

50 . Find t h e measures of t h e other two ang le s . Are two d f f fe ren t s e t s of answers p o s s i b l e ?

4. If two t r i ang les , ABC and DEF, are drawn so t h a t

m ( L BAC) = r n ( ~ EDF) = rn(L BCA) = m ( ~ EFD), what w l l l be

t r u e about angles ABC and DEF? Upon what p roper ty i s

your answer based?

5 . In each of the fo l lowing the measures of certain parts of t h e triangle ABC are given, those of the s ides in inches

and those o r the angles in degrees. You are asked to find t he measure of some o the r p a r t . I n each case, give youp

reason.

Given

( a ) m ( L ABC) = 60, m(L BCA) = 40.

( b ) m ( ~ CAB) = 52, m ( ~ BCA) - 37.

( c ) r n ( ~ ABC) = 40, m ( E ) = 2,

and in(=) = 2 .

( a ) m ( E ) = 3 , m(E) = 3, rnm) = 3 .

( e ) r n ( ~ BAC) = 100, m(L BCA) = 40,

and m ( E ) = 4 .

Find

m ( L CAB)

m ( ~ ABc)

m ( L ACB) m ( L BCA)

m ( E )

P r o p e r t y

?

?

?

?

?

f6. In the f igu re a t the r i gh t

1 and 1 are parallel. 2

(a) Does rn(L y) = m(L n)? why?

(b) Eoes m(L y) = m ( L u)? \ m y ?

( c ) Try t o prove tha t

m(L n) = m(L 4. "g,

( d ) Discuss with your classmates the things which make a good proof .

( e ) After the class discussion, rewrite your proof In accordance w i t h the points brought out by you and your

classmates.

* 7 . In problem 6, it has been shown tha t the measures of angles

a and b in the f i g u r e below are equal if line r Is - parallel to EF. Use this property t o prove t h a t the sum of t h e measures of the angles of t r i ang le DEE! shown below is 180.

i

1 10-7. Parallelograms

i Mstance Between Pa ra l l e l Lines

Look a t the f i g u r e below showing l i n e s through po in t A

and In tersec t ing line rt

Use a protractor to check the measures of t h e angles given below.

Give the following measures.

- - - Measure the segments AD, AC, and AB.

Which is the shortest?

Copy t h e f i g u r e and draw other lines through A which

in tersect r . Measure the segments on these lines from A to t he intersections

of t h e lines with r . - Ih you find any of these segments t ha t a r e shorter than AC? - Note t h a t AC is perpendicular to r . On the b a s i s of your experlence here, copy and complete t he

fallowing statement:

The s h o r t e s t segment from a po in t A to a line r is the

The length of this s h o r t e a t segment I s of ten called the distance

from A to r. - - - -

In t h e following flgure, kl and k2 represent parallel

l i n e s . Lines a, b, and c a r e perpendicular to K2 and go

t h r o u g h three p o i n t s ; D, E, and F of K1. That is, t he - - - l e n g t h s o f F A , EB, and DC a r e the distances from F , E, and D to line k2, One o f t e n draws a small square, as in the

f i g u r e a t A, B, C, to Ind ica t e that an angle is intended t o

be a r i g h t angle .

4

C

Figure 10-7-a

Drawing a line, such as C through D and perpendicular to k2, can be conveniently done by using a card or sheet of paper

with a square co rne r , The figure

at t h e r igh t i l l u s t r a t e s this.

In Figure lo-7-a above there are 21 o the r r i g h t angles besides

those marked. Can you find them?

How do you know they are a l l r l g h t \k, angles?

- - - In Pfgure 10-7-a measure the l eng ths of FAy EB, and DC.

Do they appear t o be equal Ln length? This common length is called the distance between lines kl and k2. Thus t h e d is tance

.between - two parallel lines may be described as t he leng th of any

segnent contained in a line perpendicular t o t h e two l i n e s , and having an endpoint on each of the lines.

Parallelograms

In Chapter 4 you were in t roduced to the idea of a simple closed cu rve . Any simple c losed curve which is a union of

seejnents may be called a polygon. In la ter work you may sometimes

see the word "polygon" applied to curves which are not simple, but any polygon we study In this chapter will be a simple closed

curve. Unless it is indicated otherwise, we shall also understand

t h a t a polygon l i e s in a plane.

Polygons with different numbers of sldes ( l . e . , segments)

are given spec ia l names. You already know that a polygon with

three s i d e s is c a l l e d a triangle, Similarly a polygon w i t h four

s i d e s is called a quadrilateral, and a polygon with - five sides is

cal led a pentagon.

[sec. 10-7 1

Pentagon

In a quadrilateral two sides (segments) which do n o t

i n t e r s e c t are ca l l ed opposite sides. (HOW would you describe

t w o opposite sides by r e f e r r i n g to their intersection?) Name the pairs of opposite sides in the above quadrilateral.

A particularly important kind of quadrilateral is the parallelogram. This is a quadrilateral whose opposite sides Ire

on para l l e l lines. The f igure ABCD below represents a

parallelogram. Name I t s p a i r s of opposfte s ides .

I n the f u t u r e , if two segments lie on parallel l i n e s we will

speak of the segments as parallel, Thus we can say that the

opposite sides of a parallelogram are p a r a l l e l .

Property 5. Opposite --- sides of a parallelo~ram are parallel and congruent . -

We already know opposite sides are parallel. To test for congruence, measure the s i d e s i f i the parallelogram above. Do you

find that the opposite sides have equal length? Draw several

~arallelograms and measure t h e l eng ths of the pairs of opposite

sides. Do you agree with the property (underlined) above?

[sec . 10-71

The f igure a t the right represents a rectangular sheet of paper, (Notice

that a rectangle is a specla1 kind of

parallelogram.) Take such a piece of paper and t ear or c u t it i n t o two par t s

by f i r s t folding it along the dotted

line as shown. By l ay ing one p i e c e on t h e other, show that t r i ang le ABD and

BCD have the same size. Is the area of A I \ I0

one of the triangles e q u a l to one half the

area of the rec tangle? thy?

In Problem 5 below you a r e asked to repeat the above s teps

using parallelograms of d i f f e r e n t shapes.

Exercises 10-7

1. Identify severa l pairs of parallel l l n e s in your classroom, and measure the distances between them.

2 Iden t i fy several examples of parallelograms in your classroom.

3 . k'hich of the following f igures are parallelograms, assuming that segments which appear to be parallel are parallel?

4. Draw perpendiculars to lines in approximately the positions

illustrated through points A as indicated. Use a separate piece of paper. Do not draw in your book.

5. Draw a parallelogram and cut c a r e f u l l y along its sides.

The resulting paper represents the i n t e r i o r of t he

parallelogram. Draw a diagonal ( a l i n e joining opposite

vertices) and cut the paper along Chis diagonal, Compare

the two triangular p ieces . What do you conclude about these

triangular pieces? Carry out the same process f o r two other parallelograms of d i f fe ren t shapes. Write a statement

t h a t appears to be t r u e on the basis of your experience in the problem.

6 . For each of the sets of parallel l i n e s in the figure below,

draw a line perpendicular t o one of them. (DO not write in your book. Copy the lines in approximately these positions

on a separate piece of paper .) Are the lines which are perpendicu la r to one of two parallel llnes perpendicular to the o the r also?

7 . The fo l lowing questions refer to a figure which is a quadrllateral, as suggested by t h e drawing. Each

question, however, involves a

di f f erent quadrilateral . - - -

(a) KL is parallel to OM, LM - - i s p a r a l l e l to KO, KL has - a length of 3 in. and OK

a l eng th of 6 in. $hat a r e

the lengths of LM and OM? w -

( b ) K L ~ O M I s the empty set, %$in%? i s t h e empty s e t . - LM has a length of 4 in. and I s three times - - as long as LM. Find the lengths of KL and OK. --

is the empty s e t , O M ~ K L is not the empty

! set, Can two opposite s i d e s have the same length?

Can both pairs of opposite s i d e s have t h i s proper ty?

raw figures t o i l l u s t r a t e t h i s . )

*8. (a ) Copy and complete the fo l lowing statement of a

property:

If, in a plane, a line I s perpendicular t o one of

several parallel llnea, then it is

(b) Prove the proper ty . (YOU may assume t h a t a l i n e

perpendicular to one of two parallel l i n e s does intersect the other.)

*9. In triangle ABC shown a t A

t h e right, assume that the - - - - sements AB, DL, EM, FG

are p a r a l l e l ; assume that - the segments AC and JG are parallel; and, assume - t h a t t h e segments BC and - KE are parallel.

L M G C

( a ) List t h e parallelograms

in the drawing. h here should be 10 para l -

lelograms. )

(b) IIithout measuring, list the segments in the above - f i g u r e which are congruent to AS.

( c ) k i t h o u t measuring, list the segments in the above - f i g u r e which are congruent t o BK.

(d) Without measuring, list the segments in the above - f i g u r e which are congruent to GI.

10-8. -- Areas of Paralleloarams - and Triangles

A quadrilateral is a four-sided

f i gu re . A n y quadrilateral, in which D the opposite sides lie on parallel

lines, is called a parallelogram.

What have you already discovered about the lengths of the opposl te / / sides of a parallelogram? Figure

ABCD, shown at the rfght, is a A B parallelogram. l h i c h pa l r s of sides

in this figure are congruent?

A t each vertex of a parallelogram there 1s an angle whose interior contains the i n t e r i o r of the parallelogram. A t the vertex A in the figure on t he previous page, the i n t e r i o r of angle BAD contains t h e i n t e r i o r of the parallelogram. What

are the names of the o t h e r angles which con ta in the i n t e r i o r of the

parallelogram? How many such anglea are there altogether? These

are called the angles of the parallelogram.

If the sides of a parallelogram are extended by dotted linea, as shown in the Figure 10-8-a,

Figure 10-8-a

there are several pairs of parallel lines cut by a transversal, How many transversals do you find?

Copy Figure 10-8-a. Mark the angle of the parallelogram a t vertex A as shown. We w l l l c a l l this angle of the

parallelogram "angle A." Similarly, mark angle 3 of the

parallelogram as shown. Mark a l l the angles in your f igure

which are congruent to angle A in the same way that angle A

is marked. Do the same f o r angles which are congruent to angle B. In each case give the reason why the label is correct.

If you have proceeded corpectly, a l l the angles determined by the lines in the figure ape now labeled. On the baaia of the results j u s t obtained, complete the following statements:

Property 6a. - he angles of a parallelogram a t two -- -- consecut ive 'ver t ices are ? * -- Property 6 b . The angles of a parallelogram a t two - -- -- opposite v e r t l c e s are ? . -- According t o the results above, what can you conclude about

the parallelogram if L A is a right angle? Can you be sure it is a rectangle? (Do you recall from Chapter 8 how to f i n d the

area of a rectangle?) If t h e f igure is - not a rectangle then L A and L B are not r igh t angles, and one of them is an acute

angle. Why? Suppose the acute angle is L A. From po in t D, - draw the segment perpendicular to the base AB of the parallelogram as shown in the figure

at the r i g h t . Since w e know tha t D AD and BC are congruent, imagine

C /I I

the triangle AQD moved rigidly, / I / I t h a t is, without changing its size / I / i and shape, into the p o s i t i o n of / i i / ! triangle BQt.C. Then point Q' I / - A ----

Q 6 d

liea on the extension of AB. How d do you know?

The f i g u r e QQrCD is therefore a rectangle. (HOW do you

know it h a s right angles at C and D?) Moreover the r ec t ang le QQICD and t h e parallelogram ABCD are made up of pieces of the

same size and thus have the same area according to the Matching

Property In Section 2 of Chapter 7 . To find t he area of the parallelogram it is o n l y necessary therefore to find the area of

the rectangle, which we already know how to do. If you do not

r e c a l l this, refer t o Chapter 8. Notice t h a t the base AB of the parallelogram is congruent

to the side QQf of the rectangle. How do you know t h a t is t r u e ? - The other side of t h e rectangle, QD, is a segment perpendicular

w w to the parallel lines AB and CD. Such a line is called an - altitude of the parallelogram to the base AB. The l ength of the

altitude Is the distance between the two parallel lines and, as we have already seen, is the same wherever measured. Thus it

a c t u a l l y makes no difference whether we consider the altitude from D o r that from C or that from any poin t of z. A l s o ,

either s i d e of a parallelogram may be considered as a b a s e .

On the basis of the diacusslon above copy and complete the following statement: h he number of square units of area in the

parallelogram is the - ? of the number of linear units in the ? and the number of l i nea r units in the ? to this -

base.

Notice that if a parallelogram is a rectangle, the l engths

of the base and the altitude are simply the "length" and 'width"

of the rectangle, so the method above I s the same as our earlier one in Chapter 8.

Consider any triangle ABC as

shown at the right. Through C S

and B draw lines parallel to segments and AC and meeting i n some point S. The f igure ABSC

is therefore a parallelogram. The - eement CQ thro* C perpen- A B

d i c u l a r t o l i n e AB is called the a l t i t u d e of the triangle ABC to - - the base AB. The length of altitude CQ is the distance from

f3 C to l i n e AB. Notice that AB and CQ are also a base and

an altitude of the parallelogram. In Section 7 you discovered t ha t the areas of t r iangles

ABC and SCB are the same. Since the two t r i angu la r regions

cover the whole parallelogram and its i n t e r lo r , it follows that

the area of the triangle ABC Is one half t ha t of the area of the parallelogram A 3 S C . Using the method above f o r t he

parallelogram, copy and complete the f ollowlng statement : "The

number of square units in the area of a t r iangle is the

of the number of linear units in the and the

number of linear units in the to this base, 11

Since any side of a t r i ang le may be considered as the base, t h l s rule actually gives three ways of finding t h e area of any t r iang le . This is illustrated in the following figures showing

t h e same t r i a n g l e uslng each a?? the 3 sidgs as t h e base:

BASE AB BASE AC BASE

ALTITUDE ALTITUDE ALTITUDE SA Figure 10-8-b

In the previous illustration, the same triangle has been shown in three positions, so t h a t in each case the base is shown

as horizontal. T h l s is not necessary, and you will want to

practice thinking of the bases and corresponding altitudes in

di f f erent positions.

For example, it m i g h t have been bet te r not to move the triangle but to show the three cases as follows:

BASE CB

ALTITUDE 54

Exercises 10-8

1. Find t h e areas of t h e parallelograms shown, using the

dimensions g i v e n .

2. Find the areas of the triangles shown using the dimensions given.

3 . A right t r i ang le has sides of 5 yards and 12 yards as shown.

Find the number of aquare yards .-

In its area.

4. A man owned a rectangular l o t

150 ft. by 100 ft. From one

corner , A , a fence is placed to the po in t M in the center

, o o : ' l of the longer opposite s i d e as B M . D shown.

(a ) Find t h e area of ABDC.

( b ) Find the area of AMB.

( c ) Find the area of AMDC.

5. If b Is the number of l i n e a r units in the length of the

base of a parallelogram and h the number of linear unlta

in the length of t he altitude to this base, write a number

sentence showing how to f i n d the number A of square ~nita in the area of the parallelogram.

6 If b is t he number of l l n e a r units In the length of the base of a t r iangle and h the number of' l inear units In the length of the altitude to the base, w l t e a number sentence showing how t o find the number A of square units in i t s area.

7. ( a ) In the dsawing below, measure AE and m. U s i n g

these measures, find the area of the pasallelogram. -

(b) Measure AD and B. Using these measures, find the area of the parallelogram.

( c ) Do yous results in (a ] and ( b ) agree? Since

measurement is approximate, they may not be exactly the same, but they should be close .

8. ( a ) If a figure is a quadrilateral, must it be a parallelogram? Bpla in your answer.

(b) May all parallelograms be called rectangles?

&pla in your answer.

( c ) Can a square be called a parallelogram? ExplaSn.

( d ) Must all parallelograms, rectangles, and squares be

quadrilaterals? =plain.

9 . Use the drawing below f o r this problem. Make a l l measurements to the nearest quarter of an Inch.

(a) Flnd the area of the trlmgle ABC by using the measures of and 75,

(b) FLnd the area of the triangle ABC by us ing the measures of 7% and XF.

( c ) Flnd the area of the triangle ABC by u s i n g the measures of and m.

(d) I)o your results In (a), (b), and ( c ) agree? As in problem 7, they may not be identical, but they should be close.

10. BRAINBUSTER. The

parallelogram and rectangle shown at

the r i g h t have bases

with equal lengths.

and altitudes with

equal lengths. ___l____t__l__f _ Trace t h e f igures on //:/-L another sheet of

paper, Cut out the

parall el ogram . Then

cut the parallelogram i n t o pieces which can be reassembled to form the rectangle,

11. BRAINBUSTER. Let QRST be any parallelogram not a rectangle, One possible drawing is shown below. Extend the line - segments TS and 3 as shown in the second figure below. A t the vertices Q and S, where the angles of - t he parallelogram are acute, draw the perpendiculars QV

and %. QUSV is a rectangle. Let b be the measure of - QR, h the measure of 5, and x the measure of E.

- If the measure of QU is b + x, what is the - measure of VS?

What is t h e measure of @?

Khat is t h e measure of TS?

bihat is the measure of E? \,,hat is the area of QUSV?

What is the area of the tpiangle RUS?

What is the area of the t r i ang le QVT?

Using your answers from ( e ) , (f), and ( g ) above,

show that the area of QRST is given by the sentence

A = bh.

10-9. R i g h t P r i s m s In Chapter 8 you learned about rectangular prisms. H e r e

other kinds of prisms will be introduced. Let us imagine two triangles of exactly the same size and shape l y i n g in parallel

planes. We shall say that triangles of exactly the same size and shape are congruent. Triangles ABC and A I B t C 1 in the figure below represent such triangles.

If the segments are drawn joining corresponding vertices , three quadrilaterals (four-sided polygons) are obtained. In this case

they are ABBIA', BCCtBt, and CAAfCi. If the triangles are so placed wlth respect to each other that these quadrilaterals are a l l rectangles, the resulting figure is called a trianp;ular ri&t prism.

The six points A, B, C, A ! , Bf , C8 are called the vertices of the prism, the various segments shown i n the figure are i t s edges, and the i n t e r io r s of the two t r i angu la r ends and of the t h ree rectangular sides are called its faces. To

distinguish them from the Interiors of the rectarigular sides, the i n t e r i o r s of the two triangular ends are often called the bases of the prism. How many edges, vertices, and faces are there - on a t r i angu la r prism?

Very l i k e l y you have seen a glass s o l i d whose surface is a

triangular prism. When held in sunlight such a s o l i d has the

effect of bending the l i g h t rays to produce the familiar rainbow e f fec t . In f a c t , such solids are often cal led prisms.

If, in place of us ing t r i angu la r regions for bases, we use the regions bounded by other polygons, the resulting figures are o the r kinds of prisms. ( ~ e c a l l the d e f i n i t f o n of ~ p l y g o n from

Section 7.) For example, look at the figure shown on the next

page in which the ends are pentagons (five-sided polygons) of the same s ize and shape in parallel planes, and so related t h a t

the quadr i l a t e ra l s ABBt A t , BCC W , etc . , are a l l rectangles. T h i s figure is called a pentagonal ri&t prism.

In general a r i g h t prism is a f fgure obtained from two

congruent polygons so located in parallel planes that when the

segments are drawn joining corresponding ver t ices of the polygons, the quadrilaterals obtained are a l l rectangles. The prism is the

union of the closed rectangular regions and the t w o closed polygonal regions. The rectangular regions are called the fa&s of the prism, the segments are its edges, and t h e p o i n t s where two or more edges meet are called vertices. The - bases o f the prism are the regions bounded by the or ig ina l

polygons.

As was 'indicated, t h e figures described here are c a l l e d

right prisms. La te r you will meet more general prisms for whlch

the quadrllaterals mentioned are allowed to be any parallelograms rather than necessari ly rectangles. In this chapter, however, we consider only right prisms, and whenever the word prism Is used

it will mean right prism. The rectangular prisms discussed in

Chapter 8 are, of course, simply t he right prisms whose bases are t h e rectangular r eg ions . These prisms have a very interesting

proper ty . A rectangular prism can be thought of as a prism in

three different ways, slnce any p a i r of opposi te faces can be

used as bases. No other figure can be thought of as a r i g h t

prism in more than one way.

In the drawings of the prfsms on the preceding pages, it has been convenient to show the planes of' the bases as horizontal.

However, there should be no d i f f i c u l t y in identifying such

figures when they occur in d i f f e r e n t p o s i t f a n s . For example,

the f i g u r e below represents a t r iangular prism with bases ABC

and A t B I C t , even though it is shown resting on one of its rectangular faces .

Let us consider now the problem of determining the volume of a prism. Refer to Chapter 8 and reread the discussion about

the volume of a rectangular prism. This discussion showed that

if the area of the base were 12 square unlts, then by u s i n g

a t o t a l of 12 unf t cubes of volume (some of which may be

subdivided) we could form a layer one unit t h i ck across t h e 1 bottom of the prism. If the pr ism were 3T units hlgh, it

1 would take 32 such layers t o f i l l the prism, or a t o t a l of (12)($) unit cubes, so the volume is 42 cublc u n l t s .

In this discussion it was not necessary t o conslder the

actual shape of the base. In f a c t the same reasoning app l i e s t o

finding the volume of any right prism, no matter what t h e shape

of the base, for the volume of any right prism can be considered

as consisting of a serfes of layers p i l e d on each o t h e r .

Me t hus obtain the fol lowing conclusion:

The number of cubic units of volume of a rlght prism

is the product of the number of square units of area in t h e

base and the number of linear units i n t h e he igh t .

I n this statement, the term helgh t means t h e perpendicular

d i s tance between the planes of the bases, It is the length of the sements from any vertex of one base to the corresponding ver tex of the other base. Notice especially that the height i s not measured v e r t i c a l l y unless the planes of t he bases happen to

be horizontal. In the last figure, f o r example, t h e height i s - - t he length of any one of the segments , BBf , o r C C v .

A s an example let us find t h e volume of t he triangular

prism shown below.

The bases are the triangular regions, and by Section 10-8 the

number A of square inches in this triangular base is 1

A = F(6)(8) = 24, so t h e area is 24 sq. in., but t he number of

inches i n the height of t h i s p r l s m is 25. Thus by t h e statement

above the number of cubic inches in the volume is 24 . 25 = 600,

so t h e volume is 600 cu . in.

4 53

Exercises 10-9

1. Find the number of cubic units of volume for each of t h e

prlsms shown below:

2. Find the number of cubic u n i t s of volume f o r each of the

prisms shown below:

Area of the Pen- tagon is 21 square inches.

3 . The columns in front of a bu i ld ing are in the shape of prisms 18 ft. hlgh. The bases are hexagons 15 inches

on a side. (A hexagon is a polygon w i t h six sides.) If

the columns are to be painted, f i n d t h e number of square

feet of sur face f o r each column to be painted. ( ~ o t i c e t ha t the bases--i . e . , the ends--are not to be painted . )

[aec. 10-91

4. A pup tent in t he shape of a

t r i a n g u l a r prism is 7 f t . long . The measurements of

one end are given in t h e

drawing.

( a ) If t h i s pup tent has canvas ends and a canvas floor

how many square feet of canvas is used in I t s construction? ( ~ a k e no allowance

f o r seams . )

( b ) If the t e n t has both ends but does not have a f l o o r , how much canvas is used?

( c ) How many cubic feet of air are in the t e n t ?

5 . If B stands f o r the number of square units of area in the base of a prlsm, and h is t h e number of l inear units in its height, write a sentence showing how to f ind the number

V of cubic unlts of volume in the prism.

6. A contalner in the shape of a prism is 11 inches high and

holds one ga l lon . How many aquare inches are there in the base? Do you know the shape of the baae? ( A gallon contains

231 c u . in.)

7 . A t r iangular prism ha6 a baae which is a right triangle

with the two perpendicular s i d e s 3 inches and 6 inches. If the prism is 20 inches high, what is t h e volume in

cubic inches?

4 55 A trough in the shape of a

t r iangular prism Is made by fastening two boards together at r i g h t angles and putting

on ends. If the inside

measurements are 6 inches,

as shown, and if the trough is 12 ft. long, f ind how many cubic feet of water it will hold.

Make models of triangular and pentagonal prisms from stiff

paper. Either make your own patterns, or use those on the

following pages.

How many edges, faces, and vertices are there on a triangular

prism? a pentagonal prlam? a hexagonal prism ( 6 sides)? an octagonal prism (8 s ides)?

Pattern f o r Triangular Prlam

Pa t t e rn for Pentagonal Prism

[aec. 10-91

10-10. Summary

Chapter 10 dealt l a r g e l y with some of t h e relationships

which exist between llnes on a p l a n e . Section 1 d e a l t with

properties of two l i n e s in a plane. Here you studied p a i r s

of angles ca l l ed v e r t i c a l angles and observed the following

property:

Proper ty 1. hhen two lines i n t e r s e c t , t he two angles in

each p a i r of v e r t i c a l ang l e s are congruen t .

In Sect ion 1 adjacent angles and supplementary angles were a1 so, considered .

Section 2 d e a l t with properties of three l i n e s in a plane,

i n t r oduc ing the ideas of transversals and pairs of corresponding angles. Do you r e c a l l t h e different kinds of f igu res that may be formed when three l i ne s are drawn on a plane?

In Sect ion g , information from Sections 1 and 2 was used t o

i nves t iga t e two important properties.

Proper ty 2 . kinen, n the same plane, a transversal

in te rsec t s two l i n e s and a p a i r of corresponding angles are n o t

congruent, then t he lines i n t e r s e c t .

Proper ty 2a. Idhen,in the same plane, a transversal

i n t e r s e c t s two lines and a pair of corresponding angles are congruent, then the 1 ines are p a r a l l e l .

In the language of s e t s , Proper ty 2 refers t o two lines

whose intersection is not t he empty s e t , while Proper ty 2a

refers to two lines whose intersection is the empty set.

Converse statements were considered in Sect ion 4 . Examples

were given t o show t h a t the converse of a true statement m i g h t be

t r u e , and that the converse of another true statement might be

fa lse . You were asked t o write a converse f o r Property 2 and

f o r Property 2a, as fol lows:

Converse of Property 2. If, in a plane, a transversal i n t e r s e c t s two non-parallel lines, then corresponding angles are not congruent .

Converse of Property 2a. If, in a plane, a transversal in tersects two parallel lines in a plane, then any p a i r of

corresponding angles are congruent.

You will remember tha t you found these converses were both

t r u e . In Sect ion 5 names were introduced f o r d i f fe ren t sets of

triangles: isosceles, equilateral, and scalene. Scalene triangles are those having no two sides congruent. isosceles triangles are those having a t least two sides congruent, while equi la tera l

t r i a n g l e s are the special set of isosceles triangles for which all

three sides are congruent. In this section we discovered the

following property of isosceles triangles, and found that its

converse is also t r u e .

Property 3 . If two sides of a t r i a n g l e are congruent, then the angles opposite these sides are congruent.

Converse of Property 3 . If two angles of a t r i ang le are congruent, then the sides opposi te these angles are congruent .

In Sect ion 6 we observed:

Property 4. The sum of the measures, in degrees, of the

angles o f any t r i ang le i s 180.

You obtalned thfs property by using the inductive method of

reasoning when you t o r e t he corners from a t r iangular region and placed them as adjacent angles. You also learned to prove this

property by the deductive method when you used previously proved

properties to show t h a t

rn(L X) + m(L y > + m(L z ) = 180

where L x, L y, and L z represent the angles of a trlangle.

The parallelogram studied in Sect ion 7 belongs t o the spec ia l

s e t of polygons called quadrilaterals. Do you remember what

quadrilateral means? A parallelogram belongs t o the specia l set

of quadrilaterals whose opposite segments l i e on parallel lines.

[ a e c . 10-101

Property 5. 0pposfte.sides of a parallelogram are parallel and congruent .

You also learned in this section that the shortest s e p e n t from a poin t to a line is the one perpendicular t o t h e line,

and that two parallel lines are a constant distance a p a r t .

In Sect ion 8 , two properties dqallng with the angles of a rectangle were introduced:

Property 6a. The angles of a parallelogram a t two

consecutive v e r t i c e s are supplementary.

Property 6b. The angles of a parallelogram a t t w o opposite

ver t i ces are congruent.

In this section you a l s o used c e r t a i n properties of parallelograms i n finding formulas f o r the areas of a parallelo- gram and a triangle:

( a ] The number of square units of area in a parallelogram

is the product of the number of l i n e a r unlts In the

base and the number of linear units in t he altitude to

this base.

(b) The number of square units of area in a t r iangle is

ha l f the product of the number of linear units in the base and the number of l inear units in the altitude to this base.

Section 9 d e a l t w i t h volumes of r ight prisms, and was an :xtension of the work with rectangular prisms in Chapter 8 . Here

you learned t o f ind the volume of the interior of any r i g h t prism:

The number of cubic units of volume In a right prism

is t h e product of the number of square,units of area in the base and the number of llnear units in the height.

10-11 . Historical Note - Some of the geometric ideas in Chapter 10 were discovered

by the Egyptians and Babylonians almost 4,000 years ago. For

example, they knew how to find the area of a triangle and used

t h i s knowledge in surveying and measuring fields.

males, mentioned i n Sectj-on 2, Is credited with the

discovery t h a t the measupes of t he base angles of an isosceles

t r i ang le are equa l . There is some evidence that Thales a l so knew that t h e sum of the measures i n degrees of t h e angles in a t r i a n g l e is 180.

There were many o the r famous Greek mathematicians. Their

work made ancient Greece famous as the "Cradle of Knowledge."

We will discuss only a few of these men. Pythagoras (569 ? B.C. - 500 B.C,) organized schools a t Croton In southern I t a l y which

contributed to further progress in the study of geometry. You

will learn about some of the discoveries c red i t ed to him next

year. Euclid (365 ? B.C. - 3OO ? 3.c . ) became famous by writing one of the first geometry textbooks called the Elements. This

textbook has been translated Into many languages. It has been used in teachlng geometry classes for some 2,000 years wi thout

much change. I t s f o m has been somewhat modernized to f i t

present needs. A l l of the propert-ies tire have s tudied in this

chapter may b e found in the Elements.

morn the 7th century until t h e 13th cen tu ry very little

progress was made i n mathematics. From the 13th century , however, the study of geometry and o t h e r mathematics spread r a p i d l y

throughout Europe. Mathematicians began to examine new ways of studying elementary mathematics. You will learn about the work of men such as Rene Descartes (1596 - 1650, ~ r a n c e ) ; Blaise Pascal

(1623 - 1662, prance); P i e r r e Femat (1601 - 1665, ~ r a n c e ) ; Karl Friederich Gauss (1776 - 1855, ~ e r m a n ~ ) ; and others as you continue your studies of mathematics.

A t t h e present time many new mathematical discoveries are being made in many parts of the world. There are s t i l l many important unsolved problems in geometry. As one example, suppose you have a l o t of marbles a l l exactly the shme size. How should

you pack them in a large container in the best possible way,

t h a t is, so as to get as many marblea as you can i n t o the given

volume? Nobody knows f o r sure. We know a very good way to pack the marbles, but no one has ever proved that it is the best

possible way.

Original ly geometry meant a study of earth measure. The I1 word geometry" comes from two Greek words, "geo" meaning earth

and "metry" meaning measure. Through the years geometry has been

thought of as a atudy of such elements as points, lines, planes,

space, surfaces, and solids. These elements are used to describe the shape, s i z e , position, and relations among objects in space.

[aec. 10-111

Chapter 11

CIRCLES

One of t h e most common simple closed curves is the c i r c l e .

Yo matter where you are, it is probable t h a t you w i l l be a b l e t o see one or more objects which suggest a c i r c l e . Can you find some i n your classroom now? Find as many as you can in your home. There may be a t r a f f i c "c i r c l e" near your school. Do you

think a c i r c l e is a good ffgure to use in this way?

In Chapter 4, you learned about simple closed curves, In l a t e r chapters you studled some of the characteristics of several kinds of simple c losed curves, such as parallelograms, rectangles, and t r iangles . I n t h i s chapter , you will study some of the

prope r t i e s of a c i r c l e as a mathematical figure.

You already know that a l i n e or a simple closed curve may

be thought of as a set of p o i n t s . L e t us see how we may describe

a c i r c l e as a s e t of points.

11-1. Circles and the Compass

Choose a point near the center of a sheet of paper. Label

the point P. Then, uaihg your ruler and a pencfl, mark at least

ten points, each at a distance of 3 inches, from P. mat figure do these p o i n t s suggest? If you have located the ten points In widely d i f fe ren t d i r e c t i o n s from P , they should suggest

a c i r c l e .

To draw a compl'ete c i r c l e , use a compass. Some of you are already familiar with the compass as a device for drawing c i r c l e s ,

(An instrument f o r indicating which direction is north is also

called a compass, but we a r e not concerned with that kind of

compass here. ) To draw a c i r c l e with a compass, adjust t h e arms

of the compass so that the distance

between t h e sharp point and the pencil

t i p I s the desired distance, which

in this case is 3 inches. Place the sharp po in t of the compass on the

po in t P, which you located previously on your sheet of paper. Do not move

the sharp point of the compass while making the drawing! P i v o t the compass

about the po in t P y making the penci l t i p draw a curve. Hold the compass

between the thumb and forefinger as

shown in the drawing. When doing this, lean the oompass in the

direction which you are pivoting the penci l t i p . When the pencil t i p r e tu rns to the s t a r t i ng point, you have canplated your

drawing of a c i r c l e . Is a circle a simple cloaed curve? Does your dravlng f i t t h e description of a rlmple closed curve given in Chapter 4? If you located the ten points accurately, your

drawing of the c l r c l e should pass through eaah of the points . In your circle, po in t P is the center of the c i r c l e .

Choose any one of the points you located a t a diatance of 3 inches from P. Label t h i s poin t T. Draw a me-nt Joining P - and T as shown in the figure at the r i g h t , PT ha8 a

measurement of 3 inches. This

c i rc le may be cal led "c i rc le P," meaning a c i r c l e which has as its cen te r the poin t P . Sometimes

we name a simple closed curve by

a l e t t e r , as "c i rc le C." When I I we say c i r c l e c," C is not

just one po in t of the c i r c l e . It represents t h e entire c i r c l e .

- The segment PT is called a radius of the circle. A

; radius is any l i n e segment which Jo ln s the cen te r P to a

point on the c l r c l e . Draw a second radius. Call it 5. -

When we speak of more than one radius, we say "radii ." PT - and PY are radii of the c i r c l e . How many such radii of t h e c i r c l e may there be?

We use the word " rad ius" in another way t o o . It means the distance from the center to any point on the circle. The radius

of the circle you drew is 3 inches. There is just one distance

which is "the radius of a c i r c l e , but "a" radius may be any l i n e segment having the center of the c i r c l e and a point on the c l r c l e as its endnolnts.

Look a t your drawing again.

Choose any point inside the c i r c l e and c a l l it point X. Draw the

4 4 ray FX. On the ray PX, measure a distance 3 Inches from P.

The point you obta in should be on the circle. Call this poin t S . Is PS a radius of the c l r c l e ?

P!e can now describe a c i r c l e as a s e t of points. The c i r c l e ,

with c e n t e r P and r a d i u s r units, is t h e set of a l l points in a plane at a distance r from P.

The compass has a second use. Tt is used for transferring

dis tances . D r a w a llne / and label two points, P and R,

on / . Draw a s e p e n t AB anywhere on your paper. Without - using a ruler to measure the length of AB, we wish t o find a - point Q on PR such that the length of 6 is the length of - AB. A d j u s t the arms of your compass as shown in the drawing

below, so t h a t when you put the sharp poin t a t A, the pencil

t i p is at B. Then, without changlng the opening of the arms,

lift the compass and place the sharp p o l n t at P, Draw a small 4

p a r t of a c i r c l e crossing the ray PR, still without changing

the opening of t h e arms of the compass.

Call the point of intersection Q. Then t h e length of PQ is

the length of a.

Exercises 11-1

In t h e following problems, you wlll have practice in us ing

your compass t o draw clrcles and to t r an s f e r distances.

Read the directions c a r e f u l l y . Label each point, c i r c l e , or l l n e segment in your drawing before going on t o t he

next direc t ion .

1 . (a) Label a poin t P on your paper. Draw a c i r c l e wlth center P and radius 7 cm. Call it c i r c l e C.

( b ) Label Q a p o i n t on c i r c l e C. Draw a circle w i t h

center a t Q and a rad ius 3.5 cm.

( c ) D r a w a c i r c l e with center at P and radius 3 5 cm.

( d ) khat does the intersection of t h e l a s t two c i r c l e s seem to be?

2 . In t h i s problem, you a r e to r e f e r to the drawlng below.

Follow the directions l i s t e d below t he drawing.

( a ) Near the center of a sheet of paper, draw a v e r t i c a l

l i n e about 6 inches in l eng th . Label the line rn.

Label a point A near the lower p a r t of the l i n e .

(b) Use your compass to loca te a poin t B above A on

l i n e m, so tha t AB = PQ. ( ~ o t e that "AB'~, with - no symbo l above it, means "the measure of segment AB .'I )

( c ) Locate po in t E on l i n e m, so t h a t E is above A

and AE = PR.

( d ) Locate p o i n t F above A on l i n e m so that

BF = RS.

( e ) WSth B as center and as radius, draw a c r l c l e . (f) If your drawing is accurate, there will be two labeled

points on the c i r c l e . Name the p o i n t s .

3 . (a) D r a w two intersecting lines which are n o t perpendicular,

and c a l l the lines A1 and ( ~ u t i c e t ha t we have

named both lines with t h e same letter, b u t have written numerals a f t e r the le t ters , and a l i t t l e lower. You

will recall that such numerals are cal led " subscripts. T I

li is read 'l sub-one,! or sometimes just I' 1 one.

' is read " A sub-two.'')

(b ) Call the point of intersection of 1 and 1 poin t

B. 13th B as center and radius 1 inch draw a c i r c l e

( c ) Label t h e intersections of. l i n e el w i t h the c i r c l e R and S, and the intersections of line A2 vith the c i r c l e T and Y.

- - - ( d ) Draw RT, RY, ST, and E. What kind of figure

does RTSY seem to be?

4. Draw a line A and l abe l p o i n t s X and Y about 1 inch

apart on . ( a ) Draw the c l r c l e C1 which has its center at X and

passes through Y.

(b) Draw the c l r c l e C:, which has i t s center at Y and

passes through X.

( c ) Label Z as the other intersection of c i r c l e C2 and

line A . ( d ) Draw the c i r c l e C3

whlch has i t s center a t Z and

passes through X.

( e ) What is c1 C ? 3

f5 . ( a ) D r a w t w o intersecting lines and L2 Label as

B the intersection of L1 and 4 p .

( b ) Label A a point of l1 and C a polnt of A2, so

that the length of BA is not the length of z. 4

( c ) Use your compass to mark a poin t A1 on AB such that + Al is not on BA and AIB = AB.

* + ( d ) Nark a p a i n t C1 on CB, but no t on BC, such t h a t

BC, = BC. - - - -

( e ) Draw AC, AC1, AIC, and AIC1. What kind of f i gu re

does ACAICl appear t o be?

11-2. I n t e r i o r s - and Intersections

A c i r c l e is a simple closed curve. Consequently it has an

i n t e r i o r and an exterior. Suppose we have a c i r c l e with center at P and with radius r units.

A p o i n t such as A inside the

circle is less than r units from

t h e center P, while a point

ou ts l de the c i r c l e , such as B,

is more than r units from P.

Thus It is easy, in the case of the 'B

c i r c l e , t o describe precise ly what

its interior and its exterior are. The interior is the s e t of

a l l points a t distance less than r units from P . The exterior

1s t h e s e t of a l l points at distance greater than r units

from P.

We have f requent ly worked with the notion of the intersection

of t w o sets. A c i r c l e is an example of a set of points.

Consequently we may raise questions about intersections involving c i r c l e s .

L e t us choose a po in t P and a l i n e segment of any length.

Let us draw the c i r c l e with center a t P and radius equal t o the selected segment. With your

compass, make the drawing on a

piece of paper. Your picture

should look like the f igure a t

t h e right.

Next, let us choose any

point Q on the c l r c l e . ( ~ f t e r

we make our choice, the drawing

will look l i k e the flgure at the

right. 1 How many po in t s on t he c i r c l e

_C,

are also on the ray PQ? To

answer this question, you may f i n d

i t easier to draw t h e ray on your

paper. It should look like t h e

drawing at the right.

(In general you should acquire the habit of putting on paper

whatever you need in order t o understand t he ideas you are studying. Notice how we have done t h i s , s t e p by s t ep , In ou r discussion. Follow t h i s suggestion throughout t h e remainder of the chapter. )

With the same situation as In the previous paragpaph, how _3

many po in t s on the c i r c l e are also on the ray QP? i id you -+

draw t he ray QP before you attempted t o answer?) Do you feel

the need to l a b e l one or more points which we have not yet

named? Can you descrlbe carefully (in words) the l oca t ion of the extra po in t which you believe ought to be named?

Now shade l i g h t l y the interior

of the circle as shown. hkat is the

union of t h e c i r c l e and i t s i n t e r i o r ? In Chapter 8, t h e union of a simple

closed curve and its Interior was

ca l l ed a "closed region." The union of the c i r c l e and its i n t e r i o r is a

c i r c u l a r closed region, The set of

a l l po in t s which are either on or inside the curve is the union of t h e c i r c l e and i t s i n t e r i o r .

Another way of thinking about the union of the c i rc l e and its

i n t e r io r is the following:

I The union is the s e t of a l l polnts whose distance from

the center P is either the same as or less than the

radius of the c i r c l e .

I What is the intersection of the c i r c l e and its interior?

I No point of the c i r c l e also lies in the i n t e r i o r of the c i r c l e .

We say that the Intersection of the c i r c l e and i t s i n t e r i o r is

I t h e empty s e t .

Let Y represent the union of the c i r c l e and its i n t e r i o r . C

t What is the intersection of the set Y and the line PQ. In - what way is Y n PQ quite different from t he intersection of - PQ and t h e c i r c l e ?

In the figure at the r l g h t , po in t 4 P is the center of the c i r c l e . The

four po in t s A, F, B, and G

lie on the c i r c l e . The intersection - H of AB and FG consists of the * po in t P, and AB is perpendicular

to w. Follow these directions

in maklng a copy of the diagram on a shee t of paper:

(1) Xlth compass, draw the c l r c l e . Mark the c e n t e r and label

it P.

(2) D r a w a l i n e through P. Label the points of intersection

with t h e c i r c l e A and 3.

( 3 ) Use a p r o t r a c t o r to o b t a i n a right angle , and draw a line w

perpendicular to AB. Label the points of 1ntersect:on

of t h e c i r c l e and t h e line F and G .

(4) Shade the half-plane which contains F and whose boundary M

is AB. (A colored penc i l o r crayon is u s e f u l f o r shading.

The shading should be done l i g h t l y . If you do not have a

colored p e n c i l o r crayon, shade l i g h t l y w i t h your p e n c i l . ) Let us call this p a r t i c u l a r half-plane H. Now answer t h e

f 01 1 owing :

( a ) k h a t is the Intersection of the half-plane, H, and the

c i r c l e ?

(b) Does point A belong In t h i s intersection? Does Q ?

Boes F ? Does P?

( c ) Can you count a l l the points tha t belong t o t h i s

intersection? Explain why, or why n o t .

Choose two points on t he c l r c l e which also belong t o the

i n t e r i o r of t h e angle BPF. Label these p o i n t s M and N .

Similarly, choose one point, which. we will c a l l K, which lies

on t he c i r c l e and a l s o in the I n t e r i o r of angle APF.

(d) hhat is the Intersection of t he c i r c l e and angle MKN?

( e ) t h a t is the i n t e r s e c t i o n of the l n t e r i o r of the c i r c l e and

the i n t e r i o r of the angle MKN?

Exercises 11-2

Let C be a c i r c l e w i t h center a t P and r a d i u s r units.

L e t S be any o the r po in t in the same p l a n e , r re you

drawing the f igu re , step by s t e p , as t h e problem is describing it?)

( a ) How many points belong t o the i n t e r s e c t i o n of the * c i r c l e C and the ray PS? -

(b) How many po in t s belong to the s e t C n PS? ( c ) Do your answers to (a ) and (b) depend upon where

you chose the point S?

(d) How many p o i n t s belong to the set C fl m? Does t h i s

answer depend on t he choice of S? If so, how?

2. In a plane, can there be two c i r c l e s whose intersection

c o n s i s t s of just one point?

3 Choose two points and label them P and Q. D r a w t w o c i r c l e s with cen te r at P such that Q is in t he e x t e r i o r

of one c i r c l e and in the i n t e r i o r of the other. Label the

first c i r c l e C and the second c i r c l e D. A F

4. In t he flgure at the right, let&he half -plane on t h e F-side of AB be

called H. Let J be the half-plane w

on t h e B-side of FG.

(a) What is the s e t H n J? G v

(b) What is the intersection of a l l three of the s e t s , J

and H and the c i r c l e ?

( c ) Does the diagram suggest to you t h a t t h e c irc le has

been separated i n t o what might be ca l l ed quarters? If so, can you describe several of these portions?

( d ) Can you f ind a portion which m i g h t be c a l l e d half of the c i r c l e ? Can you describe it in the language of intersections or unions of s e t s ? Can you Identify

several such por t ions? More than two?

5. Choose two distinct p o i n t s P and Q. D r a w t h e clrcle with - cen te r a t P and w i t h the segment PQ as a radius. Then

draw the c i r c l e with center a t Q and with P on the circle .

( a ) t h a t is the intersection of these two c i r c l e s?

(b ) Can you draw a line which passes through every point of

the intersection of the two circles? Can you draw more

than one such line? Why?

( c ) In your p i c t u r e shade the intersection of the interiors

of t h e two c i r c l e s . (1f you have a colored penc i l

handy, use it fop shading; if you do not, use your

ordinary pencil and shade lightly.)

(d) (1n t h i s p a r t , use a d i f f e r e n t type of shading or, if

you have one handy, use a penc i l of a different c o l o r . ) Shade the intersection of the interior of the c i r c l e whose cen te r is P and t h e exterior of t h e c i r c l e

whose c e n t e r Is Q. (Before doing the shad ing of the

intersection, you may f ind it helpfu l to mark separa te lx

the two s e t s whose i n t e r s e c t i o n is desired . )

( e ) M e another copy of the picture showing the two

c i rc les , and on it shade the union of the interiors of t he two cf rcles .

The two c i rc les shown a t t h e right

lie In the same plane and have the same c e n t e r , P . Ci rc les having the same center are called con-

centric c i r c l e s . (On an auto- mobile wheel, the rim of the wheel

and the c i r c u l a r edge of t h e t i r e

are examples of concentric c i r c l e s . If you throw a stone

i n t o a pool of water, the c i r c u l a r waves formed are a l s o examples of concentr ic circles. )

( a ) Describe t h e in t e r sec t ion of c i r c l e A and circle B.

(b) Give a word description of the shaded region, using

such words as " ln te r sec t ion , " "exterior, " and so on.

[sec . 11-21

Refer t o t h e f igure in Problem 6 . How may the i n t e r s e c t i o n

of the exteriors of the two c i r c l e s be more simply described?

In t he drawing at t ne r i g h t ,

the center of each c i r c l e lies on the o t h e r c i r c l e . Copy the f i g u r e on your paper, Shade the union of the

e x t e r i o r s o f t h e tv:o c i r c l e s . - Choose two distinct points P and Q. Draw the l i n e Pg. D r a w the c i r c l e with center a t P and with Q on the

c i r c l e . Draw the c i r c l e w i t h center at Q and- with

as a r a d i u s . D r a w a l i n e whlch passes through each po in t

of intersection of the two c i r c l e s , and c a l l the line . By inspecting t h e diagram, make an observation about a - relationship between t b - e l i n e and t h e l i n e PQ. Use your p r o t r a c t o ~ to check your observation.

I 11-3. Diameters and Tan~ents Lie have seen that the word " rad ius" can be used Ln t ~ o

d i f f e r e n t ways. By way o f revlew, a radius of a c i r c l e is one of the segments joining a p o i n t of the c i r c l e and the c e n t e r . The

length of one of these segments is t h e radius of the c i r c l e .

I The word diameter i s c l o s e l y associated with t h e word radius.

/ A diameter of a c i r c l e is a line segment which contains t h e center ! of the c i r c l e and whose endpoints l i e on the c i r c l e . For the

c i r c l e represented by the figure a t the - r i g h t , three diameters are shown; AB, - - , and ! ( A diameter of a c i r c l e Is

the longest l i n e segment that can be drawn

In the interior of a c i r c l e such that its

end-points are on the c i r c l e . ) How many

radii are shown in t h e f igure?

A set of p o i n t s which Is a diameter may be described in another way. A diameter of a c i r c l e is the union of two di f ferent radii which are segments of the same line. How does

the length of a diameter compare with the length of a radius?

The length of any diameter of a c l r c l e is also spoken of as the diameter of the c i r c l e . The diameter is a distance, and the radius is a d i s t ance .

The measure of the diameter is how many times the measure of the radius? If we choose any unit of length, and if we let r and d be the measures of the radius and the diameter of a

c i r c l e respect ive ly , then we have this important re la t ionship:

Irhat replacement for the question mark makes the following

number sentence a true statement?

The line and the circle In the

f i g u r e on the right remind us of a

t r a i n wheel resting on a track, except t h a t the flange ( o r l i p ) which guides

t h e t r a i n i s not shown. How many

points are on the circle and also >

on the line? There is only one D T E 'x . -

such polnt , the one labe led T. We say that t h e l i n e is tangent t o the c i r c l e . The single poin t

of their intersection is the point of tangency. In thfs drawing, T is the point of tangency. Another way of descrfbing a point

of tangency I s t o say t h a t It i s the only pain t of the c i r c l e

which is also on the line. Now answer the following ques t ions : - (1 ) In the figure, the l i n e DE separates t h e plane DPX i n to

two ha l f -planes. How can you describe the intersection of

t h e half-plane conta in ing X and the c i r c l e ?

( 2 ) What is the i n t e r s e c t i o n of the half-plane contalning P

and the i n t e r i o r of t h e circle?

that their intersection is the empty s e t ? Explain.

( 4 ) Can you draw a c i r c l e and a line such that their

i n t e r s e c t i o n contains exactly four po in t s? Explain.

In t h e drawing at the right, how many lines

are shown?

Hobr many of these lines are not tangent to the

c i r c l e ?

Name each point of

tangency . 6 . Suppose that we are given a c i r c l e and a point on the c i r c l e .

(a) How many r a d f i of the c i r c l e conta in the given poin t?

(b) How many l i n e s , tangent t o the c i r c l e , contain the given point?

Exercises 11-3.

1. How many tangents do you f i n d in each of the following?

2. Look for examples which represent the idea of a c i r c l e and a line tangent to the c i r c l e , t h a t is, of a l i n e and a

c i r c l e whose intersection consists of a single p o l n t . Describe these examples.

3 , The diameters of c e r t a i n c i r c l e s are l i s t e d . For each c i r c l e , find t h e distance from the center of the c i r c l e to any p o l n t on the c i r c l e .

(a) 42 cm. (a) 4 yd.

(b) 28 in. ( e ) 30 ft.

( c ) l o ft.

4 Find t h e diameters f o r each of the c i r c l e s where t h e

d i s tance from the center of the circle to a point on the c i r c l e is as fol lows:

(a) 6 in. ( d l 5 ft.

( c ) 17 cm.

5 . Draw a c i r c l e C with center at the point P , Draw three diameters of C. Draw a clrcle with center at P vrhose

radius I s equal t o the diameter of C.

6. (Warning: this problem requ i re s very ca re fu l handling of

t h e compass .)

( a ) Near the middle of a sheet of paper, mark a point Q.

(b) D r a w a circle C with center at Q, and radlus approximately 2 inches. Keep the opening of the compass arms fixed at the radius selected until the

drawing is completed.

( c ) Mark a poin t U on the c i rc l e C.

( d ) With the compass, find a poant V on the c i r c l e C - such tha t W has the same length as the radius QU.

( e ) Again, with the compass, f ind a third polnt W on C - such that has the same length as QU.

[aec. 11-31

(f) Continue on around, locating p o i n t s X, Y , 2 on C

(use compass three times) such tha t the length of each - - of the segments WX, XY, is the same as the radius of the c i r c l e .

- (g) Now compare the length of the segment ZU with the

radius of the c i r c l e .

If you had a fine q u a l i t y compass and if you were able

to per fom the construction w i t h great care, the simple closed curve UVWXYZ would represent a hexagon. (1n such a figure, if the sides are congruent and the angles are congruent, the figure is called a regular hexagon,

Your drawing should look like a regular hexagon. Note

that the segments which are sides of your hexagon are a

l l t t l e shorter than one-sixth of the c i rc le , that l a ,

the arc whfch has endpoints the same as a sement .] -

(h) What name might you give to each of the segments UX, - VY, and E?

In the diagram the poin t Q I s t h e center of t he c i r c l e and

the center of the square EFGH.

( a ) hbat I s the intersection

of the c i r c l e and the square ?

(b) What is the intersection of the line and the c i r c l e ?

(c ) What new name have we given to the point T?

( d ) How many Lines are tangent to the c i r c l e ?

( e ) Name a l l the points of tangency.

8. On your paper make a sketch of the dlagram for Problem 7 . ( A careful drawing I s not needed here. ) On your copy, draw the q u a d r i l a t e r a l RSTU . (a) How many sides of t h i s quadrilateral are segments of

llnes tangent t o the c i rc le?

( b ) What is the intersection of the i n t e r i o r of the c i r c l e and t h e exterior of the square EFGH?

( c ) Describe carefully the intersection of the e x t e r i o r of t he c i r c l e and the I n t e r l o r of the square EFGH.

(d) Irhat is the Intersection of the i n t e r i o r of the c i r c l e and the exterior of the quadrilateral RSTU? Explain

your answer by shading the correc t region on the f igure .

* 9 . Refer again to the care fu l ly drawn f igure in Problem 7 .

( a ) Do you think t h a t a l l t h r ee of the p o i n t s R , Q, T

lie on one l i n e ?

(b) Do the three p o i n t s U, Q, S appear t o l i e on one l i n e ?

( c ) Make an estimate of t h e size of t h e angle QTG.

Check your estimate with a protractor.

( d ) Make an observation about a relationship between the - - line QS and the line FG. Check your observation

with a protractor.

*lo. D r a w any c i r c l e and any l i n e tangent to the c i r c l e . maw

also the line which joins the center of t h e c i r c l e and t h e

poin t of tangency.

(a) Do you believe there is an important r e l a t i o n s h i p

between these two l ines? If so , what is it?

(b) How many radii of a c i r c l e may be drawn having one

point of tangency as an endpoint?

481

*lI. We know from the descrLption of a c i r c l e that all r a d i i

a re congruent. How can we use this f ac t to show t h a t by ou r d e f i n i t i o n of a diameter a l l diameters of a par t i cu la r c i rc le are congruent?

11-4. Arcs

In Chapter 4 we learned t h a t a s ing le point on a line

separates the line Into two ha l f - l ines . Using this idea of separation, we say that on a line, a s i n g l e point determines

two half - l i n e s . Recall t ha t if

A , B, and Q are points on a A Q 3 - - line, as shown in the ffgure at - - > the r i g h t , Q is considered

to be "between" A and B. On a line, the i d e a of "betweenness"

and the idea of "separat ioni i are closely t i e d toge ther .

Does a single point on a circle

separate t h e whole c i r c l e i n t o two par t s? D o e s point Q separate the

c i r c l e a t the r i g h t i n t o two parts?

If we start at Q and move in a

clockwise d i r e c t i o n we w i l l , in

(-J due t i m e , r e t u r n to Q. The same

I s t r u e i f we move in a counte r -

clockwise d i r e c t i o n . A single point

does not separate a c i r c l e i n t o two

I

In the figure at the r i g h t , the

two points, X and Y, separate the

c i r c l e into two parts. One of t he

I

A p a r t s contains t he po in t A . The

0 other p a r t contains B. No path from

X to Y along the c i r c l e can avoid I I at least one of the points, A and 3.

Thus, we see It takes two d i f f e r e n t

points t o separate a c i r c l e i n t o two d i s t i n c t par ts .

Refer t o the c i r c l e a t the r i g h t

showing points A , B, and Q on t he

c i r c l e . Is point A between B and Q? Is point Q between A and B?

Is po in t B between A and Q? Since 0

we can move i n e i t h e r a clockwise or counter-clockwise direction on t h e

circle, the answers t o these th ree

questions are yes. Unlike a l i n e where betweenness and separation

are c lose ly related, on a c i r c l e , or other s imple c losed curve

we can observe these t w o ideas:

( I ) A single p o i n t does no t separate the curve I n t o two p a r t s .

( 2 ) Separat ion and betweenness are not c l o s e l y r e l a t ed notions

f o r simple closed curves.

In earlier discuss ions of geometry we refer red to parts of

lines as line segnents. It w i l l be necessary for us to consider

parts of c i r c l e s . A p a r t of a c i r c l e is ca l l ed an arc. In the

drawing at the right, points A and 3 separate t h e c i r c l e i n t o two parts. Each of these p a r t s , together wi th the

points A and B, is an arc . A and

B are the endpoints of the arc. Any

two d i s t i n c t points on a c i r c l e determine two d i f f e ~ e n t arcs havlng these two points

OB as endpoints. In t h e drawing above, the arc, s t a r t i n g at point

A and moving clockwise to point B is shorter than the arc starting at p o i n t A and moving counter-clockwise t o poin t B.

K i t h only two points on a c i r c l e , such as A and 3, we

cannot easily i d e n t i f y one of t h e two arcs determined by A and

B . In the drawing a t the rl ght , we have marked and labeled a

point between each of the two

end-points on each of the arcs.

These t w o points are conveniently N

located somewhere near the middle of each of the a r c s . We use the

symbol 'm' to represent the

word ' 'arcT' . Thus, represents

the arc containing point M. represents t h e arc which n contains N. In place of we can use BMA . What o ther

f symbol represents t h e same arc as ANB? What point is contained In the arc

Exercises 11-4a.

1. Using the drawing on t h e A

right, identify t he shortest

p.rc conta in ing t h e following

p o i n t s , where t h e po in ts a re R

not endpoints of the a r c .

C OB W

(a) A ( d l R

2 . Using the drawing f o r Problem 1, name t h e po in t or points which are not endpoints, of each of t h e following a r c s .

n (b) ABC

(d) 5 2 ( e ) G 2

3. Mat are the endpoints for these arcs from the drawing in

Problem (I)?

( c ) G3 (f) n BAC

4 . In answering the previous question, was it necessary to see

t h e drawing? ExplaLn . 5 . Use the drawing a t the r i g h t

in answering the following:

(a) On the c i r c l e , I s there one specific point between

c and D? Explain. .( 6 (b) Is there one s p e c i f i c

point between C and D

on a If so, name t he

p o i n t .

( c ) Point L separates a i n t o two arca . Name these

two a rcs .

( d ) Does point L separate t h e c i r c l e i n t o two arcs? Explain why, or why not.

6 . Use your answers in Problem 5, above, to answer the

following:

( a ) Does a point on an arc separate the arc Into two arcs?

(b) On an arc, must a poin t , which is not an endpoint of

the a rc , be between two points of the arc?

(c ) Does an arc have a "s ta r t ing t t point and a "stopping"

point? If so, what are they called.

(d) For an arc a r e the no t ions of betweenness and separat ion

more like those of a line segment or of a c i r c l e ?

In the figure at the right

t h e pair of points A and B separate t h e points X

and Y. kihich po in t s , if

any, do t h e pair of points

listed below separate?

In the figure shown at the n

right, AMB is associated

with the i n t e r i o r of L APB. - --+ FG connects a point of PA

+ w i t h a point of PB.

(a) Are points A and ' F C_*

both on PA?

(b) Does point F on FQ correspond to point A

on G3 ( c ) FOP each ray from P and passlng through Is -

t he re a point on FG which is also on the ray?

( d ) Is there a one-to-one correspondence between t h e po in ts

on and the &nts on E? If s o , describe the

correspondence .

11-5. Central Anales Arcs have some properties similar to

segment. fn the figure shown at

the r i g h t , there is a natura l one-to-one correspondence between the s e t of p o i n t s of && and the

set of points of . A point may

separate a segment i n t o two par ts . Similarly, an arc may be separated

the properties of

into two arcs by a poin t which is

on the arc, bu t is not an endpoint of the arc. As with a H segment, an arc has a "starting" point and a stopping" polnt,

the two endpoints. Although arcs are parts of circles, an arc has some properties which are not the same as the properties

of a circle.

In the figure shown at the r i gh t , 3 is a diameter of circle P. Two arca are determined by the endpoints A and B. Such

arcs are very special arcs and are given a spacial name. They are

69 0

called semicircles. A semicircle

ie an arc determined by the endpoints of a diameter of the circle. n

AVB I s a semicircle in the figure shown. Can you name another semlolrcle in thls figure? Ia B?A a asrniclrcle?

The endpoints of a semicircle and the center of the circle are on a straight l ine . This, however, is n o t t rue f o r all arca. In the drawing at the right, the

n endpoints of DXE are not on a straight line passlng through the center of the clrcle. D and E are on the rays 8 and z. Angle EPD has its vertex at the center of the c i rc le . We c a l l

gL such angles central angles. A central angle is an angle

[aec. 11-51

having its ver tex a t che c e n t e r of a circle. Such angles are measured i n the same way as other angles. The unit of angle

measure is an angle of one degree. In the figure, the measure of angle EPD is about 85,

In working with arcs, we find it necessary t o compare one arc with another . Therefore, we will find it convenient to devise a method f o r measuring arcs. We th lnk of a circle divided i n t o 360 c o n g r u e n t a r c s , t h a t l s , each o f t h e 360 sucharcsin

a p a r t i c u l a r c i r c l e has the same measure. Each such arc

determines a unit of arc measure. We call t h i s unlt one degree - of arc . --

From the c e n t e r of the circle, rays which pass through the

endpoints of such an arc determine a central angle. We th ink of a degree of arc as being determined by a central angle which i s a

unit angle of one degree, I n t h e

figure a t the r igh t , i f the measure n

of AMB in degrees of arc i s 80, then the measure of L APB in angle

degrees is 80. The symbol f o r a degree of arc, is the same as

that f o r the angle degree. For measuring degrees of arc , we can use

an instrument w i t h a c i rcu la r shape contalnlng 360 congruent

arcs. Most such instruments, cal led protractors, show just half

of this scale.

On the set of numbered rays of the angle sca le , the numbers

0 and 180 correspond to opposite rays, that is, rays which lie on the same l i n e and have the same endpoint. The union of these two rays forms a l i n e whose intersection w i t h the c i r c l e is the p a i r of endpoints of a diameter. These two points determine

a speclal arc, a semicircle,, which we have mentioned earlier in

the chapter. Thinking of this angle scale we can think of the

semicircle having an arc degree measurement of 180' since it

would consist of 180 one degree arcs. Corresponding to the

special kind of arc which we ca l l a semicircle 1s a specla1 kind of cen t ra l angle w i t h a measurement of 180'. Some people f i n d it convenient t o speak of the central angle o f 180' as a

"s t ra ight angle." (Why does "s t ra ight angle1' not agree w i t h our definition of angle?)

In the f i g u r e a t the right are G

two c i r c l e s hav5ng a common center , P.

The circles are In the same plane . Such

circles we have ca l l ed concent r ic circles. n

The two arcs , ARB and 533 have the

same c e n t r a l angle, L GPH. Therefore, n n ARB and ESD must have the same a r c measure. If the angle measure of \H

L BPA is 70, then t h e arc measure n

of ARB is 7 0 . The arc measure . - n of must also be 70. However, ARB appears s h o r t e r than A %sC. Remember t h a t a r c measure is not a measure of length. Two

arcs may have the same a rc degree measure bu t have d i f f e r e n t l e n g t h s . The reason will be more apparent a f t e r you have s t u d i e d

the remainder of this chap te r .

Exercises 11-5

In the f i gu re at t h e right

determine with the use of your

p r o t r a c t o r the measure of the

fo l lowing arcs . Indica te your

~ e s u l t s w i t h c o r r e c t use of n

synbols, f o r example ~ ( A B ) = 15.

G2 P ( a ) I d ) BCD n

(b) AEcD ( e ) G 2

48 9

Constmct a c i r c l e with radius approximately 1 lF Inches. In this exercise mark off the po in t s In a counter-clockwise

path around t h e circle a f t e r starting anywhere on the circle

with A . Mark off and l abe l arcs with the following

measures :

( a ) m ( a ) = 10 ( d ) m(%) = 170

( e ) What is rn(6?)?

(a) Bow many arc degrees are in a quarter of a c i r c l e ?

(b) How many arc degrees are in one-eighth of a c i rc le?

( c ) How many arc degrees are in one-sixth of a c i r c l e ?

( d ) Row many arc degrees are in three-fourths of a circle?

4. Draw a c i r c l e wlth a radius of 2 inches. S t a r t i n g at any

poin t on the circle use a compass with the same settlng

(2 inches) to describe a series of equally spaced marks around t h e c i r c l e . CAUTION: Be ca re fu l that your compass

setting does not change.

(a) Do you get back to exactly the same point where you

started?

(b) How many arcs are marked off?

( c ) h i a t is the measure of the cent ra l angle of any of these arcs?

( d ) Faat is the degree measure of any one of these arcs?

490 0---

5. Refer t o the arc , or more b r i e f l y AF, shown in

t h e drawing below. Determine t h e fol lowing:

In the figure at the r i g h t ,

c i r c l e C and c i r c l e D are concentric and are in the same plane.

( a ) Name a diameter of

c i r c l e D.

n n (d) C D n D E

(e l 53nG

(b) Name a diameter of circle C.

( c ) khich c i r c l e has the longer diameter?

( d ) mich c i r c l e seems to be t h e longer? h he length of a c i r c l e i s cal led t h e circumference .)

(e ) Why would it be difficult to measure accurately the

length of the c i r c l e s (the circumference) by following a path around t he c i r c l e s?

An experiment to find the relationship between the diameter

and the l eng th of a c i r c l e .

(a ) Select t h r e e c i r c u l a r objects in your home, such as a

water g l a s s , a baking pan, a saucer, or a wheel from a

t o y . (Or you may cut out a c i r c u l a r piece of s t i f f

paper or cardboard.) Use a tape measure (of c l o t h o r f l e x i b l e steel tape) to find the circumference of each c i r c l e . If you do not have a tape measure, use a piece of s t r i n g , and then measure t h e length marked off on the s t r i n g .

i (b) Measure the diameters of t he same three c i r c u l a r o b j e c t s i i Since it may be d i f f i c u l t to locate the exact cen t e r of

t h e c i r c l e , measure across the c i r c l e several times to I

! obtain as good a measure of the diameter as possible. I The longest of such measurements is the diameter.

( c ) Arrange your r e s u l t s in a table as shown below. Compare

the t w o measures f o r each object. To compare two q u a n t i t i e s , we can f i n d their difference or t h e i r r a t i o . In the t a b l e , the " c - dl1 represents the difference between the measures of the circuml?erence and t h e

diameter. The " $ l1 represents t h e r a t i o af the

measures of t h e circumference and the diameter. (Bind

the ratios t o the nearest tenth.)

( d ) Do the differences in the "c - dt' column appear to be

the same?

Name of

object

water glass

p i e pan saucer

( e ) Do the ratios in the l1 2 " column appear t o be t h e

same?

[sec. 11-51

Measure of

circumference

Measure of diameter

c - d - c d

+8. Circle A has a radius of 5 inches. Circle B has a

radius of 25 inches. Explain why the a rc measure of one- f o u r t h of circle A is the same as the arc measure of one- f o u r t h of c i r c l e B.

" 9 . Demonstrate a one-to-one correspondence between the sets of

points on the two semicircles of a given circle which are determined by a diameter.

11-6. Circumference Circles

It i s difficult t o measure the circumference of a circle accura te ly . Remember t h a t the circumference of a circle I s the

length of the simple closed curve which we call a c i r c l e . From the results in Problem 7 in Exercises 11-5, you learned that

there seemed to be a relationshlp between the diameter of a c i r c l e and the circumference of t h a t c i r c l e .

11 The difference, c - d" , for one circle does not give us

a re la t ion that seems t o be true for other circles. For all circles, however, the ratio of the measures of the circumference

11 C tt a n d t h e d i a m e t e r , 3 appears t o b e t h e same. What resu l t s did you get for the r a t i o of c to d? Was

the number i n each case about the same? Has this number a l i t t l e greater than 3? If your experiment was carefully done, your results for the ratio, 8, shouldhave beenabout 1 or 3 . 2 .

It appears that the circumference of any c i r c l e is a l i t t l e more

than three times the diameter of the circle.

Mathematicians have proved that, for any circle, the ratio of the measure of the circumference of the circle to the measure of the diameter i s always the same numbeP. A special symbol i s used for t h i s number. This symbol I s written l' a" . The symbol

i s a l e t t e r from the Week alphabet. It is read "pi ." .rr i s

the first l e t t e r i n the Greek word for 'Iperimeter ." Are the

words "perimeter1' and "circumf erencet' related in meaning?

In mathematical language, we say that the r e l a t i o n of t he

measure c of the circumference of a c i r c l e to the measure d of i t s diameter is:

C - = d Or' c = f d .

The value of a is about 1 or *. use this

number fn f ind ing the circumference of a c irc l e . It is much easier to measure the diameter of a c i r c l e than

to measure the circumference. The circumference can be found by

using the measure of the diameter and the r e l a t i o n stated above. Assume the diameter of a circle i a 5 inches. Then, the length

of the circle may h e found i n t h e following wag:

(using IT w 3.14) (using 3;)

Exercises 11-6a

1 . Find the missing information about the circles described.

(use T = 3.14).

( a 1 (b 1 (c > ( d )

( e )

Giameter

?

15 ft,

?

?

5.5 ;n.

Circumference

?

?

?

?

?

Clrc le

A

B

c

D

E

Radius

LO in.

?

3.6 y d .

4.2 cm.

?

2. (a) Ghat number is obtained by dividing the measure of the circumference of a c i rc l e by the measure of the diameter of the c i r c l e?

(b) How can you determine the circumference of a clrcle if you know only the diameter?

( c ) Can you determine the diameter of a c i r c l e knowing only the circumference? Explain.

3 Copy and complete the following number sentences which show

the r e l a t i o n between the circumference of a c i rc le and the diameter of the c i r c l e .

? ( a ) T = - (b) ? = 8 ? ?

? ( c ) - = ? T

4. Find t h e missing Information about the circle described.

(use T - 3+) .

5 (a) How can you determine the measure of the radius of a

c i r c l e knowlng only the measure of the diameter?

(b) How can you determine the measure of the diameter of a

c i r c l e knowing only the measure of t he radius of the

c i r c l e ?

(a>

( b 1 ( c )

Id)

(4

( c ) How can you determine the measure of the circumference of a c i r c l e by uslng the measure of the radius of the circle?

C i r c l e

v W

x

Y

z

Circumference

22 ft.

25 in*-

16 cm.

43 yd.

88 mm.

Radius

?

?

?

?

?

Diameter

?

?

?

?

?

6 , Copy and complete t h e fol lowing number sentences which

show the r e l a t i o n between the circumference of a c i r c l e and

the radius o f . the circle.

( a ) c = a(? - r) C ( d l = r

7 . Find the missing information about the c i r c l e s described.

(use T w 3 .I).

For Problems 8 through 11, use T 3.14.

I a 1 (b

(C > ( d l

8. A c i r c u l a r lampshade 12 in. in diameter needs new binding

around t h e bottom. How l ong a s t r i p of binding will be

needed? isre regard any needed for overlap. )

9 . A s t r i p of metal 62 inches long is to be made i n t o a

c i r c u l a r hoop. What will be its diameter? Vould it be

large enough f o r t h e hoop f o r a basketball basket?

(of f ic ia l diameter is 18 inches.)

Circ l e

K

L

M

N

10. A merry-go-round in a playground has a 15-foot radius.

If you s l t on the edge, how far do you ride when I t turns once.

11. A wheel moves a distance of 12 feet along a track when the wheel turns once. What is the diameter of the wheel?

Radius

5 in,

?

17 cm.

?

Circumference

?

51 ft.

?

100 yd .

*12. A c i r c l e w i t h a diameter of 20 inches i s separated by points Into 8 a r c s of equal length.

(a) \:hat is t h e l e n g t h of the whole c i r c l e ?

(b) Irdbat is the length of each arc?

( c ) \ ;hat Is the arc measure of each arc?

( d ) On this c i rc le , how long is an arc of one degree?

OPTIONAL: - The Number T

The number represented by the symbol "a I ' is a new kind of

number. It is not a whole number. Neither is It a r a t iona l

number. Recall t h a t any decimal expansion of a rational number

is a repeating decimal. Mathematicians have proved t h a t

cannot be a repeating d e c i m a l . In an artlcle by F. Genuys In

Chiffres I (1958)~ a decimal expression for r to 10,000

decimal places was published.

Here is the decimal f o r T t o f i f t y - f i v e places.

3 .I 4159 26534 89793 23846 26433 83279 50288 41971 69399 36510 58209. . .

h he three do t s at the end ind ica te tha t the decimal

expression continues indefinitely . ) If you examine this decimal for the number u you see that

it is d i f f i c u l t t o locate the point on t h e number line to which

it corresponds. However, by examining the d i g i t s in order we can

find smaller and smaller s e p e n t s on the number l i n e which

con t a in t h e po in t f o r ?r . Study the following statements:

1. ~ = 3 + .lhl592... Therefore a ; 3, and + < 4 . thy? ( 3 < a < 4 . )

2. s = 3.1 + .041592.,. Therefore T > 3.1, and 7F < 3.2. (vhy?) (7.1 < 7~ 3.2)

3 . P = 3 . 1 4 + .001592... Therefore T > 1 and a < 3.15. (bhy?) (3 .14 < T < 3.15) Lhat should t he next statement be?

In the ri,r;ure below is shov:n a number l l n e . A c i r c l e r r i t h

d i a a e t e r one z r L t in Length is shown in its f i r s t posltion at

p o i n t 0 , 2nd a l s o Ln its position a f t e ? rolling a l o n g t h e

number l i n e . The z~proximate position of the p o i n t t :h ich

corresponds t o thc nw13er a is the po in t of 'xangcnc:~, C .

4

L i n e I Lr, t h e f i c u r e i l l u s t r a t e s Statement 1. The p o i n t

c o r ~ e s p o n d h n t o rr is on the segnent 1~1th endpoints 3 3nd k ,

The segnent on line I. ~71th endpoints ; an6 4 is shorifn

er,larr;ed ( t e n : i~ lcs as l a r g e ) on line 11. The segnent I s

s u b d i v i d e d t o show t e n t h s , so the points of division correspond

t o the numbers 3 -1, 3 2 3 3 , e t c . Statement 2 t e l l s us

t h a t t h e p o i n t for a is on the segment w i t h endpoints 3.1

and 3.2. The segment w i t h endpoints 3.1 and '3.2 is shown enlarged

( t e n times as l a r g e ) on l i n e 111. The points on line 111 subd iv ide it i n t o t e n t h s , and correspond to the numbers 3.11,

3.12, e t c . From Statement 3 we know t h a t the p o i n t corre- sponding t o a is on the sement with endpoints 3.14 and

3.15.

What numbers should be at t h e endpoints of the segment

marked IV? How should t h e p o i n t s of subdivision be labeled?

On what segnent af l i n e IV is the point f o r a ?

Look again a t line 111. Notice that t h e whole segment on

l i n e 111 was made 100 times as large as it would ac tua l ly be on t h e nunSer line I, on which the c i r c l e was rolled. Using

just three digits in the decimal for T we have shown that the

p o i n t .rr lies on a p a r t i c u l a r segment which is very small--

just 1 of t h e s e p e n t of the number llne between the po in t s

3 and 4 . If you used one more d i g i t , you could show that the endpointa

of a segment which contains a are ? and ? .

Exercises 11-6b.

1 . ( a ) Find , to three decimal places, the di f ference between

a and each of the following rational numbers:

( b ) t<hich of t h e r a t i o n a l numbers In Prgblem 2 is nearest T ? Using .a = - 22 find the following:

7 ' ( c ) The length of a c i r c l e whose diameter is 14 In.

(d) The length of a c i r c l e wlth r a d i u s 21 ft . ( e ) The diameter of a c i r c l e with circumference

132 in.

( f ) The radius of a circle with circumference 44 ft.

(g) The circumference of a c l r c l e with r ad ius 1% in.

2 . (a) Find, to four decimal places, the decimal for 2 ~ .

(b) Find, to f i v e decimal places, the decimal f o r 3a.

3 . Sometimes it is a good idea to use a as a numeral, ins tead of u s ing a decimal f o r ?r , Answer t h e following

questions us ing a as a n u m e r a l . lle say the answer is

expressed "in terms of 'Ir " . ( a ) If the leng th of a c i r c l e is 54 T in., what is i t s

diameter? I t s r a d i u s ?

(b) If the diameter of a circle is 13 in., what is the

length of the c i r c l e ?

( c ) If t h e radius of a c i r c l e is 3 . 6 em., what is the

l eng th of the circle?

4 . Suppose that the diameter of c i r c l e C is three times as

long as the diameter of c i r c l e D. What is the ratio of

measures of their circumferences? (~int: Think of lengths f o r t h e diameters, and then f i n d the circumferences. U s e

B as a numeral.)

5. Inthe f i g u r e , c i r c l e C and

c i r c l e D have the same center

P . The r ad ius of c i r c l e C is

7 in. and the radius of c i r c l e D is 5 in.

( a ) Find the l ength of each c i r c l e .

(b) If angle QRP contains 70 degrees, what Is the A

arc measure of arc ST? Of arc

( c ) Wat fractional p a r t of c i r c l e C is arc khat

f r a c t i o n a l p a r t of c i r c l e D is n n

( d ) What is t h e linear measure of arc &R? Of a r c ST?

6 . BRAINBUSTER. Suppose a t i gh t band is placed around the

earth, over t h e equa tor . Then suppose the band is made

1 foot longer which loosens the band, and t h a t the band

is the same d is tance from the earth a l l the way around.

( a ) Could a mouse crawl under the band?

(b) Could you crawl under the band? If not, how much

longer would the band have to be t o make it possible

f o r you t o crawl under it?

( c ) What mathematical property could you use to answer

these questions?

11-7. Area of a Circle

In t h e kitchen at home you may find a circular f r y i n g pan

called a nine-inch skillet. The boundary of the frying sur face

represents a c i r c l e , and "nine-inch" t e l l s the dfameter of the

c i r c l e . Some skillets are square shaped. Perhaps you can f ind

an e igh t - i nch square h y i n g pan too. A person trying to decide whether t o buy a nine- inch c i r c u l a r pan or an e igh t - inch square pan 9ight ask himse l f , "khich one I s bigger?" By "biggerr" we

mean more surface usable in cooklng. In other words, one m i g h t

want to compare the area of t h e closed region of an eight-inch

square with the area of the closed region of a nine-Inch c i r c l e . After a c a r e f u l inspection of the two skillets, you might

conclude tha t t h e areas are so nearly t h e same that you cannot

decide which I s greater. Let us try to find o u t whlch s k i l l e t

has t h e g rea te r sur face for cooking.

When speaking about a c i r c l e In everyday language, we

usual ly use t he phrase " the area of a c i r c l e " when we mean "the II area of the closed c i r c u l a r region. After we learn how to

compute the area of a clrcle, we can express the area of a c i r c l e in terms of its radius. This is f r equen t ly done when t a l k i n g

about a c i r c u l a r reglon .

Since each s i d e of t h e square is eight Inches long, we know i t s area is sixty-four square inches. But, we have not y e t

studied a method of computing the area of a c i r c l e . Fe can measure the diameter. This measure can be used to compute t he

radius o r the circumference. But it is difficult t o measure t h e

area of a c i r c l e . In the f igu re , t he po in t P 1s

t h e center of t h e c i r c l e and a l s o the A

center of the square ABEF. Let the

measure of a radius of the c i r c l e be - r . Then, each of the segments VP X Fm z

- and PZ is a radius with a measure r.

Angle VPZ is a right angle. Does E T B

t h e square ABEF have fou r t i m e s as I

much area as t h e square VAZP?

Now answer the following questions: ( ~ o t e that in the drawing t h e c i r c l e has a diameter which is equal in l ength t o

an edge of the square.)

1. ( a ) Is the area of the circle greater or less than the area of square ABEF?

(b) Is the area of the c i r c l e greater than the area of

square VAZP?

( c ) Is the area of the c i r c l e greater o r l e s s than f o u r times t he area of t h e square VAZP?

( d ) Does this tell us the area of the c i r c l e ?

Let u s t r y approximating t h e area of a c i r c l e by a rather ca re fu l measurement. On the next page, at the top, is shown a

small square. Each of the sides of t he amall square is one unit

long. The small square represents one u n l t of area. The large figure shows a c i rc le whose ~ a d i u s is ten units in

length . The large square, whose sides are tangent to the c i r c l e , is twenty units long. The region enclosed by the c i r c l e has been

covered with units of area. Use t h i s drawing to f ind the

approximate area of t h e c i r c l e . What would be an efficient way to count the number of

square units in the c i r c u l a r region? We could count the number of such units in the quarter-circle bounded by the square ABCD.

Then we could multiply this number by 4. Note, however, that

some of the units of area are p a r t i a l l y in the exterior of the circle and p a r t i a l l y in the i n t e r l o r . We must estimate the total number of square units. This can be done by counting those units

which are more than ha l f in the Interior as a whole unit. Then

we disregard t h e units f o r which the smaller p a r t is in the i n t e r i o r .

There a re about 79 units of area in the quarter-circle.

The number of unfts of area in the ent i re c i rc l e is therefore about 4 times 79, o r 316.

What is the area of t h e square ABCD? The area of AI3CD

is 100 square units. Then the area of 3 E F G is 400 square

units. Now answer the following questions:

11. (a) How does t h e area of the c i r c l e compare with the area of 3 E F G ?

(b) Is t h e area of the circle a little more than three times the apes of ABCD?

( c ) How can you quickly determine the area of the square ABCD ?

A s we have seen from the answers to t h e above questions,

t he area of BEFG is greater than the area of the circle. Since

the area of BEFG is four times the area of ABCD, then, f o u r

times t e n times t e n ( 4 - 10 10) r e s u l t s in a product greater t han t h e a rea oi' the c i r c l e . But, the area of' the c l r c l e - is a

little more than three t i m e s ten times t e n ( 3 10 ' 10).

Mathematicians have proved that f o r any c i r c l e , the area of the c i r c l e is a little more than three times t h e measure of the

radius multiplied by itself. he measure of the radius of the

c i r c l e in the drawing was 10. This was t h e same as the measure

of t h e l eng th of the square ABCD.) The area of a c i r c l e is equal to the product of w and the square of the radius. In

mathematical language, we say,

2 A = a r ,

where A is the number of units of area and r is the measure

of radius.

Let us return to the comparison of the two skillets. A 9 nine-inch circular skillet has a radius of inches

(OF in.). The area of t h e skillet may be found as fallows:

NOW we can answer the question, ''khich has the greater surface

for cooking, a nine-inch circular s k i l l e t o r an eight- inch

square s k i l l e t ? " t h a t is your answer?

Exercises 11-7.

1. Find the area of each clrcle f o r which the radius is given.

(use n = 3;)

( a ) 7 in. ( d ) 21 yd.

(b) 5 f t . (4 3.5 r"!!.

( c ) 1 4 cm, ( f ) 4.2 yd.

2. Find the area of each circle f o r which the radius is given .

(use r 3.14)

(a ) 8 ft. ( a ) 20 ft.

( b ) 10 yd.

( c ) 15 em.

( e ) 18 in.

(f) 5 yd.

3 . Information is given for f ive c i r c l e s . The l e t t e r s r, d , c, and A are the measures of the radius, the diameter, the circumference, and the area respect ive ly . Find a l l t h e

missing data. Use 3.1 as an approximation to a .

4 , khich has the greater frying surface--an eight-inch c i r c u l a r skillet or a seven-inch square frying-pan?' ( T r~ 3.14)

( a 1 (b 1 ( C 1 ( d l

(4

5 . A c i rcu la r drumhead i a twelve inches acrosa. What is the

area of the drumhead?

Circ le

A

B

C

D

E

Radius

4 ft.

111 in.

Diameter

16 cm.

20 ft.

Circumference

100 mi.

Area

6 , (a ) Vhich method is easier for fZnding t h e a rea of a

circle--to measure the radius and c a l c u l a t e t h e area, 2 w i t h t h e a i d of r , or to measure t h e area d f r e c t l y

w i t h an appropriate unit of m e a s u r ~ ?

( b ) Should both methods yield the same r e s u l t ?

A rectangular p l o t of land, 40

f ee t by 30 f e e t , I s mostly lawn.

The c i r c u l a r flower-bed has a

r ad ius of 7 f e e t . Irlhat is t he

area o f the portion of the p l o t

t ha t is planted in grass?

8. The f igure represents a simple

closed curve composed of an arc

or a c i r c l e and a dlameter of the c i r c l e . The area of t he

i n t e r i o r of this simple closed curve, measured in square

inches, is 8 s Do not use any approximation f o r a in

this problem.

(a) What is the area of the interior of t h e e n t i r e c i r c l e ?

( b ) What is t h e second power of the radius of t he c i r c l e ?

( c ) How long is a radius of the c i r c l e ?

( d ) How long is the straight portfon of t h e closed curve

represented in the f igure?

( e ) k%at is the circumference of the ( e n t i r e ) c i r c l e ?

(f) How long i s the c i r c u l a r arc represented in t h e f i g u r e ?

( g ) What is the t o t a l length of t h e simple closed curve?

* 9 . The e a r t h is about 150 milllon ki iometers from t h e sun.

The o r b i t (or path) of the ear th around the sun I s not r e a l l y

c i r c u l a r , but approximately so. Suppose t h a t t h e orblt were

a c i r c l e ; then the path would lie in a plane and the re would

be an i n t e r i o r of t h e o r b i t (in the plane) . What would be

an estimate f o r the area of this i n t e r i o r ?

"10. The c e n t e r o f t he longer c i r c l e lies on the s h o r t e r c i r c l e . The

intersection of t he two c i r c l e s is a s ingle p a i n t . This p o i n t

i ! and the cen te r s of the t w o c i r c l e s

lie on one line. If the i n t e r i o r

o f the shor t e r c i r c l e is chosen as a u n i t of measure, what would be

the measure of t he reglon inside the longer c i r c l e and outside the shor t e r c i r c l e ?

* I l . Another way to discover a r e l a t i o n s h i p between t he rad ius of a c i r c l e and the area of the same c i r c l e is t h e fol lowing.

Perform the suggested steps. On a piece of st iff paper draw

a large c i r c l e with your compass. Mark the center P of

the c i r c l e . (See left-hand figure.) Use your protractor to

ass i s t you 3n drawing 2ight lines on P which w i l l separate

the interiar o f t h e c i r c l e i n t o sixteen regions all of the

0 same area. (see right-hand f i g u r e . ) Can you ca lcu la t e how

many arc-degrees each of t h e s ix teen a rcs w i l l have?

Cut away the portion of the paper outside the c i r c l e . Cut

through the I n t e r i o r of the c l r c l e along the fully-drawn diameter. In each of the two halves , cut with great care along the d o t t e d radii from P almost to the c i r c l e itself. The eight angular portions should hang together somewhat like teeth .

With both portions c u t In a toothed fashion, fit the two

pieces together, (Only a few of the sixteen teeth are shown in

this figure. ) The upper and lower boundaries of the completed pattern have

a scal loped appearance. If they were straight, t h e entire

f l g u r e would be the I n t e r i o r of a (fill the

blank with the best choice of a name f o r this simple cloaed curve). As an application of results from Chapter 10 you may estimate the area of t he i n t e r i o r of t h i s curve by us ing

the measures of i t s apparent base and its apparent a l t i t u d e .

How does your r e s u l t compare with t h e product of and

the second power of t h e r a d i u s ?

12. BRAINBUSTER. A barn is 40 feet long and 20 feet wide.

A chain 35 feet long is attached to t he barn a t the middle

o f one of its longer sides. Another chain 35 feet long is

at tached t o the barn at one of its corne r s . E i t h e r chain

may be used t o te ther a cow f o r grazing.

(a) IfiLhich chain glves the tethered cow the greater area of

land over which t o graze?

( b ) How much difference is there between the areas of t he

two regions? (use 3 -1416 as an approximation t o T .)

11-8. Cylindrical Solids--Volume

In Chapter 8, you studied the rectangular solid, its volume and its surface area. In Chapter 10, you studled the prism, i t s volume, and its surface area. Here we shall study another s o l i d

which is frequently found in everyday l i f e . Instead of having a

rectangular base, l i k e a box, suppose a solid has a c i r c u l a r base, like a t i n can. We c a l l such a solid a cylindrical solid - (or sometimes just a cylinder). You are familiar with other examples of c y l i n d r i c a l solids such as water pipes, tanks, silos, and some drinking glasses.

The figures shown below represent cylinders. Those on the

left are c a l l e d right cylinders. Compare them with the oblfque

( o r s l a n t e d ) cy l inders on the right.

@ (D r &7~ Right Cylinders Oblique Cylinders

We r a r e l y see s lanted c y l i n d r i c a l sol ids in o rd ina ry life.

Therefore in this chapter we shall assume our solids are right

cy l indr i ca l sol ids .

We list some Important p rope r t i e s of a right cy l inde r .

(1) It has two c o n g ~ u e n t bases ( a top and a bottom) and

each is a c i r c u l a r reg ion .

( 2 Each base is in a plane and the planes of t h e two bases

are parallel.

( 3 ) If t he planes of the bases are regarded as horizontal,

then the upper base is d i r ec t l y above the lower base.

( 4 ) The l a t e r a l or s ide surface of the cylinder 5s made up

of the points of segments each joining a point of the

lower c i r c l e with the poin t d i r e c t l y above It in the

upper circle,

There are two numbers o r lengths which descrlbe a cyiindrical

s o l i d . These are the radius of the base of the cylinder and the

altitude (or height) of the cylinder. The altitude I s t h e

(perpendicu la r ) distance between the p a r a l l e l planes conta in ing the bases. For a r i g h t cy l inder , the altitude can also be thought

o f as the length of the shortest possible segments l y i n g in the

l a t e r a l su r f ace and joining the two bases. How can we f ind the volume of a cylindrical solld? In one

sense there is a fairly easy method. If the solid is like a t i n

can and will hold water (or sand) we can fill it up and then pour

I t into a standard container. However, we would l i k e to know what the answer is without having t o do this every time. For some cylinders, this method would be very impractical, perhaps

impossible.

Recall how we found the volume of a box or of a (right) prism. Ue f i r s t considered a box or prlsm one u n l t h igh . The

number of cubic u n i t s i n this box o r prism would be the same as

the number of square units in the base. Thus the measure of the

volume was clearly the measure of the area of the base tlmes one. If t h e box or prism had an altitude of two units, then the measure

of the volume would clearly be twice as much as the measure of t he

area of t h e base. That is, I t would be 2 times the measure of t h e area of t h e base .

In general , If the area of the base were B aquare units and the altitude of the box or prism were h units then the volume would be B h cubic units.

Exactly t he same situation occurs with a cy l indr i ca l s o l i d .

The measure of the volume of the cy l inde r i s simply the measure of the area of the base times the measure of the a l t i t u d e . The area of the base of a cylinder is rr2 square units. So the volume

is rr2 h c u b ~ c u n i t s . F:e now have one basic p r i n c i p l e which applies t o boxes, to

o the r prfsms, &to cyl inders . The measure of the volume'is the

measure of the area of the base times the measure of the altitude.

In mathematical terms, this is f requent ly written as fol lows:

B represents the measure of the area of the base and h

represents the measure of the height .

You should !L earn and remember how to compute t h e volume of a

cylindrical solid. To compute the volume of any solid of this

type we slmgly multiply t h e measure of the area of the base by the measure of t h e altitude, The altitude i s t h e (perpendicular) d i s t ance between t h e parallel planes whlch contain the bases. If you t h i n k of the geometrical f i g u r e and what it is you-want t o find, then most problems of t h i s type are very easy.

A s an example, assume you want to find the volume of a

cy l inde r whose radius has a measure of 7 and whose height

( o r a l t i t u d e ) has a measure of 10.

V = Bh; for a cylinder, B = 2

Theref ore, 2 V = r r h

v = T ** . 7 ' 7 '10

The volume of t h e cy l inde r is about 1540 c u b i c units.

A note on computation. Sometimes when making computations --- involving a , it is usually easier to use a decimal approxi-

mation for ?r only at the last s tep of the arithmetic. In this

way we use long decimals as little as possible. Consider 2 a 52 . 8. b:e note that 5 = 25 and tha t 25 . 8 = 200.

Therefore r . 52 8 = a . 200 (3.14) (200) = 628. This procedure is much simpler than multiplying 3.14 by 25 and then multiplying the result by 8.

Exercises 11-8

1. Information is given f o r five p i&t cylinders. The letters

r and h are t h e measures of the r ad ius of the circular

base and the height of the cylinder respect ive ly . Using

3.1 as an approximation for ?r , f i n d the volumes of each

cyl inder .

Volume

?

?

?

?

?

( a )

(b)

( C )

( d 1 ( e >

Radius ( r )

4 in.

8 ft.

10 cm.

7 ~ d .

12 in.

Cylinder

A

B

C

D

E

Height (h)

8 in.

4 ft.

30 om.

25 yd.

12 in.

. 2 . Find the volumes of t he right cy l inde r s shown here. The

dimensions glven are the radius and t h e height of each

c y l i n d e r . The f igures are not drawn to scale.

25 feet - /'

2

3 . A silo ( w i t h a flat top) is 30 feet h igh and the inside

rad ius is 6 f e e t . How many cubic Teet of graIn will it hold? (\:hat is i t s volume?) Use T * 3 .l4.

4 . A cy l indr fca l water tank is 8 feet high. The diameter

( n o t the r a d i u s ) of its base is 1 f o o t . Find t h e volume

(in cubic feet) of water which i t can h o l d . Leave your answer in terms of B . If you use an approximation f o r

P , what is your answer to the nearest (whole) cub ic f o o t ? 1 5. There are about 7T gallons in a cubic foot of water. About

how many gal lons will t he tank of Problem 2 hold?

6. Find the amount of water (volume in cubic inches) which a

100 foot length of p f p e will hold if the inside radius of a

cross-section is 1 inch. Use P " 3.14. ( A cross-

sec t ion is shaped l i k e the base . A cross-sectfon is the

intersection of the solid and of a plane parallel t o the

planes of t h e bases and between them.)

*7. Find t h e volume of a cylindrical solid whose a l t i t u d e is

10 centimeter3 and the radius of whose base is 3 centimeters. Leave y o u r answer in terns of a .

*8. In Problem 7, what would t h e volume be i f t he a l t i t u d e

were doubled (and the base were l e f t unchanged)?

"9. In Problem 7 , what would be t h e volume if the r a d i u s o f

t he base were doubled (and the altitude were l e f t unchanged)?

"10. In Problem 7, what would be the voluiie if t h e altitude were

doubled and the radius of t h e base were also doubled? ( ~ h i n k of f i r s t doubling the a l t i t u d e and then doubl ing the

r a d i u s of t h i s new cy l indr i ca l s o l i d . )

'11. In general, what is the e f f e c t on the volume of a cylindricai

solid obtalned by d.ou.bl.7.ng the altitude? 3y doubliny t h e

radius of the base? By doubling both the r a d i u s oi' t h e

base and the a l t i t u d e ?

12. BRAINBUSTER. What i s t h e amount (volume) of metal in a piece

of water p i p e 30'' long if the inside diameter of a cross - section i s 2" and t h e outside is 2.5". Use Ir " 3.1.

11-9. Cylindrical Solids--Surface Area In the previous section we considered questions about the

volume of a cylinder. We will now consider the surface area. There are two questions that m i g h t b e asked: (1) What is the

area of the curved su r face? (2 ) Fhat is t h e total sur face area?

It I s easy t o see t h e r e l a t i o n between these two if we think of the solid i t s e l f . The total area I s the area of the curved

surface plus t h e allea of the top base p l u s t h e area of the bottom

base. But the areas of t h e t o p and bottom bases are the same.

And the measure of each is ~r r2 where r is the measure of the

radius of the base. So if we know how to f ind the area of the curved surface, we can f l n d the t o t a l area.

The label o f a tin can almost covers t he curved sur face of the can . \!e w i l l assume t he label covers t he en t i r e curved

surface of a can. Then, t h e area of the l a b e l is the l a t e r a l

area of the cylinder. How are l a b e l s made? They are made and

p r i n t e d in the form of rectangular regions . The height of such

a rectangle is the he igh t of t h e cy l ind r fca l solid. The length

of the base is the circumference of the base c i r c l e of' the

cylinder. (b'hen nade, the label has this length p l u s a little

more, t o allow f o r overlapping. ) The l a t e ra l area of a cy l inde r , t hen , f s merely the area of a rectangle as shokm below.

-Circumference -7 of cylinder

I.'@ have observed that t h e lateral area o f a cylinder is the

area of a c e r t a i n rec tangle . The altitude of t h e r ec t ang le and

the a l t i t u d e of the c y l i n d e r a re the same. The length o f the

base of t h e rec tangle and t h e circumference of the base of t h e

cylinder are t h e same. Therefore t h e measure of t h e l a t e r a l

area of t he cy l inder i s the product of the measure of t h e length

of the base c i r c l e and the measure of t h e h e i g h t . Hence, the

measure is

And t h e measure of t h e total area I s t h en

2 2 a r ' h + 2T r . There are some curved sur faces , like the s u r f a c e o f a b a l l ,

which cannot be t rea ted in q u i t e this simple a way. Rectangular

reg ions , o r o t h e r f l a t surfaces, j u s t don't wrap n i c e l y around them. The areas of such surfaces can be handled in o t h e r ways.

It is fortunate t h a t cy l inde r s have "easy" curved surfaces.

Two formulas presented in this c h a p t e r are very Tmportant

in deal ing with measurements of circles. These two formulas are:

(1) c = 2 n - r (or c = P d )

(2 ) A = B r 2

The f o m u l a s f o r the volume and surface area of a c y l i n d r i c a l

solid use these t w o formulas. Therefore :t is not essential that

you memorize the formulas f o r v o l m e and sur face area, if you - can remember what the formulas represent.

You should understand and know the basic general p r i n c i p l e

f o r g e t t i n g t h e volumes of many solids. For solids, which-are

r i g h t prisms and right cylinders , this p r i n c i p l e t e l l s us that t h e

measure of t he volume is t h e product of the measure of the area

of t h e base and the measure of the a l t i t u d e , o r in mathematical syntbols, V = B h .

F i n a l l y , in many problems such as f ind ing surface areas, you

shou ld think of the geometric o b j e c t s and what you want to know.

A s an example, in finding the l a t e r a l surface of a cylindrical s o l i d think of what the l a t e r a l s u r f a c e represents i f f l a t t e n e d o u t . Then the l a t e r a l area I s the area of a rec tangle . To get

the total surface area, add to the lateral area twice the area

of the base. In mathematical symbols,

where ST represents t h e measure of the t o t a l sur face o f a

cy l inde r ,

d represents t h e measure of the diameter of the base,

r represents the measure of t h e r ad iu s of t h e base.

h represents the measure of the height of the cylinder.

To f i n d the t o t a l surface area of the cy l inde r shown at the right:

(using s 3 t ) .

ST = T;dh -t 2 8 ~ 2

22 14 2 0 ) + (2 ST = (f " ' 7 * . I

S , = 880 + 308

sT = 1188

The total su r f ace area is about 1188 square inches.

Exercises 11-9

1. Information is given f o r five cylinders, Using 7 a 3 . 1 , f ind all the missing data.

( a >

(b 1 (c

( d )

(4

Cylinder

A --

B

C

D

E

Radius

of base

Diameter

of base

H e i g h t Total Surface

Area ( A ~ )

1 ft.

?

15 cm.

?

?

16 r t . ?

8 yd.

3 ft.

17 ft.

50 cm.

12 yd.

?

?

?

?

2. Find t h e t o t a l sur face area f o r each of the cylinders shown.

(use a c~ 3.1).

HOME PROJECT: Complete the following project at home. 3 . - ( a ) Take an ordinary tin can. Measure it and make a label

for It that will f i t without overlapping. Your l abe l

will be in t h e form of a Its height w i l l be

the of the t i n can. The l eng th of I t s base

will be the of the t i n can . Try your label to see that its fits.

(b) Take another tin can of d i f fe ren t size from that of ( a ) . Make a label for it without doing any measuring.

Deterrnlne the deslred s i z e of the label by comparison

w i t h the tin can .

( c ) With a tape measure or string (to be measured) f igure out t h e l a t e r a l sur face area of the tin can of (b) by measuring the circumference of t he base and the height

d i r e c t l y . ( d ) Measure t h e diameter (and then f igure out the r a d i u s )

of t h e base of the t i n can of (b). Using this and the

height f ind the lateral surface area of t h e t i n can. Should your answer agree with t h e answer you got in

( c I ?

4. Flnd the lateral surface area (ln square cent imeters ) of a

cy l indr ica l s o l i d whose altitude is 8 centimeters and the 1 radius of whose base is l2 centimeters. Use P w 3 . 3 4 .

1 5. Flnd the total surface area of the cylinder of Problem 4.

1 16. How many square meters of sheet metal do you need t o make a

c losed cylindrical tank whose height is 1.2 meters and the rad ius of whose base is .8 meters? How many square meters do you need if the tank is to be open on top?

i "7. A small town had a large cyllndrlcal water tank tha t needed pain t ing . A gallon of paint covers abcut 400 square f e e t . How much paint I s needed to cover the whole tank If the radius of the base is 8 feet and the height of the tank is 20 feet? Give your answer t o the nearest t e n t h of a

gal lon .

11-10. OPTIONAL: Review Chapters --- 10 and 11.

1. It is often helpful to use diagrams to show relations. For example 1

may represent 2 . 3 = 6 b

6 Relations which involve sums as well as products may be

shown by diagrams.

may represent

You have studied ways of computing perimeters, areas, and

volumes for a number of geometric figures and t h e i r

interiors; and you have expressed these methods briefly in number sentences or formulas .

[aec. 11-10]

Below are twe lve diagrams for severa l of these formulas, and a lfst of the geometric figures t o which one (or more) of the formulas applies, Copy each diagram on your paper,

and beside it write the following:

( a ) The name of t he figure (or figures) to which the

diagram applies. (choose from the list of f igu res

below. )

(b) What the formula t e l l s abou t the figure: area,

perimeter, volume, circumference, l a t e r a l area. .

( c ) The formula shown by t h e diagram.

List of figures : yarall elogram rec tangul a r prism

c i r c l e triangle

rectangle t r i angu la r prism cy l inde r

In your copies of the dfagrms, draw a c i r c l e around each numeral .

Draw a square around each symbol f o r a measure you

would nave to know in order t o apply the formula to a

p a r t i c u l a r f i g u r e .

Hold a ruler horizontally across t h e diagrams (1) to

(10) in your text, to separate the numerals from t h e

symbols f o r measures. \,Thy are there more arrows below your ruler in some diagrams than in o the r s?

Each of the formulas tells how to find the measure of a

quantity which has either one dimension, two dimensions,

or three dimensions. Put a 1 , " 2 " , or "j"

beside each diagram, to show t h e dimension of the kind

of quantity t o which it relates.

Do you see any connection between your answers for Questlons 2(b) and 2 ( d ) ?

Look a t the diagrams of formulas for f indlng volumes.

h here are t h r e e . ) Put a B a t t he end of one arrow, to separate the p a r t of the formula which gives the

area of the base from t he rest of the formula.

Make up formulas relating to squares and cubes. Make

diagrams f o r the formulas, and tell what each diagram

means.

Draw the diagrams (11) and (12) in a d i f f e r e n t way.

% r i t e the formulas f o r your new diagrams.

In the above figure determine with the use of your

p r o t r a c t o r the measure of t h e rollowing arcs. Indicate

your results with correc t use of symbols.

n (L) ABC

n ( d ) CDE

4, Each side of t r i a n g l e EFG is

m e a s u r e d as 34.6 meters long.

The distance between E and t h e

cen te r P of the c i r c l e is measured as 20.0 meters. The

altitude of triangle EFG from

E to the s i d e FG is measured

as 30.0 meters long.

( a ) hhat is the area of t h e triangle EFG .

(b) \:hat is t h e area of t h e c i r c l e ? (use 3.142 as an

approximation to 8 , and round your answer t o t h e

nearest squaremeter.)

( c ) On your paper draw a sketch of the figure and shade the intersection of the i n t e r io r of the c l r c l e and the exter ior of the triangle EFG. .

(d) What is the area of this intersection?

( e ) The half -plane on the Q-side of EF and the i n t e r i o r i of the c i r c l e have an intersection. Itbat is the area

of this Intersection?

5. Below are f o u r figures representing simple closed curves. Each curve is the union of several sements and one or two

arcs of c i r c l e s . ~ach:arc either I s a semicircle oil is

measured as 90 arc-degrees. The dotted segments are not p a r t s of the simple closed curves but are useful in

indicating t he lengths .

For each curve f l n d its t o t a l length and find the area of its closed region.

I

( a )

(Use a fi: 3 .) >- I

I I

I

1 I I

24 METERS L ,------,,-----,,,,-,----- I

-I I (b) Each segment is 31 .4

centimeters long. The

d i s t ance between the

parallel segments is

17 .g centimeters. (use

55 3.14, and round final answers to nearest centimeter or nearest square centimeter. ) (0

I

I

i ( c > Unit of measurement is

foot . ( ~ e a v e your answer In terms of R . )

I I

'(d)

Unit of measurement is

, millimeter. ( ~ e a v e your answer in terms

of a . ) I

1 6 . Find the volume of a momurnent

made as follows: The base is

a rectangular block of marble

of dimensions 4' by 61 by

2 h igh . On t o p of t h i s block

in t h e c e n t e r is a c y l i n d r i c a l s o l i d of he igh t 8' and with

the radius of the base c i r c l e

1'. Use # 3.1.

' 7 . Flnd the t o t a l area of t h e exposed surface of t he monur~lent of the preceding problem. The underside of the base I s

considered as not exposed. -

Chapter 12

MATHEMATICAL SYSTEMS

A New Kind of Addition. 12-1 - - - - -

The sketch above represents the face of a four-minute c lock .

Zero is t h e s t a r t i n g point and, also, the end-point of a rotation of the hand.

With the model we might s tar t at 0 and move to a c e r t a i n position (numera l ) and then move on to another position just l i k e

the moving hand of a clock. For example, we may start with 0 2 and move of the distance around the face. We would s t o p at 2.

1 If we follow this by a rotation (moving l i k e the hand of a 2 clock), we would stop at 3 . After a rotation of from 0 w e

could fol low with a 4 r o t a t i o n . This would bring us to 1 . The f i r s t example could be written 2 + 1 gives 3 where the 2

2 indicates of a rotation from 0, the + means to follow this

by another rota t ion (like the hand of a c lock) , and t h e 1 means

ro t a t ion , thua we arrlved at the position marked 3 (or $ of a rota t ion from 0) . The second example would be 2 + 3 gives

1 where the 2 and + s t i l l mean the same as in the first

example and the 3 means a rotation of f. A common way to

write t h i s 3s:

2 + j = 1 (mod 4)

whlch is read:

Two plus three is equivalent to one (mod 4) .

The (mod 4) means that there are four numerals: 0, 1, 2, 3 on the face of t h e clock. The + sign means what we described

above - t h i s I s our new type of addition. The = between the 2 + 3 and the 1 indicates t h a t 2 + 3 and 1 are the same

{that I s , "equivalent t t ) on this clock. We call this b r i e f l y --- "addition (mod 4) .'I Of course there are other possible notations which could be used but this is the usual one. The expression

"(mod 4)" is derived from the fact that sometimes 4 is c a l l e d It the modulustt which indicates how many single Steps are taken

before repeating the pattern.

Example 1. Pfnd 3 + 3 (mod 4) .

Example 2. Bind (2 + 3) -k 3 (mod 4 ) . I

2 + 3 1 (mod 4) 1 + 3 0 (mod 41

(2 + 3 ) + 3 3 1 + 3 I 0 (mod 4)

529 I

The following t a b l e illustrates some of the addition facts

in the (mod 4) system. (Mod 4 )

We read a table of this s o r t by following across horizontally from any entry in the l e f t column, for i n s t ance 2, to the

position below some entry in the top row, such as 3 (see arrows). The entry in this position In the table is then taken as the

r e s u l t of combining the element in the left column w i t h the element in the top row (in t h a t order). In the case above we write 2 + 3 r 1 (mod 4). Use the table to check that 3 + 1 = 0 (mod 4).

Example 2. Complete the following number sentences to make them true statements.

(a) 3 + 4 = ? (mod 5 ) The mod 5 system pepresented by the face of a clock

should have f i v e positions; namely, 0, 1, 2, 3 , and 4. If you draw this c locko you will see that

3 + 4 1 2 (mod 5 ) since the 3 means a ro ta t ion of 2 from o. mi3 is followed by a 5

rotation which

ends a t 2. 5

(b) 2 + 3 = ? (mod 5) 2 + 3 = 0 (mod 5) . This Is a * rota t ion from 0 followed by a

4 2 rotation which brings us to 0. 5

( c ) 4 + 3 1 ? (mod 6 ) In the mod 6 system, the positions on the face of the

c lock are marked 0, 1, 2 , 3, 4, and 5. If you

draw t h i a c lock you wlll see that 4 + 3 = 1 (mod 6 ) .

Exercises 12-1

1. Copy and complete the table for additlon (mod 4) . Use it t o

complete the following number sentences:

( a ) 1 + 3 s ? (mod 4) ( c ) 2 -b 2 ? (mod 4 )

(b) 3 + 3 E ? (mod 4) ( a ) 2 + 3 = ? (mod 4)

2, Make a table f o r addition (mod 3 ) and for addltion [mod 5 ) .

3 . U s e t h e tables in Problem 2 to find the anawers to the

f 01 1 owing:

( a ) 1 -k 2 r ? (mod 3 ) ( c ) 2 + 2 ? (mod 3 )

( b ) 3 + 3 ' ? (mod 5 ) ( d ) 4 + 3 r ? (mod 5)

4. Make whatever tables you need to complete t h e following

number sentences.

(a) 5 + 3' ? (mod 6 ) ( c ) 3 + 6 = ? (mod 7)

(b) 5 + 5' ? (mod 6 ) ( d ) 4 + 5 s ? (mod 7)

Note: be sure to keep a l l the tables you have made. You

wfll f i n d use for them later in this chapter.

5 . Find a replacement f o r x to make each of the following

number sentences a true statement.

( a ) k + x = 0 (mod 5 ) (e ) 3 + x = 2 (mod 5 )

(b) x + 1 = 2 (mod 3 ) . (f) x + 4 = 3 (mod 5)

( c ) 1 + x = 2 (mod 3 ) (g) x + 2 1 0 (mod 3 )

( d ) 2 I- x 5 4 (mod 5) (h) 4 + x E 4 (mod 5 )

6. You have a five-minute clock. How many complete revolutions

would the hand make if you were using it to tell when 23

minutes had passed? Where would the hand be at t he end of the 23 minute interval? (~ssume that t h e hand started from

the O position.

7 . Seven hours a f t e r eight otclock is what time? mat new kind

of addition did you use here?

8. Nine days af ter the 27th of March is what date? What new

kind of addition d i d you use here?

12-2. A New Kind of Multiplication.

Before considering a new multiplication let us look a t a

p a r t of a multiplication t a b l e f o r t h e whole numbers. Here i t is:

1 2 3 4 5 5. 6

If we had thls table and forgot what 5 t i m e s 6 i s equal to, we

could look in t h e row labeled 5 and the column l abe led 6 and

find the answer, 30 , In the 5-row and 6-column (see arrows

above). Of course it is easier to memorize the table since we use it so frequently, but if we had not memorized it, it might be a

very convenient thing to have in our pocket for easy reference.

How would you make such a t a b l e if you didn't know it

already? This would be quite easy if you could add. The f i r s t 19ne is very easy - you write a r o w of zeros. For the second

l i n e you merely have to know how to count. For the third line you add 2 each time; for the four th l i n e add 3 each time, and so

f o r t h .

Now, if we use the same method, we can get a multiplication table (mod 4 ) . First block it o u t , filling in the first and

second rows and c o l m s :

( ~ o d 4)

We have Just four blanks to f i l l in. To get the 2-row (indicated

by the arrow above), we add twos. Thus the thipd entry (which I s

2 x 2 ) is 2 + 2 r 0 (mod 4) . Then our f i r a t three entries will look l l k e this:

To get the fourth entry, we add 2 to the third e n t r y . Since

0 + 2 2 (mod 41, the complete 2-row is now

For the last row we will have to add threes. Here 3 + 3 = 2 (mod 4) and 3 + 2 = 1 (mod 4 ) . So the complete table Is:

533

Now consider one way in which this table could be used.

Suppose a lamp has a four-way switch so that it can be turned t o one of four positions: o f r , low, medlum, high. We m i g h t l e t numbers correspond t o these posltions as follows:

off 1 ow medium hi&

0 1 2 3

If the l ight were at medium an; -we f l i c k e d the switch three times, the light would be a t the low posi t ion since 2 + 3 =1 1 (mod 4 ) . Suppose the l i g h t were off and three people f l i cked the switch

three times each; what would be the final positlon of the l igh t?

The answer would be "low" since 3 3 = 1 (mod 4 ) and the number

1 corresponds to "low. 11

Consider an app l i ca t ion of another mul t ip l ica t ion t a b l e . A

jug of juice las ts three days in the Willcox family. One Saturday, Mrs. Willcox bought six jugs which the family started using on the followfng day. What day of the week would it be necessary f o r her to purchase juice again? Of course It would be poss ib le to

count on one's fingers so to speak: 3 days af ter Saturday is

Tuesday, 3 days after Tuesday l a Friday, e t c . But It is much easier i f we n o t h e t h a t since there are seven days i n the week,

t h i s i s connected with multiplicatlon (mod 7) . We could l e t t h e number 0 correspond to Saturday s ince this I s the day w e s t a r t with and so on as follows:

Sat. Sun. Mon . Tue . Wed. Thur . Fri . I O 1 2 3 4 5 6

Now we need to f ind what 6 3 is (mod 7). We do not need

t h e complete multiplication table since we are trying t o f i n d a

mult iple of 6 . So we compute the 6-POW in the usual way by addlng sixes, us ing the add i t i on t a b l e (mod 7) which we

constructed f o r Problem 4 of the previous s e t of exercises.

(mod 7 )

This means t h a t 6 3 4 (mod 7) and since Wednesday corresponds to 4, it follows that Mrs. Willcox would next have to buy ju i ce on a Wednesday. Actually we d i d not really need to construct a l l the 6-row.

Exercises 12-2

1. ( a ) Make a table for multiplication (mod 5 ) .

(b) Make a table for multiplication (mod 7).

( c ) Make a table f o r multiplication (mod 6 ) .

Note: Keep these t a b l e s f o r f u t u r e use.

2. Complete the following number sentences to make them t r ue statements. You may find the tables you conatmeted in Problem 1 u s e f u l .

( a ) 3 x 2 ? (mod 5 ) ( d ) 1 + ( 3 x 4) = ? (mod 6 )

( b ) 3 x 4 ? (mod 6) ( e ) 5 + ( 6 x 5 ) = ? (mod 7)

( c ) 6 x 4 = ? {mod 7)

3 . Find a replacement f o r x to make each of the following number sentences a t r u e statement: ( D r a w the clocks which you need. )

( a ) 5 ' 10 = x (mod 11) (c) (3 4) + 2 = x (mod 8)

(b) 7 13= x (mod 15) ( d ) ( 4 7 ) + 11 = x (mod 13)

4. ( a ) Find the date of 10 weeks a f t e r December fourth.

(b) In 1957? August sixth was a Tuesday. What day of the

week was August sixth in 1959?

5 . Form a table of remainders a f t e r dlvlsion by 5, where the

entry In any row and column is the remainder after the- product is divided by 5. For instance, since the remainder

is 1 when 2 3 is divided by 5, we will have a 1 in

the 2-row and 3-column (see arrows). We have written f n a few entries to show how ft goes. For instance, t o get the

entry in the 2-row and 4-column we multiply 2 by 4 to

get 8 and since the remainder when 8 is div ided by 5 Is 3, we put 3 in the 2-row and 4-column. Complete

the table: 1

6 . Do you not ice any r e l a t i o n s h i p between the table of Problem 5 and another you have Sound? Can you give any reason for this? How can this be used to make the so lu t lon of some of the problems simpler?

*7. Use the multlpllcatlon table (mod 5) to find the

replacement f o r x t o make each of t h e following number

sentences a true statement:

(a) 3x = 1 (mod 5) ( d ) 3x = 4 (mod 5 )

(b) 3x = 2 (mod 5 ) ( e ) 3x 1 0 (mod 5)

(4 3 X ' 3 (mod 5)

+8. If it were (mod 6) instead of (mod 5) in the previous

problem, would you be able to find x in each case? If

not, which equivalences would give some value of x?

[aec . 12-21

12-3. What is an Operation? kie are familiar with the operations of ordinary arithmetic--

additfon, multiplication, subtraction and division of numbers.

In Section 12-1, a di f fe ren t operatfon was discussed. We made a

t a b l e f o r the new t y p e of addition of the numbers 0, 1, 2, 3 . This operation is completely descr ibed by the table that you made

in Problem 1 of ;?xerclses 12-1 . That is, there are no numbers to which the operation is applied except those ind ica ted and the resul ts of the operation on all pairs of these nwnbe~s are given.

The t a b l e tells what numbers can be p u t together. For Instance, the table t e l l s us t h a t the number 5 cannot be combined with any

nmber i n the new t ype of addition s ince "5" does not appear in

the left column nor in the top row. It also tells us that 2 + 3 1 (mod 4). Study t h e following tables :

1 2 3

3 1 2

1 2 3

3 2 3 1

("1 0

3 6 7 8 9

0 1 2 3 ( d ) O

0 0 1 2 3 1

1 2 3 4 5 2

2 4 5 6 7

So f a r the only operations we have had have been ca l l ed

multiplication or addition. Here in ( c ) , ( d ) and ( e ) we

have d i f f e r e n t operations and so we use d i f f e r e n t symbols:

a, and A From each one of these tables we can f i n d a c e r t a i n s e t

(the set of elements in the left column and top row) and we can p u t any two elements of this set together to get one and only one

t h i n g . For instance, in Table (a), t h e set 2s [l, 2, 3 , 4, 53 s ince these are the numbers which appear in the left column

and top row. These are the o n l y numbers which can be put together

by Table (a). I n Table (b), the set I s [ 3 , 5, 7 j 93 . Irhat

s e t 1s given by Table (c)? by Table ( d ) ? by Table ( e ) ?

Here are some examples from the tables:

3 + 5 = 3 in Table (a),

3 + 5 = 8 in Table (b),

2 1 = 5 and 2 2 = 6 in Table ( c ) . Read " " 2 square b equals 5" . 1 0 1 = 3 In Table ( d l . Read "1 c i r c l e - d o t 1 equals 3."

5 A 2 = 3 in Table ( e ) . Read " 5 t r i a n g l e 2 equals 3 ."

In each case we had a se t of elements:. i n (a) t he set i s

(1, 2, 3 , 4, 53; in ( d ) it was [l, 2, 33. We also had an operatfon: i n ( a ) it was + ; in ( d ) it was 0 . Final ly we had the r e s u l t of combining any two elements by means

of the operation; i n ( a ) t h e results were 1, 2, 3 , 4 or 5 ; in ( c ) theywere 0, 1, 2, 3 , 4, 5 , 6 , 7, 8, 9. Allof

these operations are called binary operations because they are

appl ied to two elements to get a third. So fa r the elements have been numbers but we shall see l a t e r that they do not need to be.

The two elements which we combine may be the same one and the r e s u l t of the operat ion may or may not b e an element of the s e t

but i t must be something definite - not one of several possible t h ings .

You are already familiar with some operations defined on the set of whole numbers.

Any two whole numbers can be added. Addition of 8 and 2

gives 10.

Any two whole numbers can be multiplied. Multiplication of 8 and 2 gives 16.

Addition and multiplication are two different operations defined on t h e set of whole numbers.

In discussing subtractlon, f o r instance w i t h whole numbers,

it is convenient to look ahead t o the work of the eighth grade.

The expression "6 - 9" is not the name of anything you have studied in this course . That is, it is not now possible for us to combine 6 and 9 (in tha t order) by subtraction and get

" a definite thingt' and you may wonder whether or not sub t rac t ion

is an operat ion. Next year you will learn t h a t there is "a definite thingtt (in f a c t , a number) which is cal led " 6 - 9. " With this in mind, we will consider subtraction a binary operation defined on the whole numbers (or ra t ional numbers, e t c . ) , even though we are not yet acquainted w i t h a l l the results obtained from subtraction.

Fken an opera t ion Is described by a t ab le , the elements of the set are written i n the same order in the t op row (left to

rlght) and in the left column ( t o p t o bot tom). Keeping the order the same will make some of our later work easier.

Ue must also be careful about the order in which two elements

are combined. For example,

2 ~ 1 = 5 , b u t 1 ~ 2 = 4 .

For t h i s reason, we must remember t h a t when the procedure for

readlng a table was explained, it was decided t o write the element

In the left column first and the element in the top row second with the symbol f o r the operation between them. We must examfne

each new operation t o see if it is commutative and associative.

These properties have been discussed in previous chapters ; they

are briefly reviewed here.

An operation + defined on a set is c a l l e d commutative if, for any elements, a, b y of the s e t , a + b = b + a.

An operation + def ined on a s e t is c a l l e d assoclatlve if any elements, a , b, c, of the s e t can be combined as

( a -t b) t c , and a l s o as a + (b + c) , and the two results are

the same: ( a + b) -k c = a + (b + c ) .

Exercises 12-3

l. Use the tables on pages 536 and 537 to answer the fol lowing quest ions :

( a ) 3 + 3 = ? if we use Table ( a ) .

(b) 3 + 3 = ? if we use Table (b).

( c ) 3 a 2 = ?

( d ) 2 ~ 3 = ?

( e ) 2 Q 2 = ?

(f) 1 0 1 = ?

(g) ( 2 0 3 ) 0 3 = ?

(h) 2 0 ( 3 0 3 ) = ?

{I) (I [j 1) 2 = ?

( J ) 1 (1 2) = ?

(k) 2 A ( 3 A 4) = ?

(1) (2 A 3 ) A 4 = ?

2. (a) Which of the binary operations deecribed in the tables In th i s section are commutative?

(b) Is there an easy way to tell If an operation is commutative when you examine the table for the

operation? What is it?

3 How can you t e l l If an operation is associative by examining a table for t h e operation? Do you think the operations

described in t h e tables in this section are associative?

. Are the following binary operations commutative? Make

at least a partial table f o r each operat ion. Which ones do you think are associative?

( a ) Set: A l l counting numbers between 25 and, 75. Operation: Choose the smaller number. Example: 28 combined wi th 36 produces 28.

(b) Set: A l l counting numbers between 500 and 536. Operation: Choose the larger number. Example: 520 combined w i t h 509 produces 520.

{ c ) Set : The prime numbers . Operation: Choose the larger number.

(d) Set: A l l even numbers between 39 and 61.

Operation: Choose the first number. Example: 52 combined with 46 produces 53.

46 combined with 52 produces 46.

( e ) S e t : A l l counting numbers less than 50 .

Operation: Multiply the first by 2 and then add the

second. Example: 3 combined with 5 produces 11.

Since 2 - 3 + 5 = 1 1 .

(f ) Set : All counting numbers.

Operation: Find the greatest common factor.

Example: 12 combined w i t h 18 produces 6 .

( g ) Set: A l l count ing numbers. Operation: Find the l e a s t common multiple.

Example: 12 comblned ~ 5 t h 18 produces 36.

(h) S e t : A l l counting numbers.

Operation: Raise the f i r s t number to a power whose exponent is the second number,

Example: 5 combined with 3 produces 53.

5 . Make a table f o r an operation t h a t has the commutative

proper ty .

6 . Make up a t a b l e for an operation that does not have the

commutative property.

We have been discussing binary operations. The word

"binary" indicates t h a t two elements are combined to produce

a r e s u l t . The re are other kinds of operations. A resu l t

might be produced from a single element, or by combining

three or more elements. When we have a set and, from any one element of t h e s e t , we can determine a definite thing,

w e say there i s a "unary operation" def ined on the s e t ,

If we combined three elements to produce a fourth we would c a l l it a "ternary operation". One example of a ternary operation would be finding the G . C . F . of three counting

numbers: e . g . the G . C . F . of 6 , 8, and 10 is 2.

*7. Try to show a way of describing the following unary opera t ion by some kind of a t ab le?

Set: A l l the whole numbers from 0 to 10.

Unary Operation: Cube the number.

Example: Doing t h e operation to 5 produces 5i3 = 125.

12-4. Closure. -

Study the Tables (a) and ( b ) . In Table (a) the r e s u l t & of

performing the operation are the numbera which were combined by the operation (1, 2, 3 , 4, 5 over again). E h ~ t in Table (t) the resulta of performing the operation were different numbers from those combined (6, 8, 10, e t c . instead of 3, 5 , 7, 9 ) . We have seen thia kind of difference before, and we have a name f o r i t . We have said that the s e t of whole numbers is "closed under addition1' because if any two whole numbera are combined by adding them, the resu l t is a whole number. In the same way the set El, 2, 3, 4, 51 in Table (a) is closed unde~ the new type of addition there since the results of the operation are again in the same s e t .

[aec, 12-41

However, the set of odd numbers I s not closed under addition since the result of adding two odd numbers Is not an odd number. In the same way, in Table (b) the s e t 3 , 5, 7, 9 is not closed under the operation of addition given there since the r e s u l t is not one of the set E3, 5, 7, 93.

Example - 1: The set of whole numbers: {I, 2, 3 , 4) is no t

closed under multiplication because 2 . 3 = 6 which is not one of the s e t . Of course 1 2 = 2 is in the s e t but f o r

a set to be closed the result must be in the set no matter what numbers of the s e t are combined,

Example - 2: The s e t of - a l l whole numbers is closed under

multiplication because the product of any two whole numbers is a whole number again.

Example 2: The s e t of whole numbers is not closed under subtraction.

For example, consider the two whole numbers 6 and 9 . There are two different ways we can p u t these two numbers

together using subtraction: 9 - 6 and 6 - 9. The first numeral, t tg - 6", is a name f o r the whole number 3 , but

the numeral " 6 - 9" is not the name of any whole number. Thus, subtracting two whole numbers does not always give a

whole number.

Example - 4: The s e t of counting numbers is not closed under divlslon.

8 + 2 1 s a counting m b e r , but there It is true that =

is no counting number 8. Can you give some other illustrations of closure, that is, s e t s closed under an

operation and sets not closed under an operation?

Example 5: h k a t can we say about a set S of count ing num'cers

which is closed under add i t i on and which conta ins t h e number

31 1,Ehat other numbers must it contain? Since 3 is in S,

3 + 3 , or 6 , must a l s o be in S . Since ( 3 + 3 ) and 3 are members of S, ( 3 + 3 ) + 3 = 6 + 3 = 9 must be in S .

Since ( 3 + 3 + 3 ) and 3 are in S, ( 3 + 3 + 3 ) + 3 =

9 + 3 = 12 must a l so be in S. be can continue adding 3 to the resulting numbers to see that 3k must be In S f o r

any counting number k. Thus S must contain all of the

multiples of 3. khat is the smallest s e t S of counting

numbers containfng 3 and closed under addition? As we have

seen S must contain a l l multiples of 3 . V-hat If S c o n t a h s only these numbers:

S = E3, 6 , 9, 12, ... I . Is S c losed under addition? Is the sum of any two multiples of 3 a multiple of 31 If k and m are count ing numbers, is 3k + 3m a multiple of 3? The answer Is yes, of course, since, by the d i s t r i b u t i v e p roper ty of multiplication over addition,

>k + 3m = >(k + m).

Thus, S = 3 , 6 , 9 , 12, . . . is closed under addition, and

is the smallest s e t closed under addition, ~ h i c h contains 3 . V e c a l l S the set penerated by 3 under addition.

Example - 6 : l!e could ask the same questions about multiplication

which we asked about addition in Example 5. \,bat is the

smallest set closed under multiplication and containing 3?

Such a set ce r t a in ly must contain

and so on.

k That is, every number 3 , where k I s a countlng number, 2 3 must be in the s e t . I a the set T = ( 3 , 3 , 3 , . . . I

closed under rnul t ipl icat lon? If n and m are counting numbers, is 3" . 3m a member of T?

n factors 3" = 3 - 3 . ... . 3 ,

nl f a c to r s 3m = 3 - 3 ... - 3 ,

1 -2-, n+m

= 3n + m

Thus 3" . 3" = 3 " + is a power of 3 also, so if 3n and 3m are in T so is t h e i r product . Thus

4 T = (3, 3 3 3 , ... 1

is the sma l l e s t set c losed under multiplication and containing 3 . We call T t h e s e t generated by 3 under multiplicatlon,

Following Examples 5 and 6 we say that the set generated by an element a under an opera t ion * is the s e t

( a , a * a, (a * a) * a, [(a ' a ) *a] * a, . . . ) .

Exercises 12-4 - 1 . Study again Tables ( a ) - ( d ) in this section, page 542. Which

tab le s determine a set t h a t is c losed under the operation?

Whlch tab le s determine a set t h a t is not closed under t h e

operation? How do you know?

[eec. 12-41

2 1,hich of t h e s e t s below are closed under the corresponding

operations?

( a ) The s e t of even numbers under add i t i on .

(b) The s e t of even numbers under multiplication.

( c ) The s e t o f odd numbers under multiplication.

( d ) The s e t of odd numbers under addition.

( e ) The s e t of m u l t i p l e s of 5 under a d d i t i o n .

( f ) The set of multiples of 5 under subtraction.

(g) The s e t 11, 2, 3, 41 under rnultlplication (mod 5).

(h) The s e t of count ing numbers less than 50 under t h e

operation of choosing the smaller number.

(I) The s e t of prime numbers under addition.

( ,i ) The s e t of numbers whose numerals in base five end in "3" under addition.

3 . Find the smallest s e t of count ing numbers which is;

(a) Closed under addition and contalnlng 2 . (b) Closed under multiplication and containing 2.

4 . ( a ) Find t h e s e t generated by 7 under addition.

(b) Find the set generated by 7 under multiplication.

5. L e t S be t h e se t determined by Table (d) in this section.

Flnd the subset of S which is generated by 1 under a. Find t h e subset of S which is generated by 2 under 0 .

* 6 . What subset of the s e t of rational numbers is generated by

3? Is t h i s s e t closed under division? ( 1s 3 in t h e s e t ? 1 Is - In the set? Is 3 + 3

in the set?) Does

( 3 + 3 ) + 3 = 3 + ( 3 + 3 ) ? Is t h e division operat ion associative?

*7. If an operation def ined on a s e t is commutative, must the s e t be closed under t h e operation?

*8. If an operation defined on a s e t is as soc i a t ive , must t h e

set be closed under the operation?

*g. Make up a table f o r an operation defined on the set [O, 43, 1001 so that the set is closed under the operat ion.

*lo. Make up a table f o r an operat ion defined on the s e t {O, 43, 1001 so that the set is not closed under the opera t ion .

1 2 Identity Element; Inverse -- of an Element. In our atudy of the number one i n ordinary arithmetic, we

observed t h a t the product of any number and 1 (in either order) is the same number.

For ins tance,

For any number n in the arithmetic of rational numbers,

n . l = n and 1 - n = n .

In our study of the number zero in t h e arlthmetlc of rational

numbers we observed that the sum of 0 and any number (in e l t h e r order) gave tha t same number; that is the sum of any number and 0 is the number. For instance,

For any number n in ordinary arithmetic, n + 0 = n and

O + n = n . One is the I d e n t i t y f o r multlplicatlon in ordinary arithmetic. - Zero is the i d e n t i t y f o r addition in ordinary arithmetic.

[sec. 12-53

Suppose we let * stand f o r a binary operation. Some

poasibilitiea for * are the following:

1. If * means addition of rational numbers, 0 is an

i d e n t i t y element because 0 * a = a = a * 0 fop

any r a t i o n a l number, a .

2. If + means multiplication of rational numbera, 1

is an ident i ty element because 1 * a = a = a + 1 f o r any r a t iona l number,

3 . If * means the greater of two counting numbers, then

1 * 2 = 2 because 2 is greater than 1; 1 * 3 = 3 because 3 I s greater than 1; 1 * 4 - 4 since 4 l a greater than 1, e t c , In fact,

no matter what counting number a is. So 1 is t h e

Identity for this meaning of the operation *.

We could state this formally as follows: If stands

f o r a b i n a r y operation on a set of elements and if there i a some

element, c a l l it e , which has the property tha t

f o r every element a of the set, then e I s cal led an Identity element of the operat ion *.

As another example consider the followfng t a b l e f o r an operatlon which we m i g h t call # .

[aec. 12-51

Is there an identity element for 4s ? Could it be A?

f a A # 3 = B? (Read " A sharp B equals B"). Since, from the table A # B = C, the answer to the question I s "no" and we

see t h a t A cannot be the identity. Neither can 3 be the ident l ty since A # B I s not A. However, D is an i d e n t i t y f o r # , since

A # D = D # A = A ,

B # D = D # B = B ,

C # . D = D * C = C ,

D # D = D .

Compare t h e column under D with the column under the #. Compare the row to the r l gh t of D with the row t o the right of

the # . What do you not ice? Does this suggest a way to look for an identity element when you ape given a table for the operation?

If we have an identity element then we rnay also have what is ca l l ed an inverse element. If the operation is multiplication for rational numbers, the i d e n t i t y I s 1 and we c a l l two r a t i o n a l numbers a and b inverses of each other if their product is 1,

that is, if each is the reciprocal of the other. Suppose the operat ion is addi t ion (mod 4). Here 0 I s the

ident i ty element and we ca l l two numbers inverses If their sum is 0, that is, if combining the two numbers by the operation g ives 0. To f i n d inverses (mod 4) for addition, from the table :

( ~ o d 4 )

0 is the Identity + I 0 1 2 3

2 + 2 = 0 (mod 4)

I 3 + 1 5 0 (mod 4)

1 + 3 E 0 (mod 4)

: Here 0 is its own inverse, 2 1s its own inverse, and 3 and

I 1 are inverses of each other.

[sec. 12-51 .

Def in i t ion . Two elements a and b are fnverses (or either one is the Inverse of the other) under a binary operation * with ident i ty element e if a * b = e

and b * a = e . Write again the table for # which we had in the beginning

of this section.

Remember that we showed tha t D is the Identity element for this t a b l e .

Can you flnd an element of the a e t {A, B, C, D) which

will make the statement A # - = D true? It is C : A * C = D.

A and C are inverses of each other under # . Can you f i n d

any other elements with inverses under # ?

Exercises 12-5a

1. Study tables ( a ) - (d ) In Section 12-3,

(a) Which tablea describe operations having an identity and w h a t is the identity?

(b) Pick out pairs of elements which are inverses of each other under these operations. Does each member of the s e t have an inverse?

2. For each of the operations of Problem 4, Exercfses 12-3;

(a) Does the operation have an i d e n t i t y and, if so, what I s

It?

(b) Pick out pairs of elements which are Inverses of each other under these operations.

( c ) For which operations does each element have an inverse?

*3. Can there be more than one i d e n t l t y element for a given binary operation?

If the operation is multiplicatlon we call inverses multlplicative inverses. (The multiplicative inverse of a rational number is its reciprocal.) Conaider the set of counting numbers with multiplication as the operation.' What elements have multi- plicative inverses? Does 5 have a multiplicative Inverse In the

1 s e t of counting numbere? Is 5 a counting number? The element 5 has no multiplicative inverse In t h i s set. Does 1 have a multiplicative inverse in thia set? Yes, it does, for 1 1 = 1,

It is the only element of this set which ha8 a multiplicative inverse, and I t is its own multiplicative inverae. Of course, if

in the table. There are 4 ones in the tabla. They tell us that 1 1 E 1 (mod 51, 2 3 1 (mod 5 ) , 3 - E 1 (mod 5 ) , and 4 - - 1 (mod 5 ) . (YOU supply the missing numbers. )

we expand the set under consideration to include a l l of the rational numbers except zero then each element has s multipl icat ive inverse.

4 The numbers In the p a i n 5 and $, 1 and 1, 3 and 3 are multiplicative inverses of each other. Does 0 have a multiplica-

t ive inverse? Is there any number b euch that 0 b = l? Recall the (mod 5) multiplicatlon.

X

0

1

2

3 4

0 1 2 3 4 0 0 0 0 0

0 1 2 3 4 0 2 4 1 3 0 3 1 4 2

0 4 3 2 1

How would we decide what elements of the set 10, 1, 2, 3, 4 )

have (mu1 tip1 icative) inverses in this mathematical sya tern? The identity for multiplication (mod 5) I s 1. We would be looking

for ppoducts which are the identity, so we should look for ones

Thus the multiplicative inverse of 2 in (mod 5 ) is 3 . \!hat

is the r n u l t i p l i c a t l v e Inverse of 3 in (mod 5 ) ? of 4 ? Do you see any connection between multipllcatlve inverses

and the proper ty of closure under d i v f sion?' Suppose you are

given a s e t S of numbers which is closed under multiplication and suppose a is an element of S . Bow would you know whether I t is possible to "divide by a i n s"? That is, when is it possible t o divide any element of S (including a ) by a and

obtain another element of S?

First of a l l , S must contain 1, since a + a = For

instance if S viere a s e t of r a t i o n a l numbers closed under 1 1 were in the s e t , then - + p = 1 multiplication, and if 2 2

would also have to be in t h e s e t .

Second, since 1 is in S, S must contain 1 a a = 1 a' no m a t t e r what element of S a stands fop. Thls means tha t if

1 -

2 is in S , then must also be in S. If 7 1s in S , then

2 must be in S. S cannot contain zero s fnce 1 + 0 has no

meaning.

T h l r d , I f b is any element of S and 1 is In S , then a

1 is an element of S . For instance, if 2 is in S and is J

in S, then 2 5. = * l s a l s o i n S. 3 If S is t o be closed under d i v i s i o n it must be poss ib le t o

divide any element of S by any element of S. Thus S must

conta in 1, every element of S m u s t have a multiplicative

inverse in S, and we must be able to d i v i d e every element by

every o ther element. might not be If the system were not commutative, b . -

b a b which means that , might mean two different equal t o a

things. So in this chapter we consider division only when

multiplication is commutative.

[sec. 12-51

! We can summarize what we have learned: Let S be a set of

, numbers closed under multiplication where multiplication is ' commutative . Then :

If S is closed under division, S contains the number 1,

and every element of S has a multiplicative inverse in S. 1 (1f a is in S then is also in S.)

Also, the other way around,

If S con ta in s 1 and if every element of S has its

multfplicative inverse in S, then S is closed under

divl s ion . Perhaps you can now see another reason why we c a l l division

the inverse operation for multiplication: Dividing by a number a I s the same as multiplying by the

mu1 t i p l i c a t ive inverse of a ,

For instance, if S is the s e t of rational numbers with zero 1 excluded, is the multiplicative inverse (reciprocal ) of 2

1 and hence multiplying by 2 always gives the same resu l t as

d i v i d i n g by 2. If S were the s e t of numbers 0, 1, 2, 3 , 4 and multiplication were (mod 5) as in the table above, then,

since 3 is the inverse of 2, multiplying by 3 gives the same result as dividing by 2, that is, the values of x in

the two following equivalences are the same:

3 = 2x (mod 5 ) ; 1 3 = x (mod 5 ) .

In the first case x r 1 + 2 (mod 5 ) and In the second

x r 1 3 (mod 5 ) . Everything we have said about divlaion and, multiplicative

inverses can be said about subtraction and addi t ive inverses. Here again we consider subtraction only when addition is commuta- tive in the system. Consider first (mod 4) subtraction. What

I s 3 - 1 (mod 4)? Sfnce subtraction is the Inverse operation f o r addit ion, to f ind 3 - 1 (mod 4) we must f ind the missing

number in the sentence ? + 1 e 3 (mod 4).

To find the answer from t h e Table no t i ce that , since the t ab l e

gives sums, the 3 will be inside the table, and since 1 is the

number which is added it will appear a t the top of t he t a b l e . If

we look down the 1-column until we f i n d a 3 , we see t h a t it is in the 2-row. So 2 is the number which, when you add 1 to

it, you get 3 . Since the system is commutative, the answer t o

1 + ? G 3 (mod 4) is also 2; tha t is, if we look along the

1-row until we see a 3 , it will be in the 2-column. If t he

system were not commutative 3 - 3 would have two meanings,

which would be awkward. Irhat is 2 - 3 (mod 4)? Fhat number must we add t o 3 in (mod 4) to obta in 2? We see from the

t a b l e that 3 4- 3 = 2 (mod 4), so 2 - g E 3 (mod 4 ) . What is

1 - 3 (mod 4)? Now l e t us ask another kind of question. Is there any

number which we can - add t o 2 t o obtain 2 - 7 (mod 4)? Now 2 - 3 3 (mod 4) since 2 = 3 + 3 (mod 4 ) . If we look a t the

table we see t h a t 2 + 1 r 3 (mod 4) and hence

2 - 3 = 2 + 1 (mod 4) .

This means that if we s u b t r a c t 3 from 2 we have the same

r e s u l t as if we add 1 to 2. In other words, adding 1 t o 2

gives t h e same r e su l t as subtracting its inverse, 3 , from 2.

In the same way you should show t h a t

1 - 3 E I + 1 (mod 4)

3 - 1'3 + 3 (mod 4).

555

i From the first two of these examples it appears that, in (mod 4) , : s u b t r a c t i n g 3 produces the same result as adding 1 . Is this

1 always true in t h l s system? Is 0 - 3 a 0 + 1 (mod 4 ) ? Is

[ 3 - 3 = 3 + 1 (mod 4)? I

What is t h e relationship between 1 and 3 In (sod 4 ) ? I Since 1 + 3 I 0 (mod 4), and 0 is the i d e n t l t y under addition

! i n (mod 4), what do we say about 1 and 3? They are a d d i t i v e

: inverses of each other.

I Perhaps you can guess a general pr inc ip le from this example. i We observe that:

Sub t r ac t i ng a number produces the same resu l t as adding

the a d d l t i v e inverse of the number.

Thfs p r i n c i p l e w l l l be true in any commutative system where ' we c a l l an operat ion "addition" and where the elements have i / Inverses . A l s o , similar to a proper ty which we have observed f o r

1 multiplication, we have: 1 A s e t which I s closed under addition (where addition is I

commutative) wlll be closed under subtracLion if it contains I i 0 and con ta in s the additive inverse of each of i t s members.

Notice that In addition (mod n) we have ou r first examples / of sets whleh are closed under subtraction. Nowhere in our study I !of t h e c o u n t l n g numbers, the whole numbers, and the r a t i o n a l 1 numbers t h i s year have we had a d d i t i v e inverses , except that in , a11 these systems the number zero is its own additive i nve r se .

In the following exercises you will be given t h e chance t o

1 t e s t these general p r i n c i p l e s f u r t h e r . t

Exercises 12-5b

1. ( a ) U s e the multiplication table for (mod 6 ) to find, wherever poss ib le , a replacement f o r x to make each of the following number sentences a t r u e statement:

1 . x = 1 (mod 6 ) 4x = l (mod 6) 2x =- 1 (mod 6) 5x = 1 (mod 6) 3x = 1 (mod 6 )

(b) Which elements of the set [O, 1, 2, 3 , 4, 5 1 have mu1 tiplicative inverses In (mod 6)?

2. Remember that divis ion is defined as the inverse operation for multiplication. Thus, in the arithmetic of rational numbers, the question "Six divlded by two is what?" means,

really, It Six is obtalned by multiplying two by what? " We

can def ine division (mod n) In this way:

6 + 4 = ? (mod5) means ( 4 ) ( ? ) s 6 (mod 5).

Copy and complete the following table using multiplication

and division ( m o d 5) .

3 . Here is a t a b l e f o r addition (mod 5 ) .

Copy and complete the f 01 lowing t a b l e .

m o d 5 )

4, In the arithmetic of r a t i o n a l numbers which of the following

s e t s is closed under d iv i s ion? 1

(a) (1, 2,

* 23, ... I (b) C1, 2, 2 ,

( c ) The non-zero counting numbers.

b

0

2

4

2

3

1 2 4 3

4

(d ) The rational numbers.

additive inverse of a

4

a

1

1

1

1 2

2

2

4

4

b - a

0 - 1 = 4

2 - 1 G

add1 t ive + inverse of a

0 + 4 = 4

2 + 4 ~

5 . (a) Which of the following s e t s I s closed under multfplication (mod 6)?

lo, 1, 2, 3 , 4, 53, C2, 41, to, 1, 531

53, E53?

(b) Which of the sets in (a) contain a multiplicative inverse (mod 6 ) for each of its elements?

( c ) Which of the sets in (a) is closed under d i v i s i o n (mod 6)?

6 . ( a ) Whichof the s e t s [A, B), [c, D], {B, C, PI, {A, D) is closed under the operation defined by the table below?

For instance, ( A , D) is c losed under the operation

because if we pick out that p a r t of the table we have the llttle table

which contains only A's and D's. On the other hand the set {A, C) Is not c losed s ince i t s little tab le would be

Here the table contains a D, which I s not one of the

s e t [A, C 1 . [sec. 12-53

(b) Is there an identity for *? If so, what Is it?

( c ) Whlch of the s e t s in ( a ) has an inverse under * for each of It6 elements?

* ( d l Whlch of the s e t s in (a) is c losed under the

Inverse operat ion for *? (YOU m i g h t use the

symbols 2 for this operation, so t h a t a + b = ? - * * means b+ ? = a. )

1 12-6. What I3 a Mathematical System?

I The idea of a se t has been a very convenient one in this i t book -- some use has been made of it in almost every chapter . ; But there l a really not a great deal that can be done with just !

a set of elements. It is much more interesting if something can j i be done with the elements (for Instance, if the elements are t i numbers, they can be added or multiplied) . If we have a set and

i an operation defined on the s e t , it is interesting to f i n d out

: how the operation behaves. Is it commutative? assocfa t lve?

Is there an Identity element? Does each element have an inverse? The "behavior" of the arithmetic operat ions (addition, subtraction, multiplication and division) on numbers was discussed in Chapters

3 and 6 . We have seen that different operations may "behave al ike" in some ways (both c o m t a t l v e , for ins tance) . This

suggests t ha t we study sets w i t h operations defined on them t o

see what different poeaibilities there are . It is too hard for us to l i s t a l l the possibilit~es, but some examples w i l l be given in t h i s section and the next. These are examples of mathematical systems .

Def in i t i on . A mathematical system is a set of elements

together with one or more binary operations deffned on the set.

The elements do not have t o be numbers. They may be any

objects whatever. Some of the examples below are concerned with letters or geometric figures instead of numbers.

Example - 1: Letts look at egg-timer arithmetic -- arithmetic mod 3 .

(a) There is a set of elemente the a e t of numbers

(0, 1, 23

(b) There is an operatfon + mod 3, defined on the set (0, 1, 23.

Mod 3

Therefore, egg-timer arithmetic is a mathematical system. Does this system have any interestfng properties?

( c ) The operation, + mod 3, has the commutative property. Can you tell by the table? If eo, how?

We can check some special carsee too. 1 + 2 = 0

(mod 3 ) and 2 + 1 H 0 (mod 3 ) , so 1 + 2 = 2 + 1 (mod 3 1 .

fd) There I s an Identity f o r the operation + mod 3 (the number 0).

( e ) Each element of the s e t has an inverse for the operation + mod 3 .

Study the following tables.

I Exercises 12-6 -

1. Which one, or ones, of the Tables (a), (b) , ( c ) describes

1 a mathematical system? Show that your answer is correct.

i 2 . Use the tab le s above to complete the following statements

correctly . (a) B o A = ? ( e ) Q * R = ? (i) \ - 0 = ?

(b) A - O = ? ( f ) R * S = ? (j) B O B = ?

( c ] \ \ = ? (g) P * R = ? (k) A 0 A = ?

(a) A 0 B = ? (h) - 0 = 2. ) S " S = ?

3 . Which one, or ones, of the binary operations 0, *, is

commutative? Show that your answer is correct.

4. Which one, or ones, of the binary operations o, *, - has

an Identity element? What is it in each case?

5 Use t h e t ab les above to complete the following statements

c o r r e c t l y .

( a ) P * ( Q * R) = ? (f) R * ( F * S) = ?

(b) ( P * Q ) * . R = ? ( g ) A - ( A - \ ) = ?

( c ) F * (Q * S) = ? (h ) ( A ) \ = ?

( d ) (P * Q ) * S = ? ) A = ?

( e ) (R * P ) * s = ? ( 5 ) O - ( u - ~ j = ?

6 . Does e i t h e r of the operations described by Table (b) or Table ( c ) seem to be associative? Why? How could you prove your statement? \,%at would another person have to do to

prove you wrong?

7 . ( a ) In Table ( c ) what set is generated by the element ?

( b ) In Table ( b ) what s e t is generated by the element P?

8. BRAINBUSTER. For each of the following tables, tell why it

does - not describe a mathematical system.

anumber between 3 0

and 8

( b ) ++

2

12-7. Mathematical Systems without Numbera . I n the last s e c t i o n there were some examples of mathematical

systems wi thou t numbers in them. Suppose we want to invent one.

What do we need?

1 2

the product the sum of 3 and of 2 and 6 4

563

We must have a se t of t h ings . Then, we need some kind of a binary operation -- something that can be done with any two

elements of our s e t . We have found t h a t the properties of closure, commutativity, a s s o c i a t i v i t y , e t c . &re very helpful in simplifying expressions. It would be n lce to have some of these prope r t i e s . I Let ' s start with a card. Any rectangular shaped card w i l l

d o . We w i l l use It t o represent a closed rectangular region . Lay the card on your desk and label

the corners as in t he sketch. Now pick t he card up and w r i t e the letter 1 1 ~ 1 1 on the o ther s ide (the side t h a t

was touching t h e desk) behind the " A "

you have already writ ten. Be su re t he two letters " A " are back- to-back so they are l a b e l s f o r the same corner of t h e card.

Similar ly , label t h e corners 3, C, and D on the other s lde of t h e card (be sure they're back-to-back with the B, C, and

D you have already written.) What s e t sha l l we take? Instead of numbers, l e t us take

elements which have something t o do w i t h t he card. Start with

t h e card i n the center of your desk and with t h e long s ides of the card parallel t o the front of your desk. Now move the card -- pick it up, turn it over or around in any way -- and put it back

in the center of your desk with the long sides para l l e l to the f r o n t of your desk. The card looks just the same as it did before, bu t the corners may be labe led d i f f e r e n t l y (a corner tha t started

a t the top may now be at the bottom, for instance). The position

of the card has been changed, but the closed rectangular region

looks as it did in the beginning. h he "pictur~" stays the same.

Ind iv idua l points may be moved.) The elements of our set w i l l be these changes of p o s i t i o n . We will take a l l the changes of position t h a t make the closed rectangular region look as it did

in the beginning. ( ~ o n g sides parallel t o the front of the desk.)

How many of these changes are there?

We may start with the card in some position which we will

ca l l the standard position. Suppoae it looks l i k e the f igure below.

Leaving the card on your desk, rotate it half way around ita center. A diagram of this change is:

Standard Posltlon

half way

around C gives :

Since the letters " A " , "B", e t c . are only used as a convenlence t o label the different corners of the card, we will not bother to write them upside down. The diagram below represents thia change of position, and we will c a l l the change

"R" (for rotation).

Rotate the

card ha1 f way around.

What would happen if the card were rotated one fou r th of the way around?

one fourth

of the way around C

Does the card Look the same before and after the change? No,

this change of position cannot be I n our set , since the t w o picturea are quite different.

Are there other changes of position of the closed rec tangular region which make it look t h e way it did in the beginning? Yes,

we can f l i p t he card over in two di f ferent ways as shown by the

diagrams below: H :

f F l i p the card - -------- - over, using a hor i zon ta l axis.

I v: Fl ip the card over, using a

I ver t i ca l a x i s .

Now you know why you had to label both sides of the card so

carefully. Remember, the card only represents a geometrlc figure

for us. Turning over a card makes it different -- you see the other side; but turning over the closed rectangular region would

not make i t d i f f e r e n t (of course, some of the indivldual p o i n t s

would be In di f fe ren t positions, but the whole geometric f igu re would look Just the same).

There i a one more change of position which we must consider.

It its the change which leaves the card alone (or puts each individual point back in place) . Let us c a l l It "I."

I :

Leave the card In place.

Now we have our set of elements; it I s I, B, A, R .

Let us summarize what they are f o r easy reference:

Element I: Leave the card in place

Zlernent V: F l i p the card over using a

vert ical axis

Element H : (n F l i p t h e c a r d -------

over using a

hor i zon ta l axis

Element R :

Rotate the card halfway around in the d i r e c t i on indicated

Recall the definition of a mathematical system. There were two requirements:

( a ) A s e t of elements.

(b) One or more binary operations defined on the s e t

of elements.

Our set [I, V, H, R) sa t i s f ies the first condition. Now we

need t o satisfy the second condition; we need an opera t ion . What operation s h a l l we use? How can we "combine any two elements of

: our se t " t o g e t a "deflnlte thing"? If the s e t is t o be closed

under the operation, t h e " d e f i n i t e thing" which i s t h e r e s u l t of the operation should be one of the elements again,

Here is a way of' combining any two elements or our s e t . We

w i l l do one of the changes AND THEN do t h e other one. We will

use the symbol "ANTH" for thls operation (perhaps you can think

of a better one). Thus "H ANTH V" means f l i p the card over, u s i n g a horizontal axis, and then flip t h e card over, using a

v e r t i c a l axis." Start w i t h the card in the standard position and do these changes to it. What is the f i n a l position of the card?

I s the r e s u l t of t he se two changes the same as t h e change R?

What does "V ANTH H" mean? ANTH I I V H R

operation. Some of the entries R H

are given in the table at the H ' 1 H R

Try it w l t h your card. Now we

can f i l l in the table for our I

right.

I v

Exercises 12-7

1. Check the e n t r i e s that are given in the table above and

f i n d the others. Use your card.

2 . From your table f o r the operation ANTH, or by actually

moving a card, fill In each of the blanks to make the

equations correct.

( a ) R ANTH H = ?

(b) R ANTH ? = H

( c ) ? APJTH R = H

( d ) ? ANTH H = R

( e ) (R ANTH H) ANTK V = ?

(f) R ANTH (H ANTH V) = ?

(g) (R ANTH H) ANTH ? = V

(h) (R ANTH ? ) ANTH V = H

(1) ( ? ANTH H) ANTH V = R

3 . Examine the table for the operation ANTH.

( a ) Is t h e set closed under the operation?

(b) Is the operatfon commutative?

( c ) Do you think the operation is associative? Use the operation table to check several examples.

( d ) Is there an i d e n t i t y element f o r the operation ANTH?

( e ) Does each element of the set have an inverse under the ope rat ion ANTH?

4. Here is another system of changes. I I

Cut +a triangular card with two equal sides. Label the corners

as in the sketch (both sides, back- to-back). The s e t for the system

will corisist of two changes. The

first change, ca l led I, will be:

Leave t h e card in place . The

second change, called F, wfll be: I

F l i p the card over, using the I

ver t ica l axis . F ANTH I will

mean: F l l p t h e card over, us ing

the v e r t i c a l axis, and then leave the card in place . How

wfll the card look -- as If it had been l e f t in place, I, o r as If the change F had been done? What does I ANTH F

mean? Does R ANTH I = F or does F ANTH I = I?

(a) Complete the table below:

( b ) Is the set closed under this operation?

( c ) Is the operation commutative?

( d ) Is the operation associative? Are you sure?

( e ) Is there an identity for the operation?

(f) Does each element of the set have an inverse under the

operation?

5. Make a t r iangular card with three equal sides and label t h e

corners as in the sketch (both sides, back-to-back) . The

s e t fo r this system will be made up of these six changes. I: Leave the card in place. R: Rotate the card clock- wise of the way around. 7 S: Rotate t he card clock- wise of the way around. 7 T: lip the card over, using a vert ical axis. U: F l i p the card over, using

an axis through the lower right

vertex. I

V: F l i p t he card over, using an a x l s through the lower left

vertex. Three of theae wfll be ro ta t ions about the center

A 0 , ?: -. (leave in place and two others). 4 0 x ..

The other three will be flips # \ I I

about the axe,s. (caution: the

axes are stationary; they do not r o t a t e with the card. For

example, the vertical axis remains ve r t i ca l -- it would go

through a d i f f e r e n t corner of t h e card after r o t a t i n g the card one t h i r d of the way around its center.) Make a t a b l e

f o r these changes. Examine the table. Is this operation commutative? Is there an i d e n t i t y change? Does each change have an inverse?

" 6 . Try making a table of changes f o r a square card . There are eight changes ( t ha t I s , efght elements). What are they? Is there an Identity element? Is the operation ANTH

commutative?

12-8. - The Counting Numbers and the Whole Numbera. --- The mathematical systems that we have studied so far in this

chapter are composed of a s e t and one operation. Examples are modular addl t ion or multiplication and the changes of a rectangular or t r iangular card. A mathematical system given by a set and two operations would appear to be more complicated than

these examples. However, as you may have guessed, ordinary

arithmetic is a l s o a mathematical system and we know that we can do more than one operation using the same set of numbers -- f o r examples, we can add and multiply.

To be definite, let us choose the s e t of rational numbers.

This set, together with the two operations of a d d i t i o n and multiplication forms a mathematical system which was discussed in

Chapters 6 and 8. Are there properties of this system which are entfrely different from those we have considered in systems with

only one operation? Yes, you are familiar wl th the f ac t that 2 ( 3 + 5 ) = (2 3 ) +I- ( 2 5 ) . This is an illustration of the

distributive property. More precise ly , it illustrates that

multiplication distributes over add i t i on . The dist~ibutive

property is also of in te res t in other mathematical systems.

Def;nl?lon. Supnose we have a s e t and t ~ i o binary o p e r a t i o n s ,

and o , defined on the s e t . The ogeration dLs-

t r l t x t e s over t h e operzkion o if n + (k o c > = (2 -* b) o

(a * c ) for any elements a, b, c; of t h e s e t . ( ~ n d

we can p c r f o r ~ a l l these ope ra t i ons . )

In a nzthematical system w i t h t w o operations, t he re are t h e

properties which we previogsly discussed for each of these

ogerations sepa ra t e ly . The o n l y p r o p e r t y which is concerned ~ 9 t h

both opera t ion: t oge the r is the distributive p r o p e r t y .

Exercises 12-8

L . Conslder the s e t of counting nuxbers:

( 2 ) Is t h e s e t c l o s e d under a d d i t k o n ? under multiplication?

Expl a l n .

( b ) Do t h e commu5atlve and ~ z s o c i a t i v e properties ho ld f o r addition? f o r multiplication? Give an example of' each.

( c ) PLha",s sthe identity element f o r a d d i t h n ? for

multiplication?

(d) Is $he set of counting numbers closed unde r s u b t r a c t i o n ?

under division? Explain.

The answers to (a), (b), and ( c ) t e l l u s some of the

properties of t h e rna5hematical system composed of t h e s e t of

c o u n t i n g numbers and the ope ra t l ons of addition and

multiplication.

2 . Answer the ques t ions o f Problem 1 ( a ) , (b) ( c ) for the s e t

of whole numbers. Are your answers t h e same as f o r t h e

c o u n t i n g nurnbers?

3 . ( a ) For t h e system of whole numbers, write t h r e e number sentences illustrating that multiplication dlstributes

over addition.

(b) Does addition distribute over multiplication? T r y some

examples.

4. The t w o tables below describe a mathematical system composed of the set [A , B, C, D) and the two operations * and o .

(a) Do you think * distributes over o? Try several examples.

(b) Do you th ink o di s t r ibute s over *? Try several

examples . 5 . Answer these questions f o r each of the following systems.

Is the s e t closed under the operation? Is the operation comuta t ive? associative? Is there an identity? What

elements have inverses?

(a ) The system whose s e t Is the set of odd numbers and whose operation is multiplication.

(b) The system whose set is made up of zero and the

multiples of 3 , and whose operat ion is multiplication.

( c ) The system whose s e t is made up of zero and the multiple2 of 3 and whose operation is addit ion.

( d ) The system whose set is made up of the ra t ional numbers

between 0 and 1 (no t including 0 and 11, and

whose operation is multiplication.

( e ) The system whose s e t is made up of the even numbers and

whose operat ion is addition. (zero is an even number.)

( f ) The system whose s e t is made up of the rational numbers

between 0 and 1, and whose operation is addition.

6 . (a ) In what ways are the systems of 5(b) and 5 ( c the same

(b) ~n what ways are t he ayatems of 5(a) and 5(b) different?

"7. Make up a mathematical system of your own t h a t is composed of a s e t and two operations def ined on the s e t . Make a t least

p a r t i a l tables for the operations in your system. L i s t the properties of your system.

*8. Here is a mathematical system composed or a set and two operations def ined on that s e t .

Set: A l l counting numbers.

Operation *: FZnd the greatest common factor.

Operation o : Find the l e a s t common multiple.

(a) Does the operation * seem to distribute over the operat ion o? T r y several examples.

(b) Does the operat ion o seem to distribute over the

operat ion *? Try aeveral examples.

12-9. Modular Arithmetic.

In Section 12-1 we studied a new addition done by rotating the hand of a clock. Using a four-minute clock, we said that 2 + 3 = 1 (mod 4 ) . The tables which we made described the

mathematical system (mod 4 ) . In Sectfon 12-2 we studied a new

mu1 t i p l i c a t ion u s i n g the same c lock . Modular systems are the resu l t of classifying whole numbers

in a ce r t a i n way. For example, we could classify whole numbers as

even or odd. In t h i s case, t h e even numbers: 0, 2, 4, 6 , ... are put In t h e same family and the family I s named by its smallest

member: 0. Thus the class of a l l even numbers is 0 (mod 2).

Sta r t ing from 1, the odd numbers: 1, , 5 , 7, ... are pu t

in the same family which we call 1 (mod 2). For the odds'and

evens, we then have two classes, 0 (mod 2) and 1 (mod 2). The number 5 belongs to the c l a s s 1 (mod 21, 8 belongs to the

class 0 (mod 2).

If we put every four th whole number in the same class, we

have the (mod 4) system. Here is a sketch of some of the

numbers belonging to the c l a s s 0 (mod 4) .

Every fourth whole number starting with 0 belongs to the

same c l a s s . Thus, numbers which are multiples of 4 belong t o

t he class 0 (mod 4 ) . Here is a sketch showing some of the numbers which belong to

the class 1 (mod 4).

Every fourth whole number atarting with 1 belongs to the same class, t h a t is, 1 (mod 4). Thus the numbers whfch are 1

plua a multiple of 4 belong to t h i s c lass . The two sketches below show respectively some of the numbers

which belong t o the class 0 (mod 5 ) and the class 3 (mod 5) .

The numbers belonging t o the class 0 (mod 5 ) are multiples of 5.

The numbers belonging to the class 3 (mod 5 ) are 3 plus

multiples of 5. Our first problems in a modular system used the operation of

addition. When we changed the operation t o multiplication, we got a di f fe ren t mathematical system. With both ope~ations,

modular arithmet3c is more like ordinary arithmetic than it was

w l th j u s t one operation.

For each of the modular systems we can state the number of

elements in the s e t . For instance, there are f o u r elements if

I t is (mod 41, seven elements if it is (mod 71, and so f o r t h ,

Such a set is ca l l ed a f i n i t e se t and the system l a called a

f i n i t e system. The modular systems and the systems of Section 12-7 are finite systems. On the other hand, the set of ra t iona l numbers considered i n Section 12-8 is so large that it contains more elements than any number you could name. Such a set is called an i n f i n i t e s e t and the system is called an i n f i n i t e system.

Exercises 12-9

1. Write the multiplication table (mod 8) and recall or write again the multiplication table (mod 5) which you found in

Exercises 12-2.

2. Answer each of the following questions about t h e mathematical

systems of multiplicat~on (mod 5 ) and (mod 8).

(a) Is the set closed under the operation?

(b) I8 the operation commutative?

( c ) Do you think the operation is associative?

( d ) What is the identity element?

( e ) Which elements have inverses, and what are the p a i r s of inverse elements?

(f) Is it t r u e t h a t if a product I s zero at l e a s t one of the

f a c t o r s is zero?

3 . Complete each of t he following number sentences t o make it a

true statement.

(a) 2 x 4 =- ? (mod 5 ) ( C ) 5* = 1 (mod ? )

(b) 4 x 3 r ? (mod 5 ) (d) d = 0 (mod ? )

4. Find the products:

(a) 2 x 3 = ? (mod 4) ( e ) 4 3 = ? (mod 5)

(b) 2 x 3 = ? (mod 6) ( f ) 6 * = ? (mod5)

( c ) 5 x 8 ? (mod 7 ) * ( g ) 6 256 ? (mod 5 )

( d ) 3 x 4 x 6 = ? (mod 9 )

5. Find the sums:

(a) 1 + 3 = ? (mod 5) ( c ) 2 + 4 = ? (mod 5)

(b) 4 I- 3 r ? (mod 5 ) ( d ) 4 + 4 = ? (mod 5)

6 . ( a ) Find the values of 3(2 + 1) (mod 5) and (3 2) + ( 3 - 1) (mod 5 ) .

(b) Ffnd t h e values of 4(3 + I) (mod 5) and ( 4 3 ) -k

( 4 1) (mod 5 ) .

( c ) Find the values of ( 3 2) + ( 3 - 4 ) (mod 5 ) and

3 ( 2 + 4) (mod 5 ) .

(d) In t h e examples of t h i s problem is multiplication

distributive over addition?

7 . (a) Find the values of 3 + (2 . 1) (mod 5 ) and ( 3 + 2)

3 + (mod 5 ) .

(b) Find the values of 4 + (3 1) (mod 5 ) and ( 4 + 3 ) . ( 4 + 1 ) ( m o d 5 ) .

( c ) Find the values of ( 3 + 2 ) ( 3 + 4) (mod 5 ) and

3 + ( 2 . 4) (mod 5 ) .

( d ) In t h e examples of this problem 2s a d d i t i o n distributive

over mu l t ip l i ca t ion?

Remember t h a t d i v i s i o n i s defined af ter we know about multiplication. Thus, in ordinary ar i thmetic , the

ques t1 on "Six divided by 2 is what?lt means, real ly "Six is obtained by multiplying 2 by what?" An

operation that begins with one of the numbers and the

"ar:swer1I to another binary operation and asks f o r the

other number, is called an inverse opera t ion . Division

is the inverse of the multiplication operation.

x8, Find the quotients?

(a) 2 + 3 = ? (mod 8) ( e ) 0 + 2 s ? (mod 5 )

(b) 6 + 2 = ? (mod 8) ( f ) 0 9 4 = ? (mod 5 )

( c ) 0 a 2 G ? (mod 8) (g) 7 + 3 ? (mod 10)

( d ) 3 + 4 = ? (mod 5) *(h) 7 + 6 = ? (mod 8)

9 . Find the following; remember t h a t subtraction i s the inverse operation of add i t i on .

(a) 7 - 3 (mod 8) ( c ) 3 - 4 (mod 8)

( b ) 3 -. 4 (mod 5 ) *(a) 4 - 9 (mod 12)

10. Make a table for sub t r ac t ion mod 5. Is the s e t closed under the operat ion?

11. Find a replacement for x whlch will make each of the

fol lowing number sentences a t r u e statement. Explain.

(a ) 2x = 1 (mod 5 ) ( d ) 3x = 0 (mod 6 )

(b) 3x 1 (mod 4) ( e ) x . x = 1 (mod 8)

(c) 3x = 0 (mod 5 ) (f) 4x = 4 (mod 8)

12. In Problem 11 ( d ) and (f), find at least one o ther replacement f o r x which makes the number sentence a true statement.

12-10. Summary and Review. A binary operation defined on a s e t is a rule of combination

by means of which any two elements of the s e t may be combined to

determine one d e f i n i t e thing.

A mathematical system is a set toge ther with one or more

binary operations defined on that s e t .

A s e t is closed under a binary operation if every two elements of the s e t can be combined by the operation and the

r e s u l t is always an element of the s e t .

An i d e n t i t y element for a binary operation def ined on a s e t

is an element of t he s e t which does not change any element w l t h

which it i s combined.

Two elements are inverses of each o ther under a cer ta in binary opera t ion if the r e s u l t of this operation on the two

elements is an i d e n t i t y element f o r t h a t operation.

A binary opera t ion is commutative if, f o r any two elements, the same resu l t is obtained by combining them first in one order, and then In the o t h e r .

A binary operatlon is * associative If, f o r any t h r e e elements, t h e r e su l t of combining the first w i t h the combination

of t h e second and t h i r d is the same as the r e s u l t of combining

the combination of the f irst and second wlth t h e t h i r d .

The binary operat lon distributes over the binary

operation o provided

f o r a l l elements a, b, c .

A s e t S is generated by an element b under the operatlon * if

S = b, ( b * b), (t * 13) * t, [(b * b] * b] * b, ...) {

Chapter 13

STATISTICS AND GRAPHS

13-1. Gathering Data.

If you look a t the pupi l s in your classroom it may occur to

you tha t several pupils appear to be about the same he igh t , that

some p u p i l s are taller and some are shorter. Suppose you want to

know the height of the t a l l e s t , t he height of the shortest, and

how many are the same height. Ha!, w i l l you go about finding t h i s

information? You would f i r s t measure the height of each p u p i l .

In t h i s way you will be collecting facts t o answer the questions

you have in mind. Instead of collecting f a c t s we may say you are

gathering data. h he word "data" is the p l u r a l of t h e Latin word t t datum" which means " fac t" . )

L e t us suppose t h a t you find each pupil's he igh t cor rec t to the nearest Inch. After this is done, t he measurements should be

arranged in such a way t h a t it is possible to select information

from the data as easily as poss ib l e . Making such an arrangement is done f requent ly by putting t h e data i n a t a b l e as shown i n Table 13-1. If you wished, you m i g h t list the pupils by name,

but here each p u p i l is assigned a number.

With t h e data arranged i n this way it I s very easy to answer such questions as the following. How t a l l is t h e t a l l e s t pupil?

t h e s h o r t e s t ? How many p u p i l s are 60 inches t a l l o r t a l l e r ? How many less than 60 inches t a l l ? What height occurs most

of t e n ?

Table 13-1

Heights of 15 Seventh Grade Pupils

Pupil Height 1 n Inches

This example about the heights of the pupils In a class 1s a sample of the kind of things we do In studying statistics. Statistics, In p a r t at l e a s t , has to do with the collection of data and the makfng of tables and char ts of numbers which represent the data. The tables and charts usual ly make it easfer to understand the information which is contained i n data that have been gathered. We will use the data in Table 13-1 l a t e r in this chapter t o i l l u s t r a t e some o the r thlngs t h a t we do in our study

of statistics . Many of the duties of di f fe ren t agencies in the U , S.

government could not be performed if the agencies were not able to collect a great many data t o use in their work. The Congress of the United S t a t e s has the power "to lay and c o l l e c t taxes --- to pay the debts and provide for the common defense and general

welfare of the United' Sta tes ." The amount of taxes to be col lected depends on many things. Name some of them. Certa in ly

one t h i n g on which it depends I s the number of people In the

United S t a t e s . The Congress must provide f o r counting the people "within e v e r y term of t e n years .'I The census taken in 1950 showed tha t there were about 151,000,000 people In the United

S t a t e s . Another census was taken in 1960. Ask your l i b r a r i a n

[ s e c . 13-11

to help you f ind the populat ion of the United States in 1960. Table 13-2 shows the population in rnilllons f o r every census

from 1790 through 1950. The table shows that the population in 1790 was 3.9 millions. This means there were 3,900,000

(3 .g x 1,000,000) people in the U. S. a t t h a t time. Is this an exact or approximate number? The column which Is headed Percent

of Increase shows the percent of increase in t he populat ion during the preceding ten-year period,

Table 13-2

Population Facts About the United States

-

Census Population in Increase in Percent of Years M i l l Ions Millions Increase

1790 3 99 1800 5 -3 1.4 35 0 1

1810 7 *2 1.9 36.4 1820 9 .6 2.4 33.1 1830 12.9 3 -3 33 *5 1840 17.1 4.2 32.7 1850 23.2 6 .i 35.9 1860 31.4 8.2 35 .6

1 1870 39 *a 8.4 26.6 ! 1880 50.2 10.4 26 .o

1890 62 .9 12.7 25 -5 1900 76 .o 13 .l 20.7 1910 92 .o 16.0 21 .o 1920 105.7 13 -7 14.9 1930 122.8 17.1 16 .I 1940 131 .7 8 .9 7 .2 1950 150.7 19 .O 14.5

*From Statistical Abstract of the United States, 1956.

Exercises 13-1

1. Do you see any general trends in the data shown in the table?

2. In which decade was the percent of increase the largest? Do

you know a reason for t h i s from your atudy of history?

3 . In which decade was the percent of increase the lowest? Can you explain this by history you have studied?

The Irish Famine occurred i n the years 1845, 1846, 1847. How m f g h t t h i s have affected the populat ion of the United States?

What was the percent of increase in population from 1870 to 1880?

Arrange these data in a log ica l order. Students' test scores on a mathematics t e s t were: 7&, 8@, 7 g s , 84$,

6 1 9 15$, 8@, lo@, 67$, 7 3 , 83$t 75$t 8W1 9 7% 7% 8% 8W.

Use an almanac or other reference material t o ffnd the information needed in the problems that follow.

List the populatlon of the largest city in your s t a t e for

the years 1900, 1910, 1920, 1930, 1940 and 1950. (use your city i f it is large enough to be l i s t ed , or if you can get

the figures.)

(a) L i s t the number of immigrants to the U. S. f o r each

year from 1935 to 1950.

(b) Arrange the numbers in (a) from the smallest to the

largest,

List the number of automobiles sold by the manufacturer i n the U. S . in each of these years: 1910, 1915, 1920, 1925,

1930, 1935, 1940, 1945, 1950, 19554

13-2. The Broken Line Graph. - - Such data as we talked about In the Jast section are

frequently represented by ''drawing a picture. " The "picture" f o r the data in Table 13-2 i a shown in Figure 13-1. This "picture"

is called a broken-line graph, as many of you already know. Such

graphs are uaually made on cross-section paper l i k e that used in

Figure 13-1. Broken l lne graphs are used to show change in some

item.

[aec. 13-21

POPULATION OF THE UNITED STATES

1790 - 1960

CENSUS YEARS

Figure 13-1

One hor i zon t a l line and one vertical line are used in such a way tha t they resemble a number l i n e . I n Figure 13-1, the

horizontal l i n e Is used t o show t i m e . Notlce t ha t each ten year

period is represented by the same distance along the line. The

vertical line shows the number of people. How many people does

each unit represent? These t w o l ines are t h e r e f e r ence l i ne s of - the graph. They show the scales used to draw the graph.

Mould the numbers on these scales have any meaning if the scales were not named? In fac t , would the graph mean much if it did not have a title? A good graph must be neat and "eye-

catching, " have a title that identifies the information clearly, and scales that are rneanlngful and easy to read.

Each point on the graph in Figure 13-1 represents the populatlon for the year whose number on the horizontal scale is directly below the point. The population for each point is read

by drawing a perpendicular from the point to the ver t ica l sca l e and reading the number there. Kwnbers used on graphs are usually approximate numbers so the number read for any poin t I s also an approximate number. To make the graph, the points that represent t h e population for each census year are plotted; then they are connected by. l i n e segments. The s e ~ e n t s between polnts give

estimates of t h e populatlon between census years. There is no assurance t h a t these estimates are accurate, however, since the

change may have been more rapid a t one time than at another.

Class Discussion Questions

Use Figure 13-1 t o answer the questions.

1. Did the population increase more between 1900 and 1910 than between 1800 and 1810?

2. Does the graph show a decrease In popula t ion over any ten-

year period?

3 . I f the population for 1e13 and 1820 had been the same how would this be shown i n t h e graph?

58 5

4. What was the approxlmate popula t ion I n 1945? i n 1895? How

much had it changed in the 50 years between these two dates?

5. If the population increases a t the same rate from 1950 t o

1960 as from 1940 to 1950 (if the graph climbs in a straight

l ine from 1940 to 19601, what should the populat ion be i n

1960? This portion of the graph is indicated by a dotted

I l i n e . If the census for 1960 is known at the time you answer

I the question, compare the census and your answer. I

Makin Line Graphs 2- I

When drawing graphs, the detai ls should be planned before any

, marks are pu t on paper. Look a t the information you wish t o show

and the space that is avaflable. Leave p l en ty of room for a title

and f o r l a b e l i n g scales . Graphs look better if a l l words are printed rather than written in long-hand. Graphs should be as

large as space permits. Rulers make l i n e s straight, and l i n e

segments enclosing the graph give t h e graph a finished appearance.

Line graphs require the use of scales. The biggest problem

in making a graph is deciding on the s ca l e . There are ways to tell how much each unit should represent . When graph paper is

used, the squares of the paper mark convenient u n i t s . A s an

example, we will use the data in Table 13-2. Count the number of

census years from 1790 to 1960. Since there are 18, t h e graph

must have at least 17 unlts along the hor izonta l sca le . (Why

are only 17 needed?) The horizontal scale always starts a t t h e left edge of the graph and a number is placed there . If' 3 : 7 ! i

unlts were available for use on the h o r i z o n t a l scale, what could

you do? Count the squares along the horizontal scale in Figure

13-1. Two squares were used for each census year In the graph, Next, look at the v e r t i c a l scale. It shows t he number of

millions of people i n t h e United S t a t e s . F r o m Table 13-2 f i n d the

largest number of people that needs to be shown on t h e graph.

The numbers in t h i s table should be rounded to t he nearest million before you work with them. Count the number of u n i t s from t h e

horizontal scale to the top of t h e space used. ( ~ e a v e room a t the

top for a t i t l e . ) In Figure 19-1 there are 35 units available

along t h e v e r t i c a l axis, and t h e largest nwnber of people is

l5l,OOO,OOO. Divfde t h e number of people by t h e number of u n i t s

available. I n t h i s example t h e quotient comes o u t with t he

a t roc ious number 4,314,285 and there is a remainder. According

t o this, each unlt should represent 4,314,286 people; it would be very d i f f i c u l t to work w i t h such a nwnber. Each unit of the v e r t i c a l s c a l e should represent t h e next larger, convenient numben In this case, each unit should represen t 5,000,000 people. hhy

must t h e number used for each unit be l a rge r than the quotient? Is it necessary to do t he e n t i r e division shown abovg? Since

we will use approximate numbers for the graph, the division may

also be approximate. The popula t ion was rounded t o t h e nearest million, The division needs to be carried o u t only until t h e

quotient shows the number of million. Now the problem is easy .

The r e s t o f t h e figures make no d i f f e r ence , s ince t h e number used

as a unit must be l a rger than t h e quotient.

Graphs are usually neater if only enough l i n e s of' t h e scales

are named so as t o make the scale e a s i l y readable .

Exercises 13-2

1. Find the number tha t each unit should represen t f o r each of the following situations.

Number of squares available

Largest number to be graphed

2 . Make a broken l i n e graph to represent t h e data g iven in t h i s

t a b l e :

The number of students in Frankl in Jun io r H i g h

School f o r years 1952-1957:

3 . rake a broken l i n e graph to represent the da ta in Table 13-3.

Table 13-3

Popular V o t e in h l i l l i ons Cast for Presidential Candidates

of the U. 3 . 1928 to 1956

Year Republican Party Democratic Party

Observe these instructfons:

( a ) In t h e same graph draw one broken line for the

Republican p a r t y and one f o r the Democratic par ty .

( a ) Find in your textbook, o r elsewhere, t h e name of the

president e lec t ed and the name of the unsuccessful

candidate i n each e l e c t i o n .

( c ) Use T a b l e s 13-2 and 13-3 t o find the total percent of

the popula t ion who voted f o r either the Republican o r Democratic pa r ty candidate in t h e presidential election

in 1940.

4. Make a l i n e -graph fo r the data f o r Problem 7, Exercise 1 3 - 1 .

5 . Make a l i n e graph for the data f'or Problem 8, Exercise 13-1.

6 . Make a line graph f a r the data f o r Problem 9, Exercise 13-1.

13-3. B a r Graphs.

Table 13-4 gives the p u p i l enrollment i n the seventh grade i n

t h e U. S. for t h e years 1952-1959 and the expected enrollment f o r the years 1960-1962.

Table 13-4

Seventh Grade Pupi l Enrollment in U . S. 1952-1962 Year Enrollment In thousands

"Expected enrollment fo r t h e years 1960-1962.

The data in this t a b l e are represented by t he p a p h in

Figure 13-2. This kind of graph is ca l l ed a - bar graph. Bar

graphs show comparison between similar i t e m s . Data sui table for

line graphs may also be s u i t a b l e fo r bar graphs. Data that show

~ h a n g e may also be considered as comparing similar sets; t h e

number associated with each t i m e period is then considered as a

s e t of similar items. There are some sets of data t h a t are

s u i t a b l e f o r bar graphs that are not good material f o r l i n e graphs.

An example would be a graph comparing the heights of the 10

highest mountains in North America. I n Figure 13-2 the years are represented along the hor i zon t a l base line and the enrollment in

thousands is represented along the ver t ica l l i n e at the l e f t . The bars are spaced a long the base l i n e so that the distance between

any two bars is the same. The width of each bar is t h e same also. In this graph, t h e width of the spaces and the bars are the same,

b u t this does not have to be t r u e . The name of each bar is the

number of the year p r i n t e d a t the bottom of each b a r . In some graphs, the bars will have names that are not numbers. When

planning a graph, be sure t o leave enough room t o p r i n t the name

of t h e ba r .

The number represented by each bar can be read from the

ver t i ca l scale . It is the number represented by the po in t on t h e

v e r t i c a l scale which is on the same hor i zon ta l line as t h e t op of the ba r .

In making bar graphs, be sure t o observe a11 p r i n c i p l e s f o r

constructing good graphs. Plan t o use the e n t i r e width o f available

space. Since t h e width f o r spaces may d l f f e r from the w i d t h ro r bars, thfs is possible, The number t h a t each unit on t h e v e r t i c a l

scale represents I s found in the same way as for' l i n e gtYaphs.

SEVENTH GRADE PUPIL ENROLLMENT IN U. S.

1952 - 1962

Figure 13-2

[sec . 13-31

Exercises 13-3

If each seventh grade pupil in 1960 needs a mathematics

textbook which c o s t s $3,25 what will be the t o t a l amount

spent fo r books to supply the whole seventh grade?

During which year was the enrollment about 3 million pupils?

Between which two years did the greatest change in enrollment occur? Use Figure 13-2 to obtain the answer, then use

Table 13-4 t o see if your answer is c o r r e c t .

D r a w a bar graph to represent the number of people killed in di f fe ren t types of accidents during 1956 as shown in this

table :

Motor vehicle accidents 40,000

Fa l l s 20,200

Flres and iniuries from f i res 6,500 Drownings 6,100

Railroad accidents 2,650

The hlghest altitude in each of the s t a t e s l i s t e d is given. Round the data to the nearest 100 feet, then show in a

bar graph.

Alabama 2,407 ft.

Alaska 20,320 ft.

Arizona 12,670 f t .

Arkansas 2,830 ft.

California 14,495 ft.

Colorado 14,431 f t .

At one time during t h e 1960 baseball season, the National

League teams had won the number of games shown, Show this in

a bar graph.

f i t t s b u r g h 64 San Franc 1 s c o 51 ~t . Louis 60 Cincinnati 46

Milwaukee 57 Philadelphia 42

Los Angeles 56 Chicago 39

[sec. 13-31

1 13-4. Circ le Graphs. I

I A c i r c l e graph is shown in Flgure 13-3. Such a graph is used to show t h e comparison between the p a r t s o f a whole and between

the whole and any of i t s parts. This graph shows the percent of income which a fam2ly spent fo r food, c l o t h i n g , rent and

miscellaneous expenses and the percent of income which was put into savings.

HOW ONE FAMILY SPENDS ITS MONEY

Figure 13-3

The family spent 30 percent of i t s Income for food, 20 percent

f o r c lo th ing , 20 percent f o r rent, 20 percent f o r miscel-

laneous expenses, and saved 10 pe rcen t , The family's total

income is represented by the area of t h e c i r c l e , Any length can

be chosen f o r the l ength of the r a d i u s . The length is usua l ly chosen so that t h e c i r c l e I s large enough to show c l ea r l y the p a r t s into which it is div ided and small enough to get it in the

available space . Since the par t of the income spent f o r food 1s 30 percent

we must have an area which is 30 percent of the circlets area

to represent this part of the income. To get this area we begin + by drawing the ray OP. Label with A the intersection po in t of

the ray and the c i r c l e . We know t h a t we can divide the c i r c l e into 360 equal parts by drawing 360 angles of 1 degree, each with i t s vertex at 0. T h i r t y percent of 360' is 108'. If + the protractor is l a i d along OA with t h e ver tex mark at O and

the 0' mark on OA, then the 108' mark will be on the ray 3.

The area bounded by the closed curve OAW is 30 percent o f the

area of the c l r c l e . Hence the I n t e r i o r of the closed curve OAQB

represents the pa r t of the income spent f o r food. What nercen t of the circlets area represents rent? Twenty +

percent of 360' is 72'. By placing your protractor along OB + with the vertex mark at O and the 0' mark on 03, then t h e

--* 72' mark will be on the ray OC. Continue in the same manner t o

check t h e area representing clo th lng , mlscellaneous, and savings. If the family's income is $6,000 how much is spent f o r

food? How much is put i n t o savings? In a certafn school there are 480 pupils. A t lunch time

80 pupils go home f o r lunch, 120 brlng their lunch, and 280

buy t h e i r lunch i n the school lunchroom, The circle graph i n Figure 13-4 shows the way the pupils are divided at lunch t ime . Before rnakfng thls c l r c l e graph we had to find either what percent or what fractional part of the pupils go home f o r lunch, bring their lunch, and buy their lunch . These f r ac t ions and percents

are shown in the table.

Number of P u p i l s -- Fractional Percent Degrees Part of Tota l

Go home 80 1 6 1% 60

Bring lunch 120 1 T 25 90

Buy lunch 280 7 E 21 0 -

Tota l 480

SOURCES OF PUPILSr LUNCHES

BUY LUNCH

Figure 13-4

Now we f i n d the number of degrees in the curve AB, by t ak ing 1 5 of 360 to g e t 60; the number of degrees in the curve BC

1 by t ak ing of 360 to get 90; and the number of degrees in 7 the curve CQA by t a k i n g of 360 to get 210.

Exercises 13-4

Make a c i r c l e graph t o represent the informat ion in each of t h e

fo l lowing problems . Make one graph for each problem. Round to

nearest degree .

1. In 1949 it was found that school-related accidents which

invo lved seventh grade p u p i l s in the U. S . happened aa shown.

( ~ i g u r e s given are very c l o s e approximations t o actual f igu res . )

60 percent of the accidents happened in the school

bulldings . 30 percent of the accidents happened on the school

playgrounds . 10 percent of the acc idents happened on the way t o and

from school.

2 . In 1956 the figures f o r t he accidents of Problem 1 were as

follows: gain, figures are very c lose approximations.)

36 percent of the accidents happened in t h e school

build1 ngs . 54 percent of the accidents happened on the school

playground . 10 percent of' t h e accidents happened on the way t o and

from schoo l .

j. The Drama Club earned money during the year to buy new

c u r t a i n s for the clubroom stage. About 7 of the c o s t of t h e cur ta ins was earned by selling t i c k e t s to a series of school plays . Approximately 1@ of the money needed came from s e l l i n g programs, and the remaining came from a candy sale .

4. At Washington 3r. High School 5@ of the pupils l i v e a

mile or less from school . Approximately 2w of them l i v e

more than a mile but l e s s than two miles from school, while

2 come from homes two miles or more away.

13-5. Summarizing Data. Some information can be determined e a s i l y by looking at all

the data in table form. Sometimes, however, a l a rge number of items in a tab le makes it confusing. In this case it m a y be

better t o descr ibe t h e data by using only a few numbers. Finding

anave rage of such a se t of numbers is o f t e n very helpful in studying t he data given to you.

Do you know how t o find an average? You have been dofng t h i s

Tor some time, but d i d you know there are several -- kinds of averages?

Arithmetic Mean When you c a l c u l a t e t h e average from a s e t of numerical grades

by adding the numerical grades and d i v l d i n g the sum by the number

of grades, you f i n d a niunber which you use as a representative of the numbers in the s e t . This useful average, with which you are

[sec. 13-51

already familiar, is called arithmetic mean o r t he mean. (When

the word mean is used in thfs chapter it always refers to the

arithmetic mean. ) - Let us look once more a t t h e heights recorded in T a b l e 15-1,

which is repr in ted below. Thls table g ives a lfst of the heights,

ordered from largest to smallest, of 15 pupils.

Table 13-1

Heights of 15 Seventh Grade Pupils

Pup11 Height in Inches

In describing this s e t of data, can we find one number which

we can use to represent the numbers in these measurements? One

such number would be the arithmetic mean. For this t a b l e the Sum of t h e hei tS - 874 , 58, average height (arithmetic mean) Is number of pu$ls -

T h l s commonly used measure can be found w i t h o u t a r ranging the data in any special way.

Median

Another way to obtain one number representing. the numbers in

a set of data Is to find a number so tha t ha l f of the numbers in the s e t are greater and half are less than the number found.

The median of a s e t of numbers is the m i d d l e one of the set when the numbers in the set ape arranged in order, elther from smaliest to largest o r from largest to smallest. In the s e t of

heights In Table 13-1 the middle number is 59, This Is the

median of the s e t . Half of the numbers are greater than 59 and half are less. Seven puptls are t a l l e r than 59 inches and seven

are s h o r t e r than 59 inches, If the number of elements in the set is even, there is no

middle number. Thus we must define the medfan for this situation.

If there is an even number of elements in the set, the median I s

commonly taken as the mean of the two middle numbers, For example,

in t h e s e t of numbers 8, 10, 11, 12, 14, 16, 17, 19 t he

two middle numbers a re 12 and 14. The median is 13, t h e mean - of 12 and 1 4 , although 13 is not in the set. Sometimes

several i tems are the same as the median. The se t of scores 12,

'k3, 15, 15, 15, 15, 16, 18, 19, 20 has 10 numbers in it.

The two middle ones are 15 and hence the median is 15. But the

third and four th scores are a l so 15 so that 15 is not a score such that 5 scores are smaller and 5 are larger than it.

I n the, set of salaries $2050, $2100, $2300, $2400,. $2500,

$2600, $2700, $2700, $2700, $3150 the median salary is $255Q The arithmetic mean I s $2520, The median and arithmetic mean are nearly equa l . But, if the largest salary had been $5150 instead

of $3150 the a r i t b e t 2 c mean would have been $2720 and the

median would have still been $2550. This illustrates that the

usefulness of the median in describing a set of numbers of ten lies in the fact that one number (or a few numbers in the s e t ) does - not af fec t the median as It does affect the arithmetic mean,

Mode - Which h e i a - t occurs more than any other in Table 13-l? How

many pupils have this height? T h i s height is called the mode.

In s e t s such as the natural numbers 1, 2, 3 , 4, 5 , ... no number occurs in the set more than once. But in a s e t of

data some number, o r numbers, may occur more than once. If a number occurs in the s e t of? data more often than any other number

it is called the mode. (we might say i t is the most fashionable.)

[aec, 13-51

There may be severa l modes. In Table 13-1 t he re was iust one, 61. In the s e t of sa la r les $2,050, $2,100, $2,300, $2,400,

$2,500, $2,600, $2,700, $2,700, $3,150 t h e mode is $2700.

But in the s e t of scores 19, 20, 21, 21, 21, 24, 26, 26, 26, 29, 30 there are two modes 21 and 26. ( ~ h e s e are equally fashionable.) If there had been another score of 21

in this s e t of scores , what would L h e mode have been? In Table

13-1 if t he 12th pupil were 55 inches t a l l how would this affect the mode?

Exercises 13 -5a

1. Find the mode of the following list of chapter t e s t scores:

79 , 94 , 85 , 81, 74, 85, 91, 87, 69, 85, 83.

2. From t h e scores in Problem 1, ffnd the:

( a ) Mean (b) Median

3 . The following annual salaries were received by a group of

ten employees :

$4,000, $6,0009 $12,500, 85,000, $7,000, $5,5009

$4,500, $5,000, $6,500, &5,ooQ.

(a) Find the mean of the data.

(b) How many salaries are greater than the mean?

( c ) How many salaries are less than the mean?

( d ) Does the mean seem to be a fair way to describe the typical salary for these employees?

( e ) Find the median of the s e t of data.

(f) Does the median seem to be a fair average to use f o r this data?

4. Following are the temperatures in degrees Fahrenheit at 6 p.m. for a two week period In a certain city: 47, 68, 58, 80, 42, 43, 68, 74, 43, 46, 48, 76, 48, 50

Find the ( a ) Mean (b) Median

[sec . 13-51

Grouping Data If you were listing heights of a very large group of pupils,

it m i g h t be inconvenient to list each one separately. It would be

simpler t o group the f igu res in some such way as this:

Height in Inches 62-64 59-61 56-58 53-55 50-52 47-49

Number of pupils

In order t o f i n d the median first find the t o t a l number of p u p i l s and dlvide by 2. The sum of 12 + 17 + 42 + 57 + 33 + 14 1s 175, and = 87$ so the middle person will be the

88th one, counting from the t op or bottom, If we count down from the t o p , 12 + 17 + 42 = 71. We need 17 more to reach 88. Countlng down 17 more In the group of 57 brings us to the I

! median. Since the 88th person Is w i t h i n t h a t group, we say that '

the medlan height of the whole group of pupils is between 53 and

55 Inches. Since the 88th pepson comes before we reach the

mlddle of that group as we count down, we might say tha t the median height is likely to be nearer 55 than 53.

Letts check our work and count up from t h e bottom t o the

88th person. 1 4 + 33 = 47. We need 41 more than 47 to I I

make 88, so we count 41 more and tha t takes us into the upper I par t of the group of 57 as we found when w e counted down from I t h e top . Again you find the 88th pepaon in the group of 57 1 whose height is between 53 and 55. Thus the median helght of I t h e group is between 53 and 55 inches.

Exercises 13-5b

1. Give an example in which your principal might choose to group data ra ther than use indlvldual numbers.

2 . Find the median of t he following age groups. hhat is the

median age?

Ages in Years Number In Group

j. Find the median by grouping the following data on

temperatures: (use i n t e r v a l s of 5, namely 50-54, 55-59, e tc . , to 90-94.) 62, 74, 73, 91, 68, 84, 75, 76, 80, 77, 68, 72 , 71, 86, 82, 74, 55, 72, 50, 63, 71. Label one column "~ernperature'~ and the other column " ~ r e ~ u e n c yl' .

Scatter

The mean, the mode, and t h e median are averages. Each is a

measure whfch g ives us an idea of the s ize of the measurements.

Each may be thought of as a typical o r representative number. If we want t o summarize a se t of data we can do this with

just two numbers. One of the numbers will be an average (mean, mode, or medlan) to give us a notion of the size of a typ ica l element. The other number in our summary w i l l g ive us a measure of now the data is scattered from our average, Suppose we wlsh

t o summarize the following two s e t s of numbers:

The mean and the median of set A is the same as the mean and

median of s e t B bu t the numbers in set B are spread farther apart than t h e numbers In set A . One way to measure t h i s II spread" or "scatter" is to find the dif ference between the

largest and the smallest number in the s e t . This dif ference I s

called the range. The range f o r the numbers in s e t A I s 20;

[ s e c . 13-51

the range f o r t h e numbers in set B is 2.

We can now describe s e t A as a set of numbers whose mean

is 50 and whose range is 20. These two numbers give us a shor t desc r ip t ion of the s e t . We can describe the heights of the seventh grade pupils glven in Table 13-1 by saying t h a t the mean height is 58 inches and the range is 13 inches. From this we can think that a measure of the heights l a about 58 and that a

measure of the spread of the heights is 12.

Another measure of the spread or scatter of a set of numbers

is the averaae deviat ion. To find the average deviation we must first find the deviation or difference of each number from the arithmetic mean.

Consider the set of numbers 4, 8, 10, 4, 5, 4, 7. What is the mean of this s e t ? What is the difference (deviation)

of the largest number in this set from the mean? The devia t ions of the numbers in this s e t from the mean are 2, 2, 4, 2, 1,

2, 1, These numbers t e l l us how the numbers scatter, or deviate ,

from the mean. The average deviat ion is t he mean of these

deviations. Taking the mean of 2, 2, 4, 2, 1, 2, 1 we

get 2 + 2 + 4 + 2 + 1 + 2 + 1 = 2 * L e t us use this measure, the

average deviation, In describing another s e t of data.

The total receipts of the Federal Government in the years

1946-1955 were as follows:

Year Billions Deviations from Mean

- Total 555

The arithmetic mean of these receipts is the t o t a l , 555,

divided by 10, or 55.5. (555 and 55.5 are numbers of billions.)

The t h i r d column shows the devia t ion of each yearrs recelpts

from the mean, 55.5. The mean of the deviations is 122 div ided by 10, or 12.2

We can now condense the informat ion In the table by saying:

The r e c e i p t s of the Federal Government in t h e years 1946-1955, had a mean of 55.5 billion dollars and an average deviat ion of 12 2 billion dollars.

Exercises 13-5c

h he first three questions i n these Exercises r e f e r to the daba

f o r the receipts of the federal government in 1946-1955.)

1. In which year(s) was t h e deviatlon from t h e mean t h e

greatest?

2, In which year(s) was the deviation from the mean the l eas t ?

3 . Find the mean for the years 1946-1gi\9 and the average deviat ion from the mean.

4. Find, to the nearest t e n t h , the mean and the average

dev ia t ion of the following t e s t scores:

85, 82, 88, 76, go, 84, 80, 82, 84, 8 3 .

5 . Find the mean and t h e average dev ia t ion of the t e s t scores (same test, but in another c l a s s ) :

94, 84, 68, 74, 98, 70, 96, 84, 769 96-

I +6. Another method of c a l c u l a t i n g the arithmetic mean is shown in

the following example.

I Example. Calculate the arithmetic mean of the set of scores:

! 10, 1 , 1 15, 19, 20, 21, 21, 23

Ee begln by making a reasonable guess of some number f o r the mean, Suppose we guess 18. Next we find the deviation of

each score in the s e t from 18.

Scores Deviat ions f rom 18

The sum of the deviations for the scores less than 18

1s 23. The s m of the deviations for the scores greater

than 18 is 14. We take the difference 23 - 14 = 9 and

divide this by 9 , the number of scores: 9 i 9 = 1. Since the dev ia t ions from 18 were greater f o r the scores less than 18 than f o r t h e scores greater than 18 we subtract

1 from 18 to ge t the correct mean of 17.

U s e t h e nethod of t h e above example to find the mean of i the s e t o f s c o r e s : 40, 43, 44, 47, 48, 49, 51. i

I The mean, median, and mode, for a set of data are ca l l ed I

measures of centra l tendency. This name is given to them s lnce

each of them is a number about which the data tends to "center. t I

A l l of these measures of c e n t r a l tendency are I l lus t ra ted I n the following s e t of sa la r ies of 12 people and in the graph in

r e 3 - 5 . Salar ies : $4,000, $4,500, $4,500, $5,000,

$5,000, $ 5 9 000, $5,2503 $5,250, $59250, $5,250, $5r500,

$6,000.

SALARIES OF A SELECTED GROW

NO. OF SALARIES

Figure 13-5

Calculat ion of average deviation from the mean, $5,042 (to the

neareat dol lar) .

Salarlea Deviation from $5,042 $1042 542 542 42 42 42 208 208 208 208 I

458 958 i

~ o t a1 $4500

f* = 8375, average deviation from the mean.

Range : $6000 - $4000 = $2000.

The loca t ions of the lfnea which represent the mean, median, and mode ahow that these numbers are nearly equal and the graph

shows the salaries are about equally distributed on both sides of these lines.

13-6. Smplina. We a l l know tha t a presidential election is held every four

years in the United States. In which year will the next one be held? People are very much interested In the outcome of the elections. Sometimes, long before the elections are held,

organizations make predictions concerning who wlll be elected, These organizations not o n l y predict who w i l l be elected but even predict the percent of the votes cast that each candidate w l l l receive. The candidates and the percent of vote predicted f o r

each of them in the election of 1948 by three dafferent p o l l s is

shown in the table.

Candidates Dewey Truman Thurmond Wallace

P o l l No. 1 P o l l No. 2

w 3 -3%

Poll NO, 3 4 03%

In the election t he percent of vote for each was: Truman 49.4$, Dewey 45.*, Thurmond 2.w, Wallace 2 . w . Do you see why

this election is called the "surprise election"? Although none of the p o l l s predicted the e lec t lon correctly,

their predictions were c lose . How did they do it? Did they go

about the U.S. and ask every voter how he was going to vote? Or, did they write each voter a letter? Ei the r of these ways would have been very expensive and would have taken a great deal of t h e .

Instead of either of these they used a method called samplin~.

This means that the organizations who made the predictions selected a "sample" of the population of the U . S . Then, a f t e r

H asking the people i n the samplet' how they would vote, the organizations predicted that the vote i n the entire country would

be in nearly the same rat10 as the vote in the "sample. II

If you have ever had a blood-count, the doctor took a very small amount of blood Prom the t i p of your finger, or from your

ear lobe, and then counted the red and white corpuscles in i t . This was a very s m a l l sample of your blood. The count In it; was taken as a reliable representation of the count In all the blood i n your body. Perhaps you can think of other examples of sampling.

L e t us suppose you know that a l l the employees whose names are listed in the employee's directory of a certain large

manufacturing firm are men over twenty-one years of age. Then l e t ua ask how we might uae the sampling method to make an estimate of the average height of these men. You mfght select the first and last m a n l i s t ed in t h e directory and find t h e i ~ average height. Or, you might select the first name l i s t e d under each letter of the alphabet, or the las t name under each letter, or both the '

first and last names under each letter. There are many ways a

sample could be aelected. Some would be good and some would be

[aec. 13-61

bad. Do you see any objection to any of the methods suggested? The way of selecting a sample so tha t it will be a good

representation of the group from whom the sample is selected is

a very d i f f i c u l t part of the job.

Exerc f ses 13-6

1. This problem is a sampling project f o r your c l a s s . The objective is t o f ind the mean height of your classmates. Since there Is likely to be a difference in boys1 he igh t s

and g i r l s ' heights, find t h ~ mean f o r boys and girls

separately, Boys should find the boyst mean and girls should

rind the girlsv mean. All directions are given for bays;

girls should substatute the word "g i r l s . " (a) ,Select a sample in each of these ways.

1. A l l boys whose birthdays are in March, August or December.

2. A l l boys whose first names start with G, M, or T.

3 . A l l boys who s i t in some one row in the

mathematics classroom.

(b) Find the mean height using Sample 1.

( c ) Find t h e mean height using Sample 2.

( d ) Find the mean height using Sample 3 .

( e ) Find the mean height of all boys in the class.

( f ) Which sample was t h e best representation of the hefghts

of a l l the boys in the class?

(g) In selecting samples, Is it important that the sample

be chosen so as not to give one par t t oo much

repre sent at ion?

2. There were 48,834,000 (to the nearest thousand) votes c a s t

in the 1948 e l e c t i o n f o r President of the United S t a t e s .

(a ) If P o l l No. 1 on page 505 of the t e x t had been correct, how many votes would have been cast for each candidate?

Give your answer to t h e nearest 10,000.

(b) The percent of the vote t ha t was received by each candidate 1s glven under the table of the polls. Use

these percents t o find the number of votes r e c e i v e d by

each candidate. Round thfs t o the nearest 10,000

votes .

13-7. Summary.

The subject matter of s t a t i s t i c s deals , in par t , w i th

collecting data , putting it in table form, and represent ing it

by graphs. The tabulating and graphing of the data should be done in ways such that the story t o l d by the data can be

i n t e r p r e t e d and swninarized e a s i l y . The broken line graphs, bar

graphs, and c i r c l e graphs are just a few of the kinds of graphs

t h a t may be used.

You have learned t ha t t h e r e are many d i f fe ren t measures f o r the cen t ra l tendency of the same s e t of d a t a . The ne$t time you

see graphs o r tables of a s e t of f igures in the magazines, news- papers, or your social studies book, look them over c a r e f u l l y .

If averages are mentioned, see if you can tell which average 2s

used. Whatever kind of average Is used, you have the right to

question whether the average used gives t he best representation

of a l l the data. To help you reca l l the new terms you have used i n working

with s ta t is t ics , they are listed f o r you:

Arithmetic mean o r m e a n -- the summof a l l the numbers in a

s e t divided by the number of members in the s e t .

Median -- t h e middle number when data is ordered e i t h e r

from smallest t o largest or la rges t to smallest. When there is no one middle number, the average of the two middle numbers is the median.

Mode -- t h e number occurring most in the list of data. There

may be several modes.

Range -- difference between largest and smallest number i n

a s e t .

Average devia t ion -- average of the deviations from the mean.

Chapter 14

MATHEMATICS AT WORK IN SCIENCE

14-1. - The Scientif f c Seesaw. Have you ever played on a seesaw?

If you weigh 100 pounds and your par tner on the other s i d e

of the seesaw weighs 85 pounds, where does he have to s i t to make the two sides balance? W d l l he be closer t o the centeP or farther from it than you? Can you t e l l how far?

The seesaw l a one form of simple machine that is used a

great deal around home, at work, and in sclence laboratories. 11 It belongs to the family of machines ca l l ed "levers. Of course ,

not a l l levers work l i k e seesaws! You use a lever t o open a so f t drink bottle. You use o the r forma of levers to jack up your ca r or to pry up a stone. Can you think of other examples of levers?

The sc ient i s t uses levers of rather fine construction for many purposes in his laboratory. The simpleet type is the comon laboratory balance or scale. You probably have one in your

science room. Scientists long ago studied these s c i e n t i f i c seesaws, learned how t o balance objects of different weights,

and expressed t h e i r findings in a mathematical formula. Having

the formula makes the s c i e n t i f i c seesaw much easier to use and

understand . Today you are going to play the p a r t of a s c i e n t i s t . You

wlll set up a simple seesaw experiment, make observations, try to

discover a rule and try t o s t a t e the mle in mathematical f o m . The experiment suggests how a scientist makes observations

in the laboratory, studies them mathematically and draws conclusions from them. He then tries to s t a t e the conclusions

by means of a mathematical equation, Final ly he uses the formula to predict a new result and then goes back to the laboratory to

test whether the rule works in similar situations.

14-2. A Labopat ory Experiment . Your equipment to study the scienti f ic seesaw w i l l look

something l ike this,

The materials required In your laboratory are:

A meter s t i c k or yardstick, Strong thread and two bags to hold the weights

(thin p la s t i c makes very satlafactory bags).

A s e t of objects of equal weight and convenient size (a supply of pennfes or marbles is recommended).

Procedure:

1. Balance the s t i c k by suspending it from a strong thread

t i e d at the middle of the s t i c k , If the s t i c k does not balance, a thumbtack may be placed at varfous points u n t i l the s t i c k is in

balance. Thls point a t which the s t i ck is suspended is c a l l e d the fulcrum.

2. Hang ten objects of equal weight on one side of the fulcrum and ten identfcal objects on the other side and try to

make them balance. Take two objects of unequal weight and t r y

to make them balance, Do you find tha t you muat change the

distance to achieve a balance when the welghts are unequal?

Note: Scientists usually perform some preliminary t e s t s t o

determine t h e best way t o s e t up and c a r r y out the experiment.

Their f i r s t experimental se t -up doesn't always work perfectly! You may f i n d It advisable t o make some improvements i n your

equipment and procedures a t this s t age .

When you have your equipment operating smoothly, you are

ready to take the first steps in your experiment. Scientists

usually have t he experiment carefully planned in advance, but

we s h a l l develop ou r p l an as we go along.

3 . (a ) Hang 10 pennies o r m a r b l e s (or o the r handy items

of equal weight) a t a distance of 12 centimeters from the fulcrum, and balance it with 10 identical objects on the o t h e r

1 side. (On a yardstick you may f i n d that p inch i s a convenient unit of distance.) Observe the distance of t h i s second mass from

the fulcrum when the l e v e r i s i n balance and record the dis tance

i n a table, column (a), s imi la r t o the one below. Note t h a t w and d represent the measures of the wefght and distance

respectively on one sfde of the fulcrum. W and D represent

the measures of the weight and distance on the o the r side of the

fulcrum.

TABLE I

( a ) ( b ) (4 ( d l (4 If) ( g ) Ih) (i) ( 5 )

w = 10 10 10 10 10 10 10 10 10 10

d = 12 12 12 12 12 12 12 12 12 1 2

w = 10 20 5 a 15 24 12

D =

(b) If W is 20 flnd D so t h a t the lever is i n

balance. Write the number in your table under "20," column (b).

( c ) If W Is 5 find D ao that the lever is in

balance. Write it in the table, column ( c ) under "5."

( d ) Notice that in these first three t r ia ls , w and

d remained the same. All changes were made in W and D. Use the values indicated in columns fd) - ( g ) and find the value of D. Make several other changes for W and write the corresponding results f o r D in your table -- columns (h), (I), and (j).

4 . Now, as Indicated In Table 11, l e t w be 16 and d

be 6 . Find how large W will have t o be f o r the lever t o

balance when D equals 6. How great a weight will balance the

lever 4 ern. from the fulcrum? 16 cm? T r y several other distances from the fulcrum, find what weight is needed to balance the lever, and fill In your Table 11.

TABLE I1

5. Try other values for the weights and dis tances as suggested in Tables 111 and fV and fill In similar tables of your own.

14-3. Caution: Inductive Reasoninq -- at Work! After a s c i e n t i s t has completed an experiment and col lected

data, he t r i e s t o analyze i t . He tries t o discover a l l the facts that the data present, He attempts to in te rpre t these fac ts in a connected and convenient way and to discover a general rule which the data suggest. His hope is to s ta te these results in a precise

w a y , preferably by a simple mathematical formula. Note that the s c i e n t i s t studies a number of specific

experimental reaults and from these tries to reach a conclusion which w i l l hold f o r a l l cases. By reasoning from a number of

spec i f i c experimental results, a general statement is developed

whlch will apply i n a l l similar s i t u a t i o n s . This process l a called Inductive reasoning. It should be used with caut ion.

There may be occasions when a few examples suggest a conclusion which 1s not t r ue in general. Thus, If you go to New York and meet f i v e people In succession with red hair, it is not safe to

conclude that everyone in Mew York has red hai r . Also, if you

notice t h a t = 51 = ; , = 8 , and = 5 2 you

will be in trouble if you assume t h a t you may alwaya cross off

numerals in this way.

When a general rule has been suggested, the sc i en t i s t tries to verify it by f u r t h e r experiment and, if possible, by deductive reasoning. In a l l these steps mathematics and mathematical reasoning are especially important.

Let us look at the results in our tables from this poin t of

view. We are t ry ing to determine whether there seems to be a general rmle which describes a l l these relat ionshipa . If possible, we wish to express the rule in mathematical terms, If it I s a general law, we should be able to use it to predic t where to place one object of known welght to balance a second of known weight .

1. (a ) In Table I, do you not ice any connection between the

location of masses of equal weight on opposite a i d e s of the fulcrum? What is it?

(b) IS the value of W is doubled, w and d remaining

unchanged, how does the corresponding value of D change ?

( c ) If the value of W is made half as much, how does the

corresponding distance D from the fulcrum change?

( d ) DO the values f o r n, d and W, D appear to be

related in any way? Can you s t a t e a general rule that

seems to hold concerning w, d, W, and D? State

the rule in words and then in the form of a mathematical

equation in which you use the symbols w, W, d, and D.

( e ) Check your rule byqapplylng It to some of the entries found by experiment in Tables 11, 111, IV.

2 . Use the equation suggested in the preceding exercise to pred ic t the missing entries in Table V.

615

3 . Go back to you? experiment and check t h e resul ts in Table V

to see If they actually produce a balance.

4 4 . Graahical In temre ta t f on. In Chapter 13 you learned about graphs and t h e i r usefulness

in presenting numerical information in a clear and condensed way. S c i e n t i s t s often make a graph of the observations obtained through experimentation for the he lp it can give them in summarizing and interpreting the data.

In the equation wd = WD which you obtained for yourself

in the preceding experhnent, four quantities are involved. This

equation can be interpreted in a number of ways depending on the way you followed through in the experiment. In the first pa r t of the experiment, you chose fixed values for w and d and then found the values for W and D which produced a balance. From each experiment you got one pair of va lues which sat isf ied the r e l a t i on WD = 120. A graph of WD = 120, p i c t u r e s many palrs of values which produce a balance when wd = 120.

The graph, then, suppl ies not only the Information in Table I but other possible values for W and D.

The next step is to draw the graph of the r e l a t i o n connecting a value of W and the corresponding value of D in Table I.

If you need help in drawing the graph, the following suggestions should aid you:

Use graph paper and begin with two perpendicular l ine s ca l l ed axes. The intersection of t h e axes is named poin t 0.

Label the hor izonta l axis W and the vertical axis D. 1 If you use inch squared paper, a su i tab le scale I s one

f o r each space. In Table I, the first value of W is 10 and the

corresponding value of D is 12. Locate 10 on the W axis. Follow the vert ical l i n e through 12 on the D axis. This

po in t is called (10, 12). Dotted l ines in the graph in Figure 14-4a wfll help you l o c a t e this point. Make a small dot

for the point.

[sec . 14-41

Similarly, from Table I, when W = 20, D = 6 Locate 20 on the W ax is , Follow the vertical l i n e through 20 to

the point where it meets the horizontal line through 6 on the

D axis. This po in t is called (20, 6 ) . - Before you read on, locate and mark the other points from

Table I. These are ( 5 , 24) , (8, 15), (24, 5), (12, 10).

Mark the points which correspond t o the resu l t s you found

in columns (h), (I), and ( J ) in Table I. Fill in the blanks in t he following pairs f o r (w, D) and

mark the corresponding po in t s on your graph: (4, 9 ( J 61,

(16, , ( , 18). Use WD = 120. D r a w a smooth freehand curve through the points you have

located. This curve gives you the general p i c t u r e of the relation

between wefghts and dis tances as In Table I. If any point seems to l i e to one sfde or the other of your smooth curve, check your computation. Not a l l experiments turn out perfect ly and not a l l r esu l t s f a l l into neat pa t t e rns at once. The points obtained

from the measurements you made should f a l l near the curve. Often

scientists expect no more than thfs from an experiment of this type.

The curve t h a t you have drawn I s a por t ion of a curve c a l l e d a hyperbola. You w i l l learn more about this curve in your study

of algebra.

o ( distances)

I I I I I I 1 I I l l 1 I I I I I I I I I I I I t I ' r - n t s )

Graph of WD = 120 from Data in Table 1

Figure 14-4a

Exercises 14-4

Study the graph and then answer these questions:

(a) If t h e value of w is increased, how does the

corresponding value of d change?

( b ) If t h e value of d is increased, how does t h e

corresponding value of w change?

In order t o f i n d the value oi' ? ;:!hen D is 2ki, l o c a t e

24 on the axis and i'ollow t h e horizontal line through 24

u n t i l I t meets t he curve . Read the value on the W scale directly beneath this poin t . You should have 5 Find the

va lues of W from the graph f o r the following points:

I a ) ( 3 201 I b ) ( 9 15) (4 I 3 6 )

Find t h e va lues of D from the graph:

(a) 16, 1 (b) ( 4 , 1 (4 (15, 1 Tell whether or not each of the following po in t s is on the

graph:

( a ) (10, 251 (4 (5, 5)

(b) 5 8) ( d l (20 , 15)

Estimate t h e miss ing values:

( a ) (7 , 1 ( d l (17, 1 ( b ) (8, 1 (4 ( 9 21)

(4 ( t 9 ) (f) (23, 1 Draw a graph of the r e l a t i o n between W and D given in

Table 11. Use the formula WD = 96 t o f i n d the number

p a i r s you need f o r locating points . Check the va lues you

find w i t h those in Table 11.

From the graph find D when W is 20; find U when D i s 12.

8. How does t h e value of d change when the va lue of w is

decreased? Increased?

9 . Does this graph have anything in common w i t h t he graph you

drew f o r WD = 120?

14-5. Other Kinds of Levers. --- In the introduction to t h i s chapter you read t h a t not a l l

l e v e r s are like seesaws. You may be In te res t ed in asking about this in your science class a t an appropriate t ime. Tn addition t o the automobile jack, b o t t l e opener, and crowbar mentfoned before, there a r e o t h e r examples of widely used forms of the lever. You may thi1-k o f I c e tongs, nutcrackers , scissors, claw hammers,

pliers , pruning shears, and hedge cllppers as some useful levers.

14-6. The Role of Mathematics - i n S c i e n t i f i c Experiment . Although the experiment using the l e v e r does not use a great

deal of mathematics, it does suggest how mathematics i s used i n s c i e n t i f i c a c t i v i t i e s . You saw how mathematics was used in

measuring, counting, and comparing quantities. You noted how

observat ions of data were recorded in mathematical terms. You searched f o r a p a t t e r n by studying t h e numbers in your

recorded da ta , By reasoning from a set of s p e c i f i c cases you

developed a general statement to be appl ied in a l l s imi lar s i t u a t i o n s . I n Sect ion 14 -3 t h i s kind of reasoning is ca l l ed

induct ive reasoning. It leads from a necessarily restricted

number of cases to a pred ic t ion of a general relationship. This

general relatlonshlp was stated in mathematical symbols in an equation: WD = wd. To establish t h i s general pr inc ip l e , f u r t h e r experimentation was performed.

In addition, you drew a graph of WD = 120 and of

W13 = 96 to show how these statements tell the complete story in

each case. The graph I s another ins tance of the use of

mathematics to i n t e r p r e t and t o summarize a c o l l e c t i o n of facts.

The graph also helped to revea l the general pattern which was discovered.

Many scientific f a c t s were undiscovered for thousands of years u n t f l a ler t scientists carefully set up experiments much as you have done and made discoveries on the basis of obser- va t ions . Some examples of these are the following:

(a) U n t i l t h e t i m e of Gal i leo, people assumed tha t if a

heavy object and a l ight object were dropped at the

same time, the heavy one would fall much f a s t e r than

the l i g h t one. Laok up the story of Qalileo and his experlrnent with falling obJects and see what he discovered.

( b ) From time fmerno~lal , people watched eclipses of the

sun and moon and saw the round shadow of the earth but

d i d not discover t ha t the earth was round. Eratosthenes, In 230 B.C., computed the distance around t h e world by h i s observations of the sun i n two

l o c a t i o n s i n Egypt, yet seventeen hundred years l a t e r when Columbus s t a r t e d on his journey, many people s t i l l believed the world was flat. Look up in a history of

mathematics book or in an encyclopedia the story of Eratosthenes and th i s experiment.

( c ) F eople had watched pendulums f o r many centuries before

Gal i leo did some measuring and c a l c u l a t i n g and

discovered the law which glves t h e r e l a t i o n between the length of the pendulum and the time of i t s swing. Look up this experlrnent in a book on the history of

mathematics or of 'science.

Notice that all these experiments are based on many careful measurements and observations in order t o discover the s c i e n t i f i c

law. Then the law is stated in mathematical terms. A great deal

of science depends upon mathematics in Jus t this way.

The examples which we have given here descrlbe o l d e r funda-

mental d i scover ies a11 of which used r e l a t i v e l y simple mathe-

matics, The s c i e n t i s t s of today are using more advanced

[sec . 19-61

mathematics, and many of the newer kinds of mathematics, in t h e i r

sc l e n t if ic experiments .

Exercises 14-6

1. Some seesaws are fftted s o tha t they can be shffted on t h e

support . I~~by?

Find t h e missing value In each case:

32 lb. ? 4 y d . 172lb.

100 lb. 6 ft. 8 ft.

6 , Do you suppose a 90 lb. girl could ever lift a 1000

pound box? Justify your answer.

n

?

? 2 in. 3 i n .

7. A c h i l d who weighed 5 h pounds asked his fa ther (180

pounds) t o r ide a seesaw w i t h h i m . Where should the fa ther

s i t to balance the ch i ld?

A

5 02.

8. An i r o n bar 6 f e e t long i s t o be used as a l e v e r to l i f t a

stone weighing 75 pounds. The f u l c r u m is 2 f e e t from

the stone. A force of how many pounds is needed to lift

the s tone?

A

9 . Suppose a 100 pound boy s i t s on one end of a six foot 1 crowbar, 5 feet from the fulcrum. Disregarding the weight

of the bar, how heavy an object can he l l f t a t t h e other end

of the bar?

10, L"nich po in t s lie on the graph of WD = 60?

Bi bl i ography

1. Newman, James R. THE: WORLD OF MATHEMATICS. N e w York:

Simon and Schuster, 1956.

( ~ a l i l e o , pp. 726-770; f a l l i n g b o d i e s , p . 729; pendulum, pp. 729, 741; Erathosthenes, pp . 205-207.)

2. Kline, Morris. MATHEMATICS IN WE;STERN CULTUIIE. New York: Oxford University Press, 1953.

Chapter XI11 The Quant i ta t ive Approach to Nature

Chapter XIV The Deduction of Universal Laws

INDEX

The reference i s t o the page on which the term occurs.

acute angle, 293 addend, 366 add1 tion

baae seven, 39 mod 4, 588

additive inverse, 555 ad Jacent angles, 398 Ahmes, 189 altitude of parallelogram, 440 angle, 138, 287

acute, 293 adjacent, 398 central, 486 corresponding , 407 exterior of, 287 interior of, 287, 399 meaaure of, 401 non-adjacent, 398 obtuse, 293 r i gh t , 293, 296, 299 supplernentaz , 400 vertical, 3

approximately equal, 250, 298 310, 327 approximating numbers, 376, 564 arc, 481 Archfmedes, 261 area, 245, 303

of circle, 500 of parallelogram, 438 of t r iangle, 438

arithmetic mean, 595, 607 associative property

of addition, 75, 76, 102, 197, 539 of multiplication, 76, 102, 197, 203

associativity, 563 average, 388, 594 average deviation, 600, 608 Babylonian numerals, 23 Babylonians, 397 bar graph, 588 base, 27, 30, 31

changing, 48 five, 52 seven, 33, 39, 48, 162 six, 52 twelve, 344

baae of parallelogram, 440 base of prism, 449

betweenness, 93, 131 481 binary operat ion, 536, 541 binary system, 56 62 boundary, 1 3 148 broken-line curve, 147 broken-line graph, 582 bushel, 334 center of circle 464 central angle, 486 chance, 8 changing bases, 48 circle, 463

area of, 500 center, 464 concentric, 474, 488 diameter of 475 radius of 465 semi- , 466 tangent to, 476

cfrcle graph, 591 c i r cu l a r closed re ion , 471 c i rcu la r scale, 33 circumference,

b 430 closed region, 2 5, 299

circular, 471 rectangular, 320 square, 303

closure, 84, 86, 1 W , 196, 542 collection, 85 c o ~ s s i o n , 384 common denominator, 179, 209, 225 common factor, 164, 185 common multiple, 178, 185 commutative property

of additfon, 70, 102, 196, 539 of m l t i p l f c a t l o n , 71, 99, 102, 196

commutativity, 563 compass, 463 composite number, 153, 156, 184 computers, 56 concentric circle, 474 488 congruent, 245, 251, 2h9 continuous, 243, 244, 258 converse, 412 coordinate geometry, 145 corresponding angles, 407 counting, 20, 243 counting numbers, 67 68, 85, 97, 102, 142, 151, 156, 160, 179,

cube, 319 184, 193, 198, 570

cubic centimeter , 331 foot, 344 inch, 320 millimeter, 331 uni t , 325

curve, 147 cylinder, 277, 324, 509

lateral area of, 515 oblique, 509 r igh t , 509 surface area of, 514 total area of, 514

cyllndrlcal solid, 509 volume or, 511

data, 579, 598 decimal, 365, 377, 387

expansion, 371, 372 notation, 235, 362 numeral, 28, 372 repeating, 372

decimal system, 26, 27, 160 162 deductive reasoning, 1, 3, 459 degree, 289 degree of arc, 487 denominator, 198, 279, 368 deviat ion, 600 diagram, 520 diameter of a circle, 475 d i g i t , 26, 31, 161 dimension, 332 discount, 384 discrete, 243 Msneyland, 8 distance between parallel lines, 431 distance from a point to a line, 431 distributive pro erty, 79, 102, 197, f dividend, 170, 3 9 d i v i s i b i l i t y , 151, 156, 160, 178 division, 169

base seven, 46 divisor, 170, 369 dry quapt, 334 duodecimal system, 57, 365 edge, 125

of prism, 449 E a p t , 397 m p t i a n numerals, 20 element, 86 empty set, 122 English system of measures, 264, 336

enaemble, 85 equator, 262 equilateral t ~ i a n g l e , 418 equivalent fraction, 193 Eratosthenes, 153 h c l l d , 105, 461 Euclidean geometry, 105 even number, 158, 164, 178 expanded nota t ion , 31 expanded numeral, 30 exponent, 31, 96 exponential no ta t i on , 30 exterior, 139, 148 exterior of the angle, 287 face, 316 factor, 30, 155, 163, 165, 184

greatest common, 164, 185, 199 factoring, 151 factorization, 157

complete, 156, 162, 184 fathom, 261 f i n i t e s e t , 575 finite aystem, 575 f rac t ion , 189, 193, 213

improper, 240 u n i t , 189

fractional form, 226 fulcrum, 611 Gauss, 5 geometry, 105 gram, 337 graph, 615

bar, 588 broken-line, 582 circle, 591 1 ine , 585

greatest common factor, 164, 185, 19 greatest poss ib l e e r r o r , 275, 281, 2 4, 314 Greeks, 404

8 half-line, 135, 136 half-plane, 135, 136 half-apace, 134, 136 had, 261 hexagon, 479

regular, 479 Hindu-Arabic numerals, 26 hyperbola, 616 - - ihintity element, 95, 98, 103, 547, 548 i d e n t i t y for addit ion, 197 Ident i ty for multiplication, 197

improper fraction, 240 indue t i v e reasoning, 459, 613, 619 Inequality, 72, 93, 94, 102, 118, 239, 359 i n f i n i t e set, 575 i n f i n i t e system, 575 in t e r io r , 139, 148, 469 interior of the angle, 287, 399 intersection, 469

of l i n e s , 125, 397 of line and plane, 127 of planes, 125 of sets, 122

inverse of an element, 547 inverse operation, 88, 89, 102, 216, 553 Inverses, 550 isosceles triangle, 418 kilogram, 337 Kb'nigsberg brfdges, 15 Laplace, 28 lateral area of cyl inder, 515 least common multiple, 178, 185 length, 250, 251, 300 lever, 609, 619 l ine , 105, 106, 114, 127, 135

segment, 132, 244 skew, 126

l i n e graph, 585 linear, 261 linear measurement, 270 l i q u i d quart, 334 mass, 337 mathematical system, 559 mean, 595 measure of an angle, 401 measures of central tendency, 602 median, 595, 608 meter, 262, 273 metric system, 262, 273 m~ll imeter , 274 mode, 596, 608 modular arithrnetlc, 527, 573 modular aystema, 573 modulus, 528 m l t i p l e , 151, 161, 178, 185 m u l t i p l i c a t i o n , 71 156

base seven, 44 sign, 72

multiplicative inverse, 551

National Bureau of Standards, 263 natural numbers, 68 net price, 384 non-adjacent angle, 398 non-metric geometry, 105 North Pole, 262 number

composite, 153, 156, 184 c o u n t i n g , 6 7 , 6 8 , 8 5 7,1M,142,151j156,160,179,

184 193 iJJ 570 even, 158, 164, 178 natural, 68 odd, 178 perfect , 160 prime, 151, 153, 156, 184 rational, 190, 193, 196, 203, 212, 220, 225, 235, 265, 305,

3'25, 350 whole, 67, 70, 75, 89, 93, 97, 102, 164, 184, 192, 570

number line, 93, 206, 207, 26 numeral, 20, 61, 190, 193, 3

Babylonian, 23 63

base f i v e , 52 base six, 52 decimal, 26-28, 235, 372 duodecimal, 57, 365 Egyptian, 20 expanded, 30 Iilndu -Arabic, 26 Roman, 20

numerator, 189, 198, 209 obllque cylinder, 509 obtuse angle, 293 odd number 178 one, 94, 164 one-dimensional, 332 one-to-one correspondence, 67, 68, 142, 149 operation, 536

binary, 538, 541 ternary, 541 unary, 541

opposlte faces, 316 ordering, 239 parallel planes 128 parallelogram, 433 pentagon, 433 pentagonal right prism, 449 percent, 354, 377, 382, 387

of decpease, 392 of increase, 392, 581

perfect numbera, 160

perimeter, 299 perpendicular, 295, 333 plnt, 334 place value, 363 planes, 105, 108, 127, 134 point, 105, 116 point of tangency, 476 positional notat ion, 362 possible error, 315 power, 31 precise, 276, 280, 284 precision, 275, 276, 277, 280, 310 prime number, 151, 153, 156, 184 prism

base of, 449 edge of, 449 pentagonal right, 449 right, 448 triangular rl&ht, 449 vertices of, 49 volume of, 451

probability, 8- - product, 156, 202, 213, 368 polygon, 433 proportion, 349 proportional, 349 protractor, 290 pyramid, 397 Pythagoras , 461 quadrilateral, 286, 433 quart

dry, 334 l iquid , 334

quot ient , 170, 217 radius of c i r c l e , 465 Pawe, 599, 608 rate, 231, 348 ratio, 23i, 235, 347, 348, 355 rational number, 190, 193, 196, 203, 212, 220, 225, 235, 265,

305, 325, 350 r a t i o n a l numbe~ aystem, 221 ray, 136, 138, 289, 397 reciprocal, 202, 203 rectangle, - 255, 299, 303 rectangular closed re ion, 255 rectangular prism, 31f, 322, 324 rectangular solid 320 reference l i n e s , 584 region, 148

circular closed, 471 cloaed, 245, 299 rectangular closed, 320 square cloaed, 303

regular hexagon, 479 remainder, 151, 169, 170, 372 repeating decimal, 372 Rhind Papyms, 189 r ight angle, 293, 296, 299 r ight cyl inder 509 right prism, 448 Roman numerals, 20 rounding, 375 ruler, 266 sale price, 384 sampling, 604 scale, 291, 336 scalene triangle, 419 scatter, 599 segment, 131 semicircle, 486 separation, 134, 481 set , 84, 85, 102, 109, 118, 196, 287 sides

of angle, 138 of line, 135

sieve of Eratosthenes, 153, 160 simple closed curve, 147, 245, 306 simplest form of the fraction, 199 skew lines, 126 space, 105, 106 square closed region, 303 square

foot, 344 inch, 303 u n i t , 325

standard unit 260, 289 statist lcs , 580, 607 subtraction, base seven, 42 supplementary an le, 400 surface area, 31 8

of cylinder, 514 symbol, 20, 62, 113 tangent to circle, 476 ternary operation, 541 Thales, 404 thermometer, 336 three-dimensional, 333 triangle, 138, 397 416

equilateral, 418 isosceles, 418 scalene, 419 awn of measure of the angles of, 428

t r iangular r i g h t prism, 449 twin primes, 145 two-dimensional, 332

unary operation, 541 unique factorization property, 184 u n i t f rac t fon, 189 u n i t of measurement, 251, 255 u n i t of time, 336 u n i t of weight 336 vertex, 138, 267, 317 vertical angle, 398 vertices of prism, 449 volume, 320, 322

of cylindrical solid, 511 of prism, 451

weight, 336 whole n u b e r a , 67, 70, 7 r 89, 93, 97, 102, 164, 184, 192, 570 zero, 27, 68, 95, -97, 18 ?