Page1

Math43–PreparationforLiberalArtMajorsHandout1.5–Areyou“rational”or“irrational”?

FoxtrotByBillAmend

1).Groupwork:Inyourgroupof2or3students,makealistofalltheDIFFERENTnumbersyoucancomeupwith.

Inmathematics,thenumbersystemevolvedthroughtime:

• N:Atfirst,“numbers”meantsomethingyoucancount,likehowmanysheepafarmerowns.ThesearecalledNaturalNumbers(orcountingnumbers).

• W:Atsomepoint,theideaof“zero”cametobeconsideredasanumber.Ifthefarmerdoesnothaveanysheep,thenthenumberofsheepthatthefarmerownsiszero.NaturalNumberstogetherwith“zero”makeuptheWholeNumbers.

• Z:Evenmoreabstractthanzeroistheideaofnegativenumbers.If,inadditiontonothavinganysheep,thefarmerowessomeone3sheep,youcouldsaythatthenumberofsheepthatthefarmerownsisnegative3.TheexpandedsetofnumbersthatwegetbyincludingnegativeversionsofthecountingnumbersiscalledtheIntegers.

• Q:Thenextgeneralizationthatwecanmakeistoincludetheideaoffractions.Whileitisunlikelythatafarmerownsafractionalnumberofsheep,manyotherthingsinreallifearemeasuredinfractions,likeahalf-cupofsugar.Ifweaddfractionstothesetofintegers,wegetthesetofRationalNumbers.

• H:Therearenumbersthatcannotbeexpressedasafraction,andthesenumbersarecalledIrrationalbecausetheyarenotrational.

1 -1 13 if if

tz 0.25 's 22 5 ! IT

a

IN

:/

:t

IN = { 1,2 , 3,4 ,... }

iw = { 0,1 ,2, 3,4 ,

..

. }Wholey21 = { . . .

-3,-2

,-1

, 0,1 ,i3,I

= { O,±l

,

.±2,±3,

... }

Q={¥±÷¥,±¥E,...

}

= { E / a. be Z,b±

IH : A :@ 2=0

2 @

B € z =D

52=1.4 ... :

TT=3=t.

N27= 5.196152 . . .

lT= 3.14152 .. .

O@€Real

±=Y5Y÷ns

Oleat = YAFEI

0000

¥ ftp.T#=OpPItzasquirrel in

5f= 55ft ,

Undefined

§ =3 because

2.3=6T6÷z

4g=t because 8. I=4

oI= ? because 0 . ?=5

.possible to

solve ( right now )

¥¥oie :#

¥qnkf i .

pI§sThemto Bgsas

undefined off

Page2

2.Isthefraction!"equivalentto0.333… = 0. 3?Explainwhyorwhynot?

3.Convertthefollowingtoafraction:a).1.434343…. b).25. 84.Sortthenumbersinto2groups,rationalorirrational.Writethenumbersintheappropriatebubble. 0.8 64 0 32 -19 100- 2.343443444… 7

3 75

726 76.12 121 5

12

5.Plotandlabeleachnumberonthenumberlinebelow.

A 0.75 B 3

C 9

D 212-

E 1015

-

F 6.2

G 2- H

Rational Irrational

-5 -4 -3 -2 -1 0 1 2 3 4 5

3..

..

3*0. 9¥

.

-

= 1.43

use

1001004.4343... ) .

- 143.4343 . . |O( 25.5 )= 258.8,÷, /'s:* :

- / ( 1,4343 .D= -

.4343 ...

- 1 ( 25.5 )= - 25.8

9*343.1=14 965=2331.4343 . ... 25.5 = 2¥

06kg 0-19 ft 2.343443444 :

3 ftto T

¥÷E¥ri¥..ee?.II

.

Page3

6.WorkwithapartnertodrawaVennDiagramfortherealnumbers.Makesuretoincludethefollowing:naturalnumbers,wholenumbers,integers,rational,andirrationalnumbers.

DivisionTricks:Divisibleby2:Thenumberiseven–itendsin0,2,4,6,or8 Ex:Divisibleby3:Thesumofthedigitsisdivisibleby3 Ex:Divisibleby5:Thenumberendsin0or5 Ex:Divisibleby6:Thenumberisdivisibleby2and3 Ex:Divisibleby9:Thesumofthedigitsisdivisibleby9 Ex:Divisibleby10:Thenumberendsin0 Ex:

,@@@*f@7311177534

24 : 2+4=6

111 : 1+1+1=3

37936875.

222 : z€+Tf¥ 6

ys:ztI8Ihn=i8✓

635170

Page4

7.Primenumbersarenumbersdivisiblebyonly1andthemselves.Listafewprimenumbers.8.Forthenumbers1-100below,circlealltheprimenumbers.Doyouthinktherearemoreprimenumbers?Do

youthinktherearefinitenumberofprimesorinfinitelymany?Explainbriefly.

t2 3 57

13

101/00/0/0#9=3.3

0/01/-0-0-composite- onto.oxyoo.ox-onto,¥⇐t÷⇐⇒on.ro/~r/r- / - D

I t

7. 13 5 's

whidsijspid Riemann Hypothesis

Page5

9.Whichofthefollowingnumbersisaprimenumber?Explainyourstrategytoyourpartner. 3931 5009 4100 8999

10.Thecastoftheschoolplayhadapartyatthedramateacher’shouse.Therewere20cookiesand40carrotsticksservedasrefreshment.Eachcastmemberhadthesamenumberofwholecookiesandthesamenumberofwholecarrotsticks,andnothingwasleftover.Thedramateacherdidnoteat.Howmanycastmembersmighthavebeenattheparty?Explainyouranswer.

11.Giventhefollowingclues,whatismynumber?

• Mynumberisaperfectsquare.• Theonlyprimenumberisinitsprimefactorizationis2.• Mynumberisafactorof80.• Thesumofitsdigitsisodd.

Prime # it 0end Prime

2/7#(31 5=961

31 #236## F99t95 !!* HK Lily

.

20 cookies

40 carrot sticks

# Cast Members : 20,10 , 5,4 ,

2,1

l6:l+6=@ ←

T3/6=3.3-2.24=2.24/9=7.791=3.364=2.2-2.2-2.216=2.2-2.28/1=3.3-3.32/5=5.51/00=5.5-2.2

Ex : Find the Prime Factorization

64 11}Fa¥Ee

. \20.4@ - 41 ' 1 '

�2�. @20 .@

64=2-2-2.21500= 5. 5. 5. 3. 2.2

n -

15 100 Multiply to

051.130 /\check

• . 10 .

10*5.0*0

Page6

HOMEWORK:1).Describethedifferencebetweenrationalandirrationalnumbers.Youcanuseexamplestohelpsupportyour

answer.

2).Identifyeachnumberasrationalorirrational.

3.676776777…. 76.12 32 49 57-

3).Isitpossibletoconvert3.676776777….toafraction?Ifyes,findthefraction?Ifno,whynot?4).Isitpossibletoconvert 76.12 toafraction?Ifyes,findthefraction?Ifno,whynot?5).Statewhetherthefollowingstatementsaretrueorfalse.a).Arealnumberiseitherarationalorirrational.

b).Anintegerisalsoarationalnumber.

c).Zeroisnotaninteger.

d).1.625isanirrationalnumber.

e).0.333...isarationalnumber.

f).3.31662479...isairrationalnumber.

g).9/11isnotawholenumberbutitisarationalnumber.

h).20%isawholenumber.

1.5