multiply by 1

7/25/2019 Multiply by 1

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Multiply by 1Multiply by 1Multiply by 1Multiply by 1

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Contents

Introduction..........................................................................1

Multiplying Fractions Part I..............................................5

Make a 1...............................................................................6

Solve for a Variable .............................................................8

Divide by 1...........................................................................9

Multiply by 1 Part I.........................................................10

Changing a Fractions Denominator..................................12

Decimals to Fractions ........................................................13

Decimals to Percents..........................................................14

Multiply by 1 Part II .......................................................15

Comparing Fractions Part I.............................................17

Comparing Fractions Part II............................................19

Fractions to Percents..........................................................20

Change Signs in a Fraction ................................................21

Simplify Fractions..............................................................22

Multiplying Fractions Part II ..........................................24

Simplify Algebraic Expressions ........................................26Simplify Algebraic Rational Expressions..........................28

Unit Conversions Part I...................................................29

Unit Conversions Part II .................................................33

Long Division ....................................................................36

Negative exponents............................................................37

Negative exponents in the Denominator............................39

Roots in the Denominator ..................................................40

Complex Numbers .............................................................41

Imaginary Numbers in the Denominator ...........................42


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InInInIntroductiontroductiontroductiontroduction

Three Dog Nights hit song in the sixties said

One is the loneliest number that there ever was.

I am not sure I agree with that.

However, I would agree that One is the Rodney Dangerfield of math:

Onedont get no respect.

Multiplying by 1 is one of the most useful techniques we have in math. We use the

technique of multiplying by 1 everywhere. As we shall see, much of what we do

when solving a wide variety of problems involves just multiplying by 1.

The principle of multiplying by one is deceptively simple:

Any quantity multiplied times 1 is equivalent to itself.

Translated into a mathematical equation, it reads:

x

1 = x

At first glance, the students initial response often is "Well, duh! Of course!" The principle

seems trivial. How could something so easy be useful?

But, think about E=mc2. This very simple formula (or equation) has very significant

use: it was the basis for developing the atom bomb. Along the same lines, these

simple appearing formulas are among the foundations of science and business:

A=LW relates area to the length and width of a rectangle;

C=D relates a circles circumference to its diameter through a constant

value, ;

D=RT relates distance traveled to rate of travel and time;

F=ma relates force, mass and acceleration in physics. A man was put on the

moon using this seemingly simple equation.


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E=IR relates voltage, current and resistance in electricity. All electricity and

electronics hinges on this relationship.

I=PRT relates the amount of interest to the amount of principle, the interest

rate and time. The worlds banking industry is built on this.

All of the examples above look pretty simple. But each one is a fundamental pillarof the science or industry that uses it.

Multiplying by 1 has wide application to what we do and where we will go in our

mathematics study. This booklet examines the technique as it is applied to different

types of problems. By understandingthe principle in depth then applyingthe princi-

ple we will be able to solve many types of problems.

This allows us to not have to memorize the solutions of different types of problems.

Actually, we do not even wantto memorize the solutions of different types of prob-

lems. Memorizing all the solutions is impossible as there are an infinite number of

different problems. Rather, we will understand a basic principle then learn how to

apply it to a wide range of problem types. This, then, produces the solutions to the

problems.

Multiplying by 1 is a first step in building a mathematical tool chest stocked with

mathematical problem-solving tools.

Think of carpentry.There are quite a few different kinds of carpentry tools: ham-

mers, nails, saws, chisels, squares, etc. If we have a hammer (a tool) and we un-

derstand the hammers use (the principle) we can hammer any nail that comes

along (the application of the principle). Of course it is not good enough to just know

only about the hammer. Using only a hammer we are limited in what we can build.

Therefore we have a variety of tools about which we must learn. Then, through the

appropriate use of each tool, we can build a wide variety of useful things: chairs,

tables, houses, etc.

We can think of our studies in math along the same lines. Just a carpenter accumu-

lates a carpentry tool chest we can accumulate a mathematical tool chest. We add

[mathematical] tools and we learn how to use them. Then, when we want to solve

a problem, we go to our tool chest and pull out the right tool and apply it appropri-ately.

As you step through the various lessons, we will be identifying and examining a ba-

sic math principle (tools) then learning how to apply it.


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1. Consider the carpentry example above. We are building a tool box of useful

tools. However, if we owned the tools but always just told someone else how

to use them, we would never really learn how to use the tools ourselves. We

must use the tools to master them. Math is no different. We must use the

math tools in order to really learn them. We must think through the problems

and apply the tools we have in a logical sequence. This means thinking!

Thinking is hard work. The main reason most people have problems with

math is that they never really learned how to think through a problem.

Students often try to memorizethe process not understandthe process. As stated

above, there are an infinite number of problems. We cannot memorize them all.

There is very little memorization in math (there is some, more on that below) but

there is a huge amount of understanding required. Understanding is easier than

memorization. And more useful, too!

2. There is some memorization and it is very important. The weakest math area

most people have is themultiplication facts. The multiplication facts are 3

x 4 = 12, 5 x 8 = 40, etc. If we do not know our multiplication facts we will

always have trouble with math.

Without knowledge of the multiplication facts we do not really understand

how the numbers work. If we do not understand something then it is essen-

tially magic. We say a few incantations, wave our arms around, throw some

eye of newt, bat wings and mouse tails into the cauldron and presto a

magic formula appears! If multiplication is "magic" then the rest of math will

be magic also.

The multiplication facts are often discussed (but with the widespread use of

calculators much less so now than they used to be). The student should be

very familiar with the multiplication facts at the minimum from 1 x 1 through

12 x 12. We should know these backwards and forwards.

We must know the divisionfacts.

What are the division facts? We rarely talk about the division facts but they are

just as important as the multiplication facts. Division facts are the reverse of the

multiplication facts: 4 divided by 2 is 2 (4 / 2 = 2); 45 divided by 9 is 5 (45 / 9 =

5); 72 divided by 8 is 9 (72 / 8 = 9), etc. There are two division facts for every

multiplication fact. As example, for the multiplication fact 3 x 9 = 27, we should

know 27 / 9 = 3 and 27 / 3 = 9. We should know both division facts for every

multiplication fact.


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Multiplication facts and division facts can be found on page Error!

Bookmark not defined.of this document. Practice them until you know

them instantaneously. This will greatly aide your study of multiplying by 1.

An excellent way to learn something is to teach it to someone else. Corner

your elementary-school child, significant other (if he/she REALLY loves youthey will help you here use the guilt option if necessary) or your dog, cat,

gold fish, what ever, and drill them on the math facts. In the process of help-

ing them learn you will also.

3. I have observed in the course of teaching math over a span of four decades

that most people view their study of math as simply learning how to push

numbers around.

Math is more than numbers.

Math includes numbers but also terminology, reading and understanding the

essence of a problem and the logical thinking necessary to reach a solution.

Mathematics is a precise language. If we do not understand the language we

cannot properly communicate. In this booklet, terms are presented. The student

is strongly encouraged to learn the definitions presented and use the terms

properly as they think about the math problems.

With those thoughts in mind lets get started on learning about

Multiplying by 1


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Multiplying FractionsMultiplying FractionsMultiplying FractionsMultiplying Fractions Part IPart IPart IPart I

a

b

c

d =

a c

b d

( "" "" 0)

In the previous lesson we saw that a fraction represents a part of an entire amount

or quantity.

If we want to take a fraction of a fraction we multiplythe two fractions together.

"Why," you ask, "would we want to deal with a fraction of a fraction?"

Consider the situation where you would want to divide3

4of a gallon of gas equally

among four people. Each person would receive1

4of the available quantity of gas. We

would do this mathematically by multiplying the3

4by

1

4:

3

4 x1

4

When we multiply fractions, we multiply straight across: numerator times numera-

tor and denominator x denominator like this:

a c ac

b d bd =

Using numbers in our example above, this

would look like this:

3

4 x

1

4 =

3 x 1

4 x 4 =

3

16

What we have done is take of . We do this by multiplying the fractions.

Note that the word of means multiply. of means to multiply the two frac-

tions.

The "ac" and "bd" means that the

value represented by "a" (some

number) is multiplied by the value of

"c" (some other number). Likewise

for "b" and "d".


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MMMMake a 1ake a 1ake a 1ake a 1

aa

=1

("a" 0)

The principle stated above should be pretty

obvious as

4

4=

1

56

56=1

857.3

857.3=1

So, we can represent the number 1 in many, many different ways.

Use the space below to write some of your own ones:

Let's expand on this a bit.

Notice that the top quantity (the

numerator) and the bottom

quantity (denominator) are the same.

But, they do not have to be the same

actual number!But they must be

equivalent amounts.

For instance, consider this:7 =1

This is true, is it not? Yes, it is.

Seven days is the same as one week. The numbers can be different as long as the

units are correct.

A fraction indicates a mathematical division

operation.

3

7 is read 3 divided by 7

We typically do not use the symbol. We

would rather use fractional symbols or re-

place the division operation with a multipli-

cation.

Math Language

In a fraction, the top number is thenumerator

and the bottom number is thedenominator:

Numerator

Denominator


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The unitsare the key to making the quantities equivalent.

These two amounts are the same. So, we could write:

7days

1week = 1

Notice here that the numerators and denominators are not the same number but

that they are equivalent amounts or quantities. And that is really all that counts!

(Its the unitsof measure! Pay attention to the units.) Write some more examples

of 1 where the numbers are not the same but the numerators and denominators

are equivalent.

Considering the above example:7days

1week= 1

does it make any difference if we write:1week

7days

= 1

No it does not! The quantity on the top (the numerator) is still equivalent to the quan-

tity on the bottom (the denominator) and the fraction is still equal to 1.

Similarly,

= 1gallon

4quarts = 1

4quarts

1gallon = 1

4quarters

1dollar

= 1 1dollar

4quarters

= 1

When we are writing "1" we can put either value in the numerator (on top) and the

other value in the denominator (on the bottom). We will use this idea later.

Use the examples you made up, above, and rewrite them another way:


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Solve for a VariableSolve for a VariableSolve for a VariableSolve for a Variable

We have seen how to make a 1: Divide a quantity by itself.

That is,3

13

= ,0.5

10.5

= , and

12 1

1

2

= , etc.

We can use this idea to solve for an algebraic variable in an equation.

Suppose we have 5 15x = .

We are looking for some number (represented by x) that when multiplied by 5 is

equal to 15. We should be able to rapidly come up with the solution that x must be

equal to 3 because 5 3 = 15.

But we must have a better way of finding the unknown value because not all prob-

lems are this easy and can be guessed.

So, lets make the coefficient of x into a 1.

We do this by dividing both sidesof the equation by

5.

So we would have

5 15

5 5

x= or

53

5x= and

we recognize that5

15

= so

3x =

Dividing both sides of the equation by the coefficient of

the unknown variable (in this case, x) will make the

coefficient equal to 1.

We have solved for the value of a variable by making a 1.

We must divide both sides ofthe equation by the same

amount if the equation is to

remain true. Consider:

8 = 8

If we divided only 1 side by 2

we would have

4 = 8

which is clearly not true. But

dividingbothsides of the

equation by the same amount

(2) keeps the equation true:

4 = 4


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Divide by 1Divide by 1Divide by 1Divide by 1

Lets take a look at what happens if we divide by 1. We have a principle to con-

sider:

1

a= a

("a" )

In this principle, the letter "a" represents any number. So you could substitute 4 or

279 or any other number you want for "a".

A :

4

1 = 4

279

1 = 279

121

= 12

45.6

1 = 45.6

Does the third example, above, seem a little strange to you?

One half () divided by 1 is one half. As the principle states:Anyquantity divided

by 1 is equal to itself. That includes fractions and, as shown in the last example

above, decimals also.

We use this idea to make fractions when we need one. For instance, the number 5

can be made into a fraction by dividing by 1:5

51

= ,23

231

= , etc.

Math language

A quotientis the result of dividing two numbers.

If we divide 6 by 3 the quotient is 2.

62

3

=


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Multiply by 1Multiply by 1Multiply by 1Multiply by 1 Part IPart IPart IPart I

In an earlier lesson we learned how to make a 1.

Now let's see how this might be useful. What can we do with the 1 we made?

We will start by learning another principle:

:

1 =

In this principle, "a" represents any quantity (we will stay with numbers for now). So

you could substitute 4 or 279 or any other number you want for "a".

A :

4 1 = 4

279 1 = 279

1 =

45.6 1 = 45.6

Lets put a couple of things together and see

what happens.

Multiply3

7 times

5

5 5

5 is equivalent to 1 so we are multiplying by 1

3 5 3 5 15

7 5 7 5 35

= =

Multiply numerators (straight across) and

multiply denominators (straight across)

The symbol "" indicates

multiplication or "times."

The "" is the same as the

"x" many people use in mul-

tiplication. In algebra, "x" iscommonly used for an un-

known value. Using them

both will be confusing. So

we will use the "".

However, be careful, that the

4 1 does not morph into

4.1. Handwriting neatness

counts!


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Our principle states: Any number multiplied by 1 is equivalent to the original num-

ber. Therefore, since all we did was multiply by 1 (i.e.5

5) we see that

3 15

7 35=

What we actually accomplished here is to convert 37

into an equivalent quantity

with a denominator of 35.

We have a method of changing a fractions denominator by simply multiplying by 1!


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Changing a Fractions DenominatorChanging a Fractions DenominatorChanging a Fractions DenominatorChanging a Fractions Denominator

We will use the Multiply by 1 Principle to change the denominator of fractions.

Lets change the fraction5

8into the equivalent number of

1

16ths.

We want to change the denominator of the5

8(the 8) into a 16.

If we multiplied the 8 by 2 the product is 16 ( 8 2 = 16 ).

So, we multiply like this:5

8

2

2

Notice that we are multiplying by2

2. Why?

(Answer: Because we can only multiply by 1 and not change a quantitys value and

2

2 = 1).

When we multiply fractions we multiply straight across, so:

5 2 10=

8 2 16

We have converted5

8into the equivalent number of

10

16. These two fractions repre-

sent the same value and are equal.

Lets try a few more. Convert each of the fractions in column (1) to an equivalent

fraction with the denominator shown in column (2). Multiply by 1 to obtain thesolution. Show your work in column (3)

(1) (2) (3) E

3

4

?

12

5

6

?

24

4

11?

55


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Decimals to FraDecimals to FraDecimals to FraDecimals to Fracccctionstionstionstions

We multiply by 1 to convert decimals to fractions.

Consider 0.125.

Multiply 0.125 by1000

1000 The number of decimal places (in this case 3

decimal places for the 0.125) tells you the

power of 10 to use when making your 1. For 3

decimal places we would use310 or 1,000

(which is 10 10 10). Another way of stating

this is that the number of zeros in the 1 is the

same as the number of decimal places. The

0.125 has three decimal places so we use

1,000 to make our 1 because it has 3 zeros.

0.125 1000 125

1 1000 1000 = Multiply across the numerator and the denomi-

nator. (Remember how we can make any num-

ber into a fraction by dividing by 1? We looked

at this in the section Dividing by 1 on p. 7.)

Another example: lets convert 0.75 into a fraction.

0.75 100 75

1 100 100 = Multiply 0.75 by

100

100.(The 0.75 has two deci-

mal places so we use a multiple of 10 that has

two zeros: 100).

.


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Decimals to PercentsDecimals to PercentsDecimals to PercentsDecimals to Percents

We multiply by 1 to convert decimals to percents.

Per cent means per hundred. The word cent show up in the English language

when 100 is meant: century = 100 years; 100 cents = 1 dollar; there are 100 cen-

timeters in a meter, etc.

The word per is a mathematic code word meaning divided by.

So when we say per hundred we are really saying (after the decoding) divided by

100.

When we change a number so it is divided by 100,

we can go directly to percent.

We define a symbol to represent the fraction1

100.

% = 1100

So then, starting with 1: 1 =100 1

100 100 % 100%100 100

= = = = 1

Convert 0.523 to a percent

0.523 100% 53.2% = Multiplying by 100% is the same as multi-

plying by 1. (1 = 100% )

So we do not change the value of thenumber, only its form.

Consider all the steps for the above problem:

0.523 100 52.30.523

1 100 100= = Recognize that

52.3 153.2

100 100=

153.2 53.2%

100 = 1

100= %


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Multiply by 1Multiply by 1Multiply by 1Multiply by 1 Part IIPart IIPart IIPart II

Now we can convert a fraction to a different

denominator.

Let's see how to use this idea.

Is4 16

9 36= ?

If so, how can we demonstrate that it is?

We must represent both items (the

4

9and the

16

36) with a common

reference. To do this, both fractions must have the same denominator. That

would be a 36. We will determine how many1

36

ths each fraction represents.

Why did we pick 36?

Let's look at the4

9. If we were to multiply the denominator by 4, we would then

have a fraction with a bottom number of 36. So, this fraction starts to look like the

other fraction that also has a denominator of 36. Both fractions would be represented

with a common reference (or common denominator). In amount, both fractions

would represent the quantity of1

36sized pieces.

But, we cannot just arbitrarily multiply only one part of a fraction by some valuewithout changing the value of the fraction. We must end up with an equivalent

amount. So, to not change the value of the number, we will

multiply by 1. (Why can we do this?) But we will pick a special "1":4

4. (Principle:

Any number dividedby itself is 1.)

So if we multiply4

9by

4

4 we get:

4 4 16

9 4 36 =

So then, the two numbers are equivalent even though they do not look the same.

Math Language

Numerator

Denominator

TheNumeratoris the number on top of the

fraction; theDenominatoris the number on

the bottom of the fraction.

When we multiply frac-

tions, we multiply the

numerators together and

we multiply the denomi-

nators together


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How did we do this? We multiplied by 1 (4

4). Observe that the 1 was carefully se-

lected. How did we select the one? We found a number we could use to multiply the

smaller denominator (the bottom number of the fraction) to get the larger denomi-

nator. Then we used that number to multiply both the top and the bottom of the

fraction [to make it equivalent to 1].

Let's look at another example:

Is2

3=

13

18?

We must represent both fractions with some com-

mon denominator. This would be the number of1

18

ths that each fraction represents. The denominator

for both fractions must be 18 for us to compare

them. Why1

18?

(Important point: the denominator for both fractions will always be equal to

or larger than the larger of the two original denominators).

2

3

6

6 =

12

18 To determine how many

1

18ths are the equivalent of

2

3we multiply by 1.

What "1" would we use? Use a 1 to change the de-

nominator of the2

3(the 3) to an 18. Multiply the 3

times a 6. But our rule states that we can only

multiply by 1.

So we would have to multiply both the numerator

and the denominator by 6. Thus we would multiply

by6

6.


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Comparing FractionsComparing FractionsComparing FractionsComparing Fractions Part IPart IPart IPart I

We will use our fundamental principles to compare two (or more) fractions to de-

termine if the fractions are equal or, if not equal, which fraction might be larger

compared to the other.

The fundamental principles are

"Making a 1"

a

a =1

("a" 0)

and

"Multiply by 1"

This is stated mathematically as:

a 1 = a

Suppose we must determine if two fractions are equal, and, if not which of two

numbers is larger:11

12or

8

9?

We would go about this by establishing a common reference by which the numbers

can be compared.

What is meant by a common reference?

Consider a situation where you have several types of currency -- British Pounds,

French Francs and Dutch Marks -- and you must determine the relative value of

each sum of money. How would you determine which currency represented the

most value?


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If you were used to U.S. Dollars, it would be reasonable to convert each of the curren-

cies into Dollars then compare their values in this common reference.

In a like manner, when we compare fractions we must transform them into a com-

mon reference. The common reference is that each fraction has the same de-

nominator.

In the numerical example above, we would change both fractions to have the samedenominator.

Begin by picking a value for the denominator that both 12 and 9 will divide into

evenly.

Pick the value 36. (Both 12 and 9 will divide evenly into 36).

So then, multiply both fractions by a specially chosen 1 to convert the denomina-

tors into 36.

11 3 33

12 3 36

= and8 4 32

9 4 36

=

Write the two division facts for the following multiplication facts:

Multiplication fact Division Facts

7 x 4 = 28 )

)

3 x 9 = 27 )

)

Above, we saw that multiplication is associative (The associative property of multi-

plication). Is there an associative property of division? Why or why not?1How about

subtraction?

Learn the multiplication and division facts completely. Before each mathematics

lesson, spend time reviewing the math facts. In time, as the student reviews the

facts and uses them, the student will learn them. Learning the facts cannot be over

emphasized. They are one of the keys to success in math.

1 There is no associate property for division.32

8is not equal to

8

32. Likewise, sub-

traction is not associative: 10 7 is not equal to 7 - 10


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Comparing FractionsComparing FractionsComparing FractionsComparing Fractions Part IIPart IIPart IIPart II

Sometimes it is not convenient to convert the denominator of one fraction to

the same value as the fraction to which it is being compared.

Consider the two fractions

5

6and

7

8

We cannot multiply 6 by some number to get 8. And we cannot multiply 8 by

some number to get 6. We will have to change both fractions denominator to

some different value.

Both 6 and 8 will divide into 24. We can use 24 as a common denominator.

5 4 20

6 4 24

= Multiply by4

4

(or 1) to convert the original

fraction to a denominator of 24

7 3 21

8 3 24 = Multiply by

3

3(or 1) to convert the original

fraction to a denominator of 24

Now we can easily see that the fraction7

8is larger than the fraction

5

6.

21 20 1

24 24 24 = And we can then subtract the smaller frac-

tion from the larger fraction to determine

the difference. Subtraction requires common

denominators in fractions which we already

have!.


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Fractions to PercentsFractions to PercentsFractions to PercentsFractions to Percents

We multiply by 1 to convert fractions to percents.

We will use the concept presented earlier of changing the value of a fractions

denominator.

A fraction, as we have seen, is a portion of a whole: means three parts of

a total of four.

Likewise, a percentage is so many parts of 100.

The basics of converting a fraction to a percentage is changing the fractions

denominator to 100. We accomplish this by multiplying by 1.

To change3

4to a percentage multiply by 1 where 1 =

25

25:

3 25 3 25 75

4 25 4 25 100

= =

Earlier we noted that % is the symbol representing1

100

Then75 1

75 75%100 100

= =

Other examples:

17 5 17 5 85 185 85%

20 5 20 5 100 100

= = = =

7 12.5 7 12.5 87.587.5%

8 12.5 8 12.5 100

= = =

Notice that the 1 is

12.5

12.5

When we convert a fraction to a percent we are really just converting the

fractions denominator to 100 by multiplying by 1.

Convert the following fractions to percents:


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Change Signs in a FractionChange Signs in a FractionChange Signs in a FractionChange Signs in a Fraction

a

b b

=

("" 0)

It is often useful to change the sign of a fractions denominator or numerator.

Using Multiply by 1, we can easily do this.

Consider the fractiona

b(where b 0) and we desire to not have a negative

sign in the denominator.

Using1

11

=

we can multiply the fraction and change its signs:

1 ( 1)

1 ( 1)( )

a a a

b b b

= =

Using this in an example:

3 3 1 3

4 4 1 4

= =

So we can change the signs in

a fraction by multiplying by 1.

Multiplying Signed Numbers

Remember the simple rule:

Multiplying like signs yield a + product

Multiplying unlike signs yield a product.Or

(+)(+) = + (+3)(+2) = +6

()() = + (3)(2) = +6

(+)() = (+3)(2) = 6

()(+) = (3)(+2) = 6

This applies to division in a like manner:

Dividing like signs yield a + quotient

Dividing unlike signs yield a quotient.


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Simplify FractionsSimplify FractionsSimplify FractionsSimplify Fractions

At this time, we have learned two fundamental principles:

a

a =1

("a" 0)

:

1a a =

We use these principles to simplify

fractions.

Simplify the number21

35

Factor 21 into 3 7

Factor 35 into 5 7

We start by factoring both the 21 and the 35.

Math language:

To factor means to find two or more

numbers that when multiplied together

produce the original number.

Example: we can factor 6 into 2 3, or

2 3 = 6.

We can factor 12 into 2 2 3 = 12 or

2 6 = 12 or 4 3 = 12


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21 3 7

35 5 7

=

Using the factors, rewrite the original fraction.

3 7

5 7 = Group the factors so we have 1s.

Recognize that7

7 = 1

3 31

5 5 =

Any number times itself is 1

We have simplified the fraction21

35to

3

5using Multiply by 1.

We simplified the fraction with the following process:

1. Factor the numerator and denominator

2. Identify the "1"s (Any number divided by itself = 1)

3. Remove the "1"s (Any number multiplied by 1 is equivalent to

the original number.)

Let's try another one: simplify30

42.

Factor 30 into 5 6 and

Factor 42 into 7 6

We can factor 30 a number of different

ways: 2 15 or 3 10 or 5 6

Same thing with 42: 2

21 or 3

14 or 7

6Notice that we picked factors that have a

common number: 6. There is a reason for

this. Why is this important?

30 5 6

42 7 6

=

Using the factors, rewrite the original fraction.

Recognize that there is a common factor of 6

in the numerator and denominator that will

divide with a quotient of 1.

5 51

7 7 =

Any number times itself is 1

We have simplified the fraction30

42to

5

7using Multiply by 1.


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24

Multiplying FractionsMultiplying FractionsMultiplying FractionsMultiplying Fractions Part IIPart IIPart IIPart II

1ac a c a a

bc b c b b

= = =i i

As we noted earlier, when we multiply fractions together, we simply multiply

the numerators together and we multiply the denominators together.

Just multiply them straight across.

So, using the above, let's consider the fraction

ac

bc

"a", "b" and "c" just represent num-

bers. So, the principles we have

learned still apply.

Let's rewrite the above as

1ac a c a a

bc b c b b

= = =

Notice that in the above 1cc

=

Try these:

a c

a b

=

x y

y z

=

In the second example re-write y z as z y (it makes no difference what

order we multiply numbers; e.g. 2 3 = 3 2)

then you can group the ys intoy

y.

The examples above are referred to as "algebraic fractions". But they follow

the same rules as arithmetic.

There are a couple of things to understand

about the fraction as written here:

ac a c= i

The number represented by "a" is

multiplied by the number represented by

"c".

For example, if a = 3 and c = 5, then the

value ac would = 15.

But, in this general case we do not know

what the values of a and c are, so we

cannot multiply them together to obtain a

numerical product.

Therefore, we leave the numerator as ac

and likewise in the denominator, bc.


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25

Effectively learning to simplify fractions is extremely valuable. Generally, we can

make our computations much easier and less error prone if we simplify first.

Consider multiplying these fractions:20 25

45 16

The way to go about this is NOTto start multiplying things together.

(I.e. 20 x 25 and 45 x 16).

Rather, we would simplify first.

20 = 4 5

45 = 9 5

25 = 5 5

16 = 4 4

By combining our knowledge of Math Facts

and factoring, we determine the factors of

each element of both fractions.

If there are more than two fractions multi-

plied do this for the numerators and de-

nominators of each fraction.

20 25

45 16 =

(4 5) (5 5)

(9 5) (4 4)

=

The parenthesis shown here make it easier

to see where the factors come from. You

would not include them when actually doing

this problem.

4 5 5 5

4 5 4 9

=

Rearrange the factors to identify the 1s.

Numbers can be multiplied in any order,

2 3 = 3 2, so rearranging them helps

see what is going on.

5 5 251 1

4 9 36 = Simplify the fractions to 1 and multiply the

numerators and denominators.

Did you understand how the process moved from one part to the next?

We simply factor (using our Math Facts), then re-group, then find equivalents

of 1:4

4and

5

5. When we finish, we know there are no more factors of 1 in

the fraction and we have simplified it as much as possible.


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26

Simplify Algebraic ExpressionsSimplify Algebraic ExpressionsSimplify Algebraic ExpressionsSimplify Algebraic Expressions

Simplifying algebraic expressions employ the same methods as those used to

simplify arithmetic fractions.

For example consider:

ab

ac . Note that we have a common factor in both the

numerator and the denominator.

So, we can consider this algebraic fraction as

a b

a c

which can be rewritten as

a b

a c

Recognizing that 1a

a=

Our original expression is simplified to 1

b

c

or just

b

c

For example, lets simplify2

5

xy

yz

2 2 2

5 5 5

xy y x y x

yz y z y z= =

Rearrange the fraction to identify the ratios of 1

2 21

5 5

y x x

y z z =

2

5

x

z The fraction is in its simplest form with no

common factors in the numerator and de-

nominator.

More complicated expressions are handled the same way.

Starting withab ac

ad ae

+

+we use the Distributive

Property to obtain

( )

( )

a b c

a d e

+

+

which can be rewritten asa b c

a d e

+

+

or 1b c

d e

+

+i

The Distributive Property

( )ab ac a b c+ = + As example:

Let a=2, b=3 and c=5

2(3) 2(5) 2(3 5)+ = + 6 10 2(8)+ =

16 16=


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Which then is simplified to our final simplified solution ofb c

d e

+

+

To illustrate, lets simplify2 3

5 7

xy x

xy x

+

2 3

5 7

xy x

xy x

+

=

(2 3)

(5 7)

x y

x y

+

Factor the common factors using the Dis-

tributive Property: ( )ab ac a b c+ = +

2 3

5 7

x y

x y

+

=

2 31

5 7

y

y

+

Identify the ratios of 1

2 3

5 7

y

y

+

The fraction is in its simplest form with no

common factors in the numerator and de-

nominator.

Multiplying by 1 allows us to simplify algebraic expressions.


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Simplify Algebraic RationalSimplify Algebraic RationalSimplify Algebraic RationalSimplify Algebraic Rational

ExpressionsExpressionsExpressionsExpressions

Expanding on the previous section, we use multiply by 1 to simplify more

complicated algebraic expressions such as

2

2

6 8

7 10

x x

x x

+ +

+ +

2

2

6 8 ( 2)( 4)

7 10 ( 2)( 5)

x x x x

x x x x

+ + + +=

+ + + +

As before, we begin by factoring the nu-

merator a denominator2:

( 2) ( 4)

( 2) ( 5)

x x

x x

+ +=

+ + We recognize that

( 2)1

( 2)

x

x

+=

+

4 41

5 5

x x

x x

+ += =

+ +

The expression in simplest terms.

Thus we have used multiply by 1 to simplify rational expressions.

2 Demonstrating factoring algebraic polynomials is beyond the scope of thistext.


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Unit ConversionsUnit ConversionsUnit ConversionsUnit Conversions Part IPart IPart IPart I

Let's see how to use the idea of multiplying by 1 with dimensions or units.

Recalling an earlier principle

:

1 =

Notice that the principle says that the product is "equivalent" to the original

number. This means that theproduct'smagnitude (a magnitude is an

amount or quantity represented) is equal to the original number. But,

equivalent amounts may not look the same.

We use this sort of thing all the time.

We know that 12 inches is equivalent (i.e. the same length as) 1 foot:

12 inches = 1 foot.

We know that 3 feet is equivalent (i.e. the same length as) one yard:

3 feet = 1 yard.

We know that 365 days is equivalent (i.e. the same time as) to one year:

365 days = 1 year.

Notice that in the examples above, we are NOT saying that 12 = 1 or that 3

= 1 or 365 = 1. We must be very careful of our units here: inches, feet,

yards, days, years. Part of learning math effectively is to realize that the

units are part of the number. So we deal with this concept of two things hav-

ing equivalent measures even though they do not look the same.3

3 Here is a fun thing you can pull on your friends: Bet them that you can prove

that 1 = 7. Write it like this: 1 = 7 . Leave a space between the 1 and the=. Of course, they will dispute the fact that 1 = 7. After the appropriate amount ofwrangling, in which they refuse to believe that 1 = 7, you simply write the words week

and days in the appropriate spots: 1 week = 7 days. With the appropriate units 1 does


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30

Math is more than just numbers. The numbers actually represent something.

So, just as we saw in the previous lesson, consider the three examples

above, we can make a "1" in the following ways:

12inches1

1foot=

3feet1

1yard=

365days1

1year=

The equations above are derived from true mathematical statements:

12 inches = 1 foot Start with a true statement.

12inches 1foot

11foot 1foot= =

Dividing both sides of the equation by 1

foot. It is very important to not loose the

units. We could have used 12 inches to di-

vide by also.

121

1

inches

foot=

This gives us a conversion factor between

inches and feet.

How do we use this?

Consider the question: How many inches (in) are in 4 feet (ft)?

4ft = 4ft 1i

12in

4ft = 4 ft1ft

i 12in

1 =1ft

4ft 12in 4 12in f

1 1ft 1 f

t

t=

i i

i

4 12in f 4 12in f

1 f 1 ft

t t

t=

i i i

i ft

1ft

=

4 12in48in

1=

i

, 4 = 48

indeed equal 7. You can use this with days and years, gallons and quarts, dollars and

dimes, etc.

The mathematical "sentences" to the left are equa-

tions. Notice that an equation has an =(equal sign)

in it. If no equal sign is present, it is referred to as anexpression.


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Consider another example we all can understand: US money.

How many quarters are equivalent to one dollar (USD)? Of course, we know

the answer is 4 quarters = 1 USD. But lets examine it in detail.

We first think of them in a similar way:

A quarter is 25 cents; a dollar is 100 cents.

So we can describe them both with a common reference: cents.

1 quarter = 25 cents Start with a true statement.

1quarter 25cents=

25cents 25cents

Divide both sides of the equation by 25

cents. It is very important to not loose the

units. Cents will be in the denominator.

1quarter1

25cents=

251

25

cents

cents=

1 quarter 125 cents

=

100 =1 D A

100cents1

1 USD=

Divide both sides of the equation by 1 USD.

Cents will be in the numerator.

100 cents= 1

1 USD

1 quarter 100 cents= 1

25 cents 1 USDi

(1 1 1=i

)

.

1

.

100 cents quarter= 1

25 cents USD

4 quarter= 1

1 USD

D ; cents

= 1cents

!

4 = 1 D 1D

Clearly, this is a long drawn out process. We generally do not consciously

think about each of the above steps when we consider how many quarters


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are in one US Dollar. We are quite familiar with U.S. money. However, these

are the steps involved.

Lets look at another example using units we may not be familiar with:

How many ounces (oz) are in 3 kilogram (kg)?

16 oz = 1 lb A truth.

16oz= 1

1 lb

Make a 1 by dividing both sides by 1 lb.

2.2 = 1 Another truth.

2.2 lbs= 1

1 kg

Divide both sides by 1 kg.

16oz 2.2 lbs 16oz 2.2 lbs=

1 lb 1 kg 1 kg 1 lbi i

Multiplying these together and rearranging

16 2.2oz 35.2oz= = 1

1 kg 1 kg

i

We now have a conversion factor between

ounces and kilograms. Notice thatlb

= 1lb

.

Units behave just like numbers.

3kg 35.2oz 3 35.2oz kg=

1 1 kg 1 kg

i

i For 3 kg, we multiply using our conversion

factor which is equal to 1.

kg= 1

kg

3 = 105.6 So we can convert between kilograms and

ounces by multiplying by 1.


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Unit ConversionsUnit ConversionsUnit ConversionsUnit Conversions Part IIPart IIPart IIPart II

We have learned that:

Any quantity divided by itself = 1.

The product of any number and 1 is equivalent to the original

number.

We can use these ideas to convert from one set of units to another.

As examples:

How many quarts are in 3 gallons?

How many Y (Japanese Yen) can we get for 4 USD (U.S. Dollars)?

How many feet are in 29 yards?

The three questions above are examples of unit conversions. There are many

more daily examples.

Let's look at how we can apply the idea of 1 to these types of problems.

Consider: Any number divided by itself = 1.

We saw earlier if we have two quantities that equal the same amount they

are equal even if they do not look the same. For instance 4 quarters = 10

dimes. Both 4 quarters and 10 dimes are each worth 100 cents. Even though

they do not look the same both represent the same amount.

So, we can say that because they represent the same amount:4quarters

= 110dimes

To answer the question of how many quarters can we get for 30 dimes, we

would start with the 30 dimes and multiply by our representation of 1:

30dimes 4quarters=

1 10dimesi 4quarters = 1

10dimes

.

.

3 4 quarters 10 dimes=

1 10 dimes

i i

i i 30 3 10

. :

!

101

10= and

dimes= 1

dimes 1

.


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3 4 quarters = 12quartersi Dividing common factors and units

, 30 = 12

Of course this makes sense because 30 dimes = $3.00 and 12 quarters = $3.00

In this process we have used unit conversions to convert from dimes to quar-

ters.

Let's take a look at how many quarts are in three gallons:

4 quarts = 1 gallon These two quantities are equivalent and can

be used to express 1

4quarts= 1

1gallon

3gallons 4quarts1 1gallon

=i Start with 3 gallons and multiply by our 1

3 4quarts gallons=

1 1 gallonsi i Rearrange and recognize

gallons= 1

gallons. The

gallons will divide just like the numbers do

3 4 quarts = 12 quartsi

We have used multiplying by 1to convert 3 gallons into 12 quarts.

Let's determine how many Y (Japanese Yen) can we get for 4 USD (U.S. Dollars):

1 USD = 106 Y The exchange rate as of this writing. Source:

http://www.xe.com

1061

1USD=

Make a 1.

4 USD 106

1 1USD

= Multiply 4 USD by our special 1

4 106 USD

USD =

i

1

We grouped the multiplication elements (fac-

tors) so that we could divide the USD ratio to

obtain a 1. Note thatUSD

1USD

=

4 106 424 = Do the multiplication


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4 USD = 424

Let's determine how many feet are in 29 yards:

1 yard = 3 feet

3ft= 1

1yd

Make a 1.

Divide both sides of the equitation by 1 yard.

This puts the units yards in the denomina-

tor so it will divide into yards in the nu-

merator in a subsequent step.

29 yds 3ft=

1 1yd

Multiply 29 yds by our special 1

29 3 ft yd= 87ft

1 yd

We grouped the multiplication elements (fac-

tors) so that we could divide the USD ratio to

obtain a 1. As we have done before:yd

= 1yd

29 yds = 87ft

We have converted yards to feet using Multiplying by 1.


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Long DivisionLong DivisionLong DivisionLong Division

Long division involves something like 23 1897 . We know how to do long divi-

sion with whole numbers.

However, consider if you are dividing by a non-whole number such as

4.3 245

In order to divide, we must have a whole number for the divisor. The divisor

is the number we are dividing by - the 4.3 in this case. Notice the 4.3 has a

decimal in it. We have to change it to be a whole number, i.e. 43.

Lets consider this problem in light of what we know about fractions. We

know that fractions are just division problems. So the division problem above

can be written as

2454.3

In order to have a whole number in the denominator, we will change the de-

nominator by multiplying by 10 (in the case of long division such as this we

will always multiply by a factor of 10: 10, 100, 1000, etc.). But, as before,

we can only multiply by 1 (a 1 = a) so we do not change the value of the

number, only its appearance. So when we multiply both numerator and de-

nominator by 10 (that is, we are multiplying the original fraction by 1 written

as10

10

), we get

245 10 2450

4.3 10 43 =

Now when we re-form our long division problem we have

43 2450

Notice what happened. By multiplying both numbers involved in the division

by 10 (effectively multiplying by 1) we move the decimal in the divisor to

make the divisor a whole number. When we did that we also increased the

number being divided into (the dividend) by a factor of 10. We generally takethe short cut of simply moving the decimal in the divisor to the right to make

a whole number then moving the decimal in the dividend the same number

of places. But, as you see here, we are really just multiplying by 1 to achieve

the results.


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Negative eNegative eNegative eNegative exponentsxponentsxponentsxponents

Let us first consider some exponent basics. Exponents are really short-hand

notation for successive multiplying. An exponent tells us how many factors to

use when a base is multiplied times itself. As examples:

a2= a a

a3= a a a

a4= a a a a

To multiply exponential terms with like bases we simply add the exponents:

ama

n= a

m+n

To see how this works consider

a2a

3= a

2+3= a

5

a2= a a and a

3= a a a

So, multiplying the two together we see

a2a

3= (a a) (a a a)

Which is simply

a a a a a = a5

It is certainly possible to have a situation such as4a . So what does this mean?

Using the multiplication rule for exponents indicated above we can changethe -4 exponent into a positive value by multiplying by 1:

4

41

a

a=

4 4 4 4 4 4 4 04

4 4 4 4 4

1

1 1

a a a a a a aa

a a a a a

+

= = = = = =i (note that

0 1a = )

So,4

4

1a

a

= . This can be generalized to

1mm

aa

=

We converted a term with a negative exponent to a term with all positive ex-

ponents by multiplying by 1.

You might have noticed in the above example that0 1a = . Lets consider why.

0 ( )m ma a + = 0 = m + (-m)

( )m m m ma a a+ = i Using the rules of exponents: m n m na a a +=i

Math Symbology

exponentbase


42/46

1m m mm

a a aa

= =i i Where

1mm

aa

= as shown above

1m

m

a

a=

So0 1a =

This leads us to another general rule for exponents. Lets consider what

happens when we divide two variables such asm

n

a

a

1

1

m m

n n

a a

a a= =

Use our rule for multiplying fractions.

m na a = Where1mm

aa

= as shown above (replace the m with

n to fit our example).

m n

a

Combining like basesm

n

a

a=

m na So, when we divide like bases we subtract the expo-

nents.

Multiplying by 1 has lead us to a new exponent rule for dividing like bases:

m

n

a

a=

m na


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Negative exponents in the DenominatorNegative exponents in the DenominatorNegative exponents in the DenominatorNegative exponents in the Denominator

Sometimes we run into a situation like this: 41

a. In this case we must deal

with the negative exponent in the denominator. A negative exponent in the

denominator is a mathematically undesirable thing. We do not allow negativeexponents in the denominator or the numerator of a final answer.

We want only positive exponents in our terms. So we would multiply the

fraction by a special 1:

4

41

a

a=

4

11

a

So 4 4

4 4 4 4

1 1a a

a a a a =

i

i

i

Using our exponent rule for multiplication:

4 4 4 44

4 4 4 4 0

1

1

a a a aa

a a a a = = = =

i

i

(remembering that0 1a = ).

Rework the following such that it has no denominator (other than 1):

5 3

4 2a x

a xy z= Make our 1: 1 =

4 1 2 1

4 1 2 1a x y za x y z

We simply applied the opposite sign to the expo-

nents in the denominator.

5 3 4 1 2 1

4 2 4 1 2 1

a x a x y z

a xy z a x y z

=i

Multiply by our 1

5 -4 3 -1 -2 -1

4 -4 -1 2 -2 -1

a a x x y z=

a a x x y y z z

Multiply numerators and denominators. Regroup

for clarification. Note that x = x1and z = z1. (We

do not write exponents of 1.)

5 4 3 1 2 1

4 4 1 1 2 2 1 1

a x y z

a x y z

=

Combine exponents according the exponent rule

for multiplying like bases:m n m na a a +=i

1 2 2 11 2 2 1

1 1 1 1

a x y za x y z

=

i i i

Simplifying and noting that

0 1a =


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Roots in the DenominatorRoots in the DenominatorRoots in the DenominatorRoots in the Denominator

Sometimes, we run across the situation where we have a root in the denomi-

nator of a fraction:7

5.

It is poor mathematical form to leave roots in the denominator. The thorough

mathematician will rationalize the fraction to remove the offending root.

We do this by multiplying by 1:5

5

So then,

7 5 7 5 7 5

55 5 5 5= =i

We have eliminated the root in the denominator by multiplying by 1.


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Complex NumbersComplex NumbersComplex NumbersComplex Numbers

Complex numbers are of the form a bi+ where i is the imaginary number 1 .

It is poor form to have a complex number in the denominator of a fraction

such as

1

3 2i+

To eliminate the imaginary number

from the denominator we would multi-

ply by a specially selected 1 comprised

of the complex conjugate of 3 2i+ which

is 3 2i .

The 1 would be3 2

3 2

i

i

So1 3 2 3 2

3 2 3 2 (3 2 )(3 2 )

i i

i i i i

=

+ +

Evaluating the denominator we have:

2

3 2 3 2

9 6 6 4 9 4( 1)

i i

i i i

=

+

Notice here that 2 1 1 1i = =

So

3 2 3 2

9 4 13

i i

=+

From here, of course, we would simplify this fraction as much as possible.

But this example requires no further simplification although we could write it

as

3 2

13 13i to put it in the form a bi+

By multiplying by 1, we have eliminated the complex expression from the

denominator.

Complex Conjugate

A complex conjugate is a complex num-

ber where the sign of the imaginary term

is reversed. For example the complex

conjugate of 6 5i is 6 + 5i.

In general the complex conjugate of

a + bi is a bi.


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Imaginary Numbers in the DenominatorImaginary Numbers in the DenominatorImaginary Numbers in the DenominatorImaginary Numbers in the Denominator

As in an earlier section, a problem also occurs when the root of the negative num-

ber is in the denominator of a fraction. Like before, we do not leave fractions with a

root in the denominator. We must rationalize the fraction.

We use Multiply by 1 to accomplish this.

Consider the fraction3

5.

We will multiply by 1:5

5

3 5 3 5

55 5

=

Then3 5 3 5 1 3 5

5 5 5

i = =

It is considered poor form to leave a negative sign in the denominator (-5), so

3 5 3 51

5 1 5

i i =

or

3 5

5i

multiply by 1

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