UNIT 1.6 MULTIPLYING AND UNIT 1.6 MULTIPLYING AND

DIVIDING REAL NUMBESDIVIDING REAL NUMBES

Warm UpMultiply or divide.

1. 6(7)

3. 42 7÷

4.2.

6. Write as an improper fraction.

42 6

4

5. Write as a mixed number.

Multiply real numbers. Divide real numbers.

Objectives

reciprocalmultiplicative inverse

Vocabulary

When you multiply two numbers, the signs of thenumbers you are multiplying determine whetherthe product is positive or negative.

Factors Product

3(5) Both positive

3(–5) One negative

–3(–5) Both negative

15 Positive

–15 Negative

15 Positive

This is true for division also.

WORDS

Multiplying and Dividing Numbers with the Same Sign

If two numbers have the same sign, their product or quotient is positive.

NUMBERS 4 • 5 = 20

–15 ÷ (–3) = 5

Multiplying and Dividing Signed Numbers

WORDS

Multiplying and Dividing Numbers with Different Signs

If two numbers have different signs, their product or quotient is negative.

NUMBERS

6(–3) = –18

Multiplying and Dividing Signed Numbers

–18 ÷ 2 = –9

Find the value of each expression.

–5 The product of two numberswith different signs is negative.

Example 1: Multiplying and Dividing Signed Numbers

A.

12 The quotient of two numberswith the same sign is positive.

B.

Find the value of the expression.

The quotient of two numberswith different signs is negative.

Example 1C: Multiplying and Dividing Signed Numbers

First substitute for x.

Find the value of each expression.

–7 The quotient of two numberswith different signs is negative.

Check It Out! Example 1a and 1b

1a. 35 ÷ (–5)

44 The product of two numberswith the same sign is positive.

1b. –11(–4)

Find the value of each expression.

–6x = –6(7) First substitute 7 for x.

Check It Out! Example 1c

1c. –6x for x = 7

= –42 The product of two numbers with different signs is negative.

Two numbers are reciprocals if their product is 1.A number and its reciprocal are called multiplicative inverses. To divide by a number, you can multiply by its multiplicative inverse.

Dividing by a nonzero number is the same as Multiplying by the reciprocal of the number.

10 ÷ 5 = 2 10 ∙ = = 215

105

Multiplicative inverses

Dividing by 5 is the same as multiplying by thereciprocal of 5, .

You can write the reciprocal of a number by switching the numerator and denominator. A whole number has a denominator of 1.

Helpful Hint

Example 2 Dividing by FractionsExample 2A: Dividing by Fractions

Divide.

To divide by , multiply by .

Multiply the numerators and multiply the denominators.

and have the same sign, so the quotient is positive.

Example 2B: Dividing by Fractions

Divide.

Write as an improper fraction.

To divide by , multiply by .

and have different signs,

so the quotient is negative.

Check It Out! Example 2a

Divide.

Write as an improper fraction.

To divide by , multiply by .

and have the same signs,

so the quotient is positive.

Check It Out! Example 2b

Divide.

Write the reciprocal of .

To divide by , multiply by .

and have different signs,

so the quotient is negative.

Check It Out! Example 2c

Divide.

Write as an improper fraction.

To divide by multiply by .

The signs are different so the quotient is negative.

No number can be multiplied by 0 to give a product of 1, so 0 has no reciprocal. Because 0 has no reciprocal, division by 0 is not possible. We say that division by zero is undefined.

Properties of Zero

WORDS

NUMBERS

ALGEBRA

Multiplication by ZeroThe product of any numberand 0 is 0.

13 · 0 = 0 0(–17) = 0

a · 0 = 0 0 · a = 0

Properties of Zero

WORDS

NUMBERS

ALGEBRA

Zero Divided by a NumberThe quotient of 0 and any nonzero number is 0.

06 = 0

0 ÷ a = 0

0 ÷ 23 = 0

0a = 0

Properties of Zero

WORDS

NUMBERS

ALGEBRA

Division by Zero

Division by 0 is undefined.

a ÷ 0 a0

12 ÷ 0Undefined

–5 0

Undefined

Example 3: Multiplying and Dividing with Zero

Multiply or divide.

A. 15

0

B. –22 ÷ 0undefined

C. –8.45(0)0

Zero is divided by a nonzero number.

The quotient of zero and any nonzeronumber is 0.

A number is divided by zero.Division by zero is undefined.

A number is multiplied by zero.

The product of an number and 0 is 0.

0

Check It Out! Example 3a

Divide:

Multiply by the reciprocal.

Any number multiplied by0 equals 0.

Turn into an improperfraction.

Check It Out! Example 3b and 3c

Divide:

3b. 0 ÷ 0

0 ÷ 0 = undefined Any number divided by 0 is undefined.

3c. (–12.350)(0)

(–12.350)(0) = 0 The product of any number and 0 is 0.

rate

334

times

•

time

113

Example 4: Recreation Application

Find the distance traveled at a rate of 3 mi/h for 1 hour.To find distance, multiply rate by time.

34

13

The speed of a hot-air balloon is 3 mi/h. It travels in a straight line for 1 hour before landing. How many miles away from the liftoff site does the balloon land?

13

34

Example 4: Recreation Application

3 34 •1 1

3 = 15 4 • 4

3 Write and as improper fractions.343 1 1

3

15(4) 4(3) = 60

12

= 5

Multiply the numerators andmultiply the denominators.

3 34 and have the same sign, so

the quotient is positive.1 1

3

The hot-air balloon lands 5 miles from the liftoff site.

What if…? On another hot-air balloon trip, the wind speed is 5.25 mi/h. The trip is planned for 1.5 hours. The balloon travels in a straight line. How many miles away from the liftoff site will the balloon land?

Check It Out! Example 4

5.25(1.5) Speed times time equals distance.

= 7.875 mi Distance traveled.

Find the value of each expression.

1. 35–7 –5 2. 2x for x = –6 – 12

Lesson Quiz

Multiply or divide if possible.

3. –3 ÷ 134 (0)4. –21

3 5. – ÷ 034

– 12 7

0 undefined

15 miles

6. A cyclist traveled on a straight road for 1 hours at a speed of 12 mi/h. How many miles did the cyclist travel?

14