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Real Numbersand Their Basic Properties1

Copyright © Cengage Learning. All rights reserved.

Section 1.51.5

Multiplying and Dividing Real Numbers

3

Multiply two or more real numbers.

1. Divide two real numbers.

2. Use signed numbers and an operation to model an application.

Objectives

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22

33

4

Multiplying Real Numbers1.

5

Multiplying Real Numbers

Because the times sign, , looks like the letter x, it is seldom used in algebra.

Each of the following expressions indicates the product of x and y.

x y (x)(y) x(y) (x)y xy

6

Multiplying Real Numbers

What does this expression mean?

5 4

Definition: 5(4) = 4 + 4 + 4 + 4 + 4 = 20

5(-4) = (-4) + (-4) + (-4) + (-4) + (-4) = -20

7

Multiplying Real Numbers

Since

(5)(4) means adding the number 4 five times,

You can think of (-5)(4) as subtracting the number 4 five times

(–5)4 = –(4) – (4) – (4) – (4) – (4)

= (–4) + (–4) + (–4) + (–4) + (–4)

= –20

Because xy = yx,

(-5)4 = 4(-5).

8

Multiplying Real Numbers

Likewise, the expression (–5)(–4) indicates that –4 is to be used as a term in a repeated subtraction five times.

(–5)(–4) = –(–4) – (–4) – (–4) – (–4) – (–4)

= –(–4) + [–(–4)] + [–(–4)] + [–(–4)] + [–(–4)]

= 4 + 4 + 4 + 4 + 4

= 20

9

Multiplying Real Numbers

The expression 0(–2) indicates that –2 is to be used zero times as a term in a repeated addition. Thus,

0(–2) = 0

Finally, the expression (–3)(1) = –3 suggests that the product of any number and 1 is the number itself.

10

Multiplying Real Numbers

Rules for Multiplying Signed Numbers

To multiply two real numbers, multiply their absolute values.

1. If the numbers are positive, the product is positive.

2. If the numbers are negative, the product is positive.

3. If one number is positive and the other is negative, the product is negative.

4. Any number multiplied by 0 is 0: a 0 = 0 a = 0.

5. Any number multiplied by 1 is the number itself: a 1 = 1 a = a.

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Your Turn

Find each product: a. 4(–7) b. (–5)(–4) c. (–7)(6) d. 8(6) e. (–3)2 f. (–3)3 g. (–3)(5)(–4) h. (–4)(–2)(–3).

Solution:

a. 4(–7) = (–4 7)

= –28

b. (–5)(–4) = +(5 4)

= +20

c. (–7)(6) = –(7 6)

= –42

12

Your Turn

d. 8(6) = +(8 6)

= +48

e. (–3)2 = (–3)(–3)

= +9

f. (–3)3 = (–3)(–3)(–3) = 9(–3)

= –27

cont’d

13

Your Turn

g. (–3)(5)(–4) = (–15)(–4)

= +60

h. (–4)(–2)(–3) = 8(–3)

= –24

cont’d

14

Divide two real numbers2.

15

Dividing Real Numbers

= 2, because 2 4 = 8

= 3, because 3 6 = 18

These examples suggest that the following rule

= c if and only if c b = a

is true for the division of any real number a by any nonzero real number b.

16

Dividing Real Numbers

For example,

= +5, because (+5)(+2) = +10.

= +5, because (+5)(–2) = –10.

= –5, because (–5)(–2) = +10.

= –5, because (–5)(+2) = –10.

17

Dividing Real Numbers

Furthermore,

is undefined, because no number multiplied by 0 gives –10.

However,

= 0, because 0(–10) = 0.

18

Divide two real numbers

Rules for Dividing Signed Numbers

To divide two real numbers, find the quotient of their absolute values.

1.If the numbers are positive, the quotient is positive.

2.If the numbers are negative, the quotient is positive.

3.If one number is positive and the other is negative, the quotient is negative.

4.Division by 0 is undefined.

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Your Turn

Find each quotient: a. b. c. d.

Solution:

a.

b.

The quotient of two numbers with like signs is positive.

The quotient of two numbers with unlike signs is negative.

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Example 4 – Solution

c.

d.

The quotient of two numbers with unlike signs is negative.

The quotient of two numbers with like signs is positive.

cont’d

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Use signed numbers and an operation to model an application

3.

22

Your Turn– Stock Reports

In its annual report, a corporation reports its performance on a per-share basis. When a company with 35 million shares loses $2.3 million, find the per-share Loss.

Solution:

A loss of $2.3 million can be represented by –2,300,000. Because there are 35 million shares, the per-share loss

can be represented by the quotient

The company lost about 6.6¢ per share.

Use a calculator.