Section 1.5

Homework questions?

Section

Concepts

1.5 Multiplication and Division of Integers

1. Multiplication of Integers2. Multiplying Many Factors3. Exponential Expressions4. Division of Integers

Section 1.5 Multiplying Integers

The multiplication rules for integers are determined from a pattern of products.Consider the following problems:2 • 3 =2 • 2 = The pattern decreases 2 • 1 = by two with each row.2 • 0 = 2 • (-1) = The product of a 2 • (-2) = positive number and a 2 • (-3) = negative number is

negative.

Section 1.5 Multiplying Integers

Now start the pattern with a negative integer.

-2 • 3 = The pattern increases -2 • 2 = by two with each row.-2 • 1 = -2 • 0 = The product of a -2 • (-1) = negative number and a -2 • (-2) = negative number is -2 • (-3) = positive.

PROCEDURE Multiplying Signed Numbers

1. The product of two numbers with the same sign is positive.

Examples: (5)(6) = 30(-5)(-6) = 30

2. The product of two numbers with different signs is negative.Examples: 4(-8) = -32

-4(8) = -323. The product of any number and zero is zero.

Examples: 5(0) = 00(-7) = 0

Example

Multiply.

1

-8(-6) -5(10) (14)(-2)

7(9) -11(0)

Example

-2(-6) 3(-10) -12(4)

8(5) -7(0)

Multiply.

2 You Try.

Section 1.5 Translating to a Mathematical Expression

Operation Words for Multiplication

ProductTimes

MultiplyOf By

Twice (times 2)

Example 3 Translate to a mathematical expression. Then simplify.

The product of 7 and -8.

-3 times -12

5 multiplied by -6

Example 4 You Try – Translate to a mathematical expression. Then simplify.

The product of -12 and -6

-4 times -11

Multiply 5 by -9

Example

Multiply the factors from left to right.

5 Multiplying Many Factors

(-2)(-2) (-2)(-2)(-2) (-2)(-2)(-2)(-2)= 4 = 4(-2) = 4(-2)(-2)

= -8 = -8(-2) = 16

These products indicate the following rules:

•The product of an even number of negative factors is positive•The product of an odd number of negative factors is negative

Example 6 Multiply.

(-3)(-2)(-4) You Try

(-5)(-2)(-7)(-1) (-3)(-4)(-8)(-1)

(-4)2(-1)(5) (-1)(-4)(-6)(5)

PROCEDURE Exponential Expressions

2( 4) 24

“negative 4 squared” “negative 1 times 4²” (-4 -4) -1 (4)(4)∙ ∙ = 16 -1 16∙

= -16

Example 7 Simplify.

2( 6)

23

3( 2)

28

Example 8 Simplify. You Try

27 3( 3)

29 4( 2)

DEFINITION Division of Integers

The sign rules for dividing integers are the same as the sign rules for multiplying integers, because we can check our division answers by multiplying.

24 ÷ 6 = 4 Check: 4 6 = 24∙-24 ÷ 6 = -4 Check: -4 6 = -24 ∙ 24 ÷ (-6) = -4 Check: -4 (-6) = 24∙-24 ÷ (-6) = 4 Check: 4 (-6) = -24 ∙

PROCEDURE Dividing Signed Numbers

1. The quotient of two numbers with the same sign is positive.

2. The quotient of two numbers with different signs is negative.

3. Zero divided by any nonzero number is zero.

4. Any nonzero number divided by zero is undefined.

Example 9 Divide.

50 ÷ (-5) 0 ÷ (-8)

-36 ÷ 3 -9 ÷ 0

42

7

Example 10 Divide. You Try

-40 ÷ 10 18 ÷ -2

0 ÷ 3 -4 ÷ 0

36

12

Example 11 Translate to a mathematicalexpression. Then simplify.

The quotient of 25 and -5

-14 divided by -7

4 divided into -24

Example 12 You Try. Translate to a mathematical expression and simplify.

The quotient of -100 and 10

-56 divided by -8

6 divided into -12

Example 13 Applications

During a 10-min period, a SCUBA diver’s depth changed by -60 ft. Find the average change in depth per minute.

Stephanie wrote five checks to the employees of her business, each for $225. If the original balance in her checking account was $890, what is her new balance?

Example 14 Applications. You Try

When a severe winter storm moved through Albany, New York, the change in temperature between 4:00 pm and 7:00 pm was -27⁰F. What was the average hourly change in temperature?

Sarah’s checking account had $320 when she wrote two checks for $150, and one check for $82. What is her new balance?

DEFINITION Multiplying Signed Numbers

1. The product of two numbers with the same sign is positive.

Examples: (5)(6) = 30(-5)(-6) = 30

2. The product of two numbers with different signs is negative.Examples: 4(-8) = -32

-4(8) = -323. The product of any number and zero is zero.

Examples: 5(0) = 00(-7) = 0

DEFINITION Dividing Signed Numbers

1. The quotient of two numbers with the same sign is positive.

2. The quotient of two numbers with different signs is negative.

3. Zero divided by any nonzero number is zero.

4. Any nonzero number divided by zero is undefined.