© William James Calhoun

Multiplying Rational Numbers

© William James Calhoun

To multiply rational numbers.

The rules for multiplying numbers are different from adding and subtracting numbers.

You need to keep the rules for adding and subtracting numbers in your head.

Keep those rules separate from the multiplying numbers rules we are about to discuss.

Remember multiplying numbers is actually a quick way of adding numbers by grouping them.

© William James Calhoun

The product of two numbers having the same sign is positive.The product of two numbers having different signs in negative.

2.6.1 MULTIPLYING TWO RATIONAL NUMBERS

The short-and-sweet is that multiplying rational numbers is just the same as all the multiplying you have done before.

The only new additions to the rules-of-old are the following:

1) A positive times a positive is a positive.

2) A positive times a negative is a negative.

3) A negative times a negative is a positive.

Commit these three new additions to memory.

© William James Calhoun

EXAMPLE 1α: Find each product.a. (-9.8)4 b. 3 2

4 3

EX1EX1ββ

Negative times positive yields negative.All that is left is to multiply 9.8 by 4 and put a negative sign on the result.

(9.8)4 = 39.2 -39.2

Negative times negative yields a positive.Multiply the numbers.

3 2

4 3

6

12

Reduce.

1

2

EXAMPLE 1β: Find each product.

a. b. (-1.4)7 4

205

© William James Calhoun

EXAMPLE 2α: Evaluate if a = 2.

EX2EX2ββ

EXAMPLE 2β: Evaluate if a = 3.

25

a6

Plug 2 in for a. 25

26

Exponent.5 5

26 6

252

36

2 25

1 36

Multiply.

Multiply again.

50

36

Reduce.

25

18

32

a3

© William James Calhoun

EXAMPLE 3α: Simplify each expression.a. (2b)(-3a) b. 3x(-3y) + (-6x)(-2y)

EX3EX3ββ

EXAMPLE 3β: Simplify each expression.a. (-2a)(3b) + (4a)(-6b) b. (5x)(-3y) + (-7x)(4y)

Positive times negative yields a negative.

Multiplying with letters, so letter configuration will change.

Multiply the numbers.

First term will be negative; second positive.Multiply the numbers in both terms.

2 * 3 = 6

a’s and b’s form new letter configuration: abSo, the answer - keeping the sign in mind - is:

-6ab

3 * 3 = 9 and 6 * 2 = 12Now handle the changes in letter configurations.

x * y = xy and x * y = xy

Bring it all together.Combine like terms.

-9xy + 12xy = 3xy

© William James Calhoun

Question: What is -1 times 5?

Answer: -5

Question: What is -1 times -14?

Answer: 14

Question: So what does multiplying by -1 do to any number?

Answer: Multiplying by -1 changes only the sign of a number.

The product of any number and -1 is its additive inverse.-1(a) = -a and a(-1) = -a

2.6.2 MULTIPLICATIVE PROPERTY OF -1

© William James Calhoun

EXAMPLE 4α: Find

EX4EX4ββ

3 1 24 3 4 1 .

4 3 5

Handle the first pair.3 1

44 3

3 13

4 3

39

1213

4

We now have: 13 23 4 1

4 5

Handle the first pair.13 2

34 5

13 17

4 5

221

20No reduction is needed.

We now have: 2214 1

20

Handle the first pair. 2214

20

884

20

221

5No reduction is needed.

We now have: 2211

5

221

5 Finally the answer!

© William James Calhoun

EXAMPLE 4β: Multiply.

a. b.2 1 9 1

3 6 5 2

5 1 114 7

6 5 12

© William James Calhoun

17-31 odd35-39 odd, 43-51 odd