Multiplying Negative and Positive Numbers

Multiplication is a short way of doing repeated addition. Just as 3 × 8 means 3 addends of 8: 8 + 8 + 8 = 24
3 × (-8) means 3 addends of -8.
-8 + (-8) + (-8) = -24; therefore, 3 × (-8) = -24
When we multiply (a positive number) × (a negative number), the product is a negative number. The Commutative Property of Multiplication is also used with signed numbers.
3 × (-8) = -8 × 3 = -24

Multiplying negative and positive numbers results in negative number. Let us see sample problems for multiplying negative and positive numbers.

Multiplying Negative and Positive Numbers:

MULTIPLICATION RULES:

From these patterns we can write the rules for multiplication of two numbers. (The rules in your text should emphasize that they apply to two numbers.)

1. Multiplication of two numbers with the same signs:

a. Multiply each  absolute values

b. The answer is positive

RULE 1:

4 × 6 = 24 -8 × (-7) = 56

2. Multiplication of two numbers with different signs:

a. Multiply each absolute values

b. The answer is negative

RULE 2:

             -8 × 5 = -40 3 × (-1) = -3

Now that you have used the rules for multiplication, you may find that the addition rules confuse you! It helps to remember how

we added on the number line to derive the addition rules. Then think about these patterns if you forget rules for multiplication.

There are several ways to write multiplication:

             6 × 4;                6 × 4 and 6(4)    or     (6)(4)

We will continue to put parentheses around negative numbers that follow the operations:

a. 8 × (-4) 8 × (−4) 8(-4)

b. -5 × (-2) − 5× (−2) -5(-2)

The first factor can be written with parentheses, but it usually is written as line b above.

Now let's see what happens when more than two factors are multiplied:

(REMEMBER no symbols between parentheses means to multiply).
          -3(-4)(2)                            -6(-2)(-1)(-4)                              -5(6)(-2)(3)

            12 × 2                          12 × (−1)(−4)                             -30(-2)(3)

                24                                     -12(-4)                                     60(3)

                                                           48                                          180

(2 negative factors)                   (4 negative factors)                     (2 negative factors)

When we had an even number of negative factors the products were positive!


            -3(4)(2)                               -6(-2)(-1)(4)                               -5(6)(-2)(-3)

            -12 × 2                                12 × (-1)(4)                               -30(-2)(-3)

               -24                                     -12(4)                                        60(-3)
                                      

                                                          -48                                          -180


           (one negative factor)        (3 negative factors)                   (3 negative factors)

When we had and odd number of negative factors, the products were negative.

Multiplying Negative and Positive Numbers:

RULE for multiplying any number of factors:

1. Multiply the absolute values

2. Count the negative factors

a. The product is positive for an even number of negative factors.


b. The product is negative only If the number of negative factors is odd..

When you divide, you think of the related multiplication problem:

              20 ÷ 4 = 5 because 4 × 5 = 20

dividend ÷ divisor = quotient, and divisor × quotient = dividend

a. 27 ÷ (-3) = ?
       

        -3 × ? = 27 The product 27, is positive; therefore,

the two factors must have the same sign. Since the first factor is negative, the second factor is also negative

Since -3 × (-9) = 27, we know

          27 ÷ (-3) = -9

b. -12 ÷ 2 = ?                 The product, -12, is negative; therefore, the two factors must have

2 × ? = -12                     different signs. Since the first factor is positive, the second factormust be negative.

c. -28 ÷ (-4) = ? The product, -28, is negative;

therefore, the two factors must have -4 × ? = -28 different signs. Since the first factor is negative the second factor must be positive.

Since             -4 × 7 = -28

we know -28 ÷ (-4) = 7

	PROBLEMS:	ANSWERS:
1	-6(-9)	54
2	8(-7)	-56
3	-9(4)	-36
4	-3(-4)(2)	24
5	(-5)(6)(3)	-90
6	-5(-2)(5)(-3)	-150
7	-2(-1)(7)(3)	42