MULTIPLICATION AND DIVISION OF SIGNED NUMBERS. You learned from an earlier chapter that multiplication is a short way of doing repeated addition.

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1 Tallahassee Community College 46 MULTIPLICATION AND DIVISION OF SIGNED NUMBERS You learned from an earlier chapter that multiplication is a short way of doing repeated addition. Just as 3 8 means 3 addends of 8: = 24 3 (-8) means 3 addends of (-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 It does not make sense to say we have -8 addends of 3, but the pattern of multiplication (factors and products) will help you to learn the rules for multiplying signed numbers. 5 4 = 20 Notice the first factor is = 15 As the second factor decreases by 1, 5 2 = 10 the product decreases by = = 0 5 ( 1) = 5 When the pattern is continued, the 5 ( 2) = 10 product of a positive number and a 5 ( 3) = 15 negative number is a negative number. The order of the factors will not change these products. 2 5 = = = = = 10 Now let -5 be the first factor and study the resulting patterns = -25 When the first factor is -5 and the -5 4 = -20 second factor decreases by 1, the -5 3 = -15 the product increases by = = = 0-5 (-1) = 5 When the pattern is continued, the -5 (-2) = 10 product of (a negative number) x (a negative number) is a positive number. MULTIPLICATION RULES

2 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.) RULE 1 1. Multiplication of two numbers with the same signs: a. Multiply the absolute values b. The answer is positive 4 6 = 24-8 (-7) = 56 RULE 2 2. Multiplication of two numbers with different signs: a. Multiply the absolute values b. The answer is negative -8 5 = (-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) ( 1)( 4) -30(-2)(3) 24-12(-4) 60(3) (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) (-1)(4) -30(-2)(-3) 2

3 -24-12(4) 60(-3) (one negative factor) (3 negative factors) (3 negative factors) When we had and odd number of negative factors, the products were negative. RULE for multiplying any number of factors: 1. Multiply the absolute values 2. Count the negative factors a. If there is an even number of negative factors, the product is positive. b. If the number of negative factors is odd, the product is negative. I. PROBLEMS: I. ANSWERS: 1. -6(-9) (-7) (4) (-4)(2) (-5)(6)(3) (-2)(5)(-3) (-1)(7)(3) 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 3

4 b =? The product, -12, is negative; therefore, the two factors must have 2? = -12 different signs. Since the first factor is positive, the second factor must negative not defined Since 2 (-6) = - 12, we know not defined = -6 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. be DIVISION RULES RULE 1 Since -4 7 = -28 we know -28 (-4) = 7 We see that when two numbers with the same signs are divided, the answer is positive. -28 (-4) = = 5 RULE 2 When two numbers with different signs are divided, the answer is negative. 27 (-3) = = -6 Study the Properties of Zero and One in Division in your text. II.PROBLEMS: (-4) (-3) II.ANSWERS: Look at the numbers of the problems and the answers. Check 7-10 carefully! 4

5 In 8 and 10, REMEMBER for an operation to be defined, there must be one and only one answer that would check. Be sure your answers check. Multiply divisor quotient. The answer should equal the dividend. NEVER DIVIDE BY ZERO! 5