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Real Numbers and Their Basic Properties Copyright © Cengage Learning. All rights reserved. 1 Section 1.5 Multiplying and Dividing Real Numbers Copyright © Cengage Learning. All rights reserved. Objectives 1 Multiply two or more real numbers. 21. Divide two real numbers. 2. Use signed numbers and an operation to model 3 an application. 3 1. Multiplying Real Numbers 4 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. xy (x)(y) x(y) (x)y xy 5 Multiplying Real Numbers What does this expression mean? 54 Definition: 5(4) = 4 + 4 + 4 + 4 + 4 = 20 5(-4) = (-4) + (-4) + (-4) + (-4) + (-4) = -20 6 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). 7 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 8 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. 9 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. 10 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 11 Your Turn cont’d d. 8(6) = +(8 6) = +48 e. (–3)2 = (–3)(–3) = +9 f. (–3)3 = (–3)(–3)(–3) = 9(–3) = –27 12 Your Turn cont’d g. (–3)(5)(–4) = (–15)(–4) = +60 h. (–4)(–2)(–3) = 8(–3) = –24 13 2. Divide two real numbers 14 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. 15 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. 16 Dividing Real Numbers Furthermore, is undefined, because no number multiplied by 0 gives –10. However, = 0, because 0(–10) = 0. 17 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. 18 Your Turn Find each quotient: a. b. c. d. Solution: a. The quotient of two numbers with like signs is positive. b. The quotient of two numbers with unlike signs is negative. 19 Example 4 – Solution cont’d c. The quotient of two numbers with unlike signs is negative. d. The quotient of two numbers with like signs is positive. 20 3. Use signed numbers and an operation to model an application 21 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 Use a calculator. The company lost about 6.6¢ per share. 22