Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 1

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 2 The Real Number System Chapter 1

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide Multiplying and Dividing Real Numbers

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 4 Objectives 1.Find the product of a positive number and a negative number. 2.Find the product of two negative numbers. 3.Use the reciprocal of a number to apply the definition of division. 4.Use the rules for order of operations when multiplying and dividing signed numbers. 5.Evaluate expressions involving variables. 6.Translate words and phrases involving multiplication and division. 7.Translate simple sentences into equations. 1.6 Multiplying and Dividing Real Numbers

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide Multiplying and Dividing Real Numbers Multiplication Property of 0 For any real number a, Finding the Product of a Positive and Negative Number Since multiplication can also be considered repeated addition, the product 3( – 1) represents the sum –1 + (–1) + (–1) = –3. Add –1 three times. The product of a positive number and a negative number is negative.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 6 Example 1 Find each product using the multiplication rule. (a) 9(–3) = 1.6 Multiplying and Dividing Real Numbers Finding the Product of a Positive and Negative Number –(9 · 3) = (b) –6(8) =–(6 · 8) = (c) –16(⅜) =–6 (d) 2.9(–3.2) =–9.28 –27 –48

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 7 Example 2 Find each product using the multiplication rule. (a) –5(–7) = 1.6 Multiplying and Dividing Real Numbers Finding the Product of Two Negative Numbers 35 (b) –6(–12) =72 (c) –2(3)(–1) =–6(–1) = (d) 3(–5)(–2) =–15(–2) = The product of two negative numbers is positive. 6 30

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide Multiplying and Dividing Real Numbers Reciprocals Pairs of numbers whose product is 1 are called reciprocals of each other. Using a Reciprocal to Apply the Definition of Division 0 has no reciprocal.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide Multiplying and Dividing Real Numbers Division The quotient Using a Reciprocal to Apply the Definition of Division Note of real numbers a and b, with b ≠ 0, is

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 10 Example 3 Find each quotient. 1.6 Multiplying and Dividing Real Numbers Using a Reciprocal to Apply the Definition of Division

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 11 Example 4 Find each quotient. 1.6 Multiplying and Dividing Real Numbers Using a Reciprocal to Apply the Definition of Division

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide Multiplying and Dividing Real Numbers Dividing Signed Numbers The quotient of two numbers having the same sign is positive. The quotient of two numbers have different signs is negative. For any positive real numbers a and b, Using a Reciprocal to Apply the Definition of Division

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 13 (b) –6(–2) –3(–4) = Example 5 Simplify. 1.6 Multiplying and Dividing Real Numbers Using the Order of Operations with Signed Numbers Find all products, working from left to right. (a) –9(2) – (–3)(2) = –9(2) – (–3)(2) = –18 – (–6) = – = –12 12 – (–12) = = 24

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 14 Example 5 (concluded) Simplify. 1.6 Multiplying and Dividing Real Numbers Using the Order of Operations with Signed Numbers Simplify the numerator and denominator separately.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 15 Example 6 Evaluate each expression, given that x = –1, y = –2, and m = – Multiplying and Dividing Real Numbers Evaluating Expressions Involving Variables Substitute the given values for the variables. Then use the order of operations to find the value of the expression. (a) (3x + 4y)(–2m) (3x + 4y)(–2m) = [3(–1) + 4(–2)][–2(–3)] = [–3 + (–8)][6] = (–11)(6) = –66

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 16 Example 6 (continued) Evaluate each expression, given that x = –1, y = –2, and m = – Multiplying and Dividing Real Numbers Evaluating Expressions Involving Variables (b) 2x 2 – 3y 2 =2(–1) 2 – 3(–2) 2 = 2(1) – 3(4) = –10 = 2 – 12

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 17 Example 6 (concluded) Evaluate each expression, given that x = –1, y = –2, and m = – Multiplying and Dividing Real Numbers Evaluating Expressions Involving Variables

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 18 Translating Words and Phrases Word or Phrase Example Numerical Expression and Simplification Product ofThe product of –5 and –2–5(–2) =10 Times13 times –413(–4) = –52 Twice (meaning “2 times”) Twice 62(6) =12 Of (used with fractions) ½ of 10½(10) =5 Percent of12 % of –160.12(–16) =– Multiplying and Dividing Real Numbers

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 19 Translating Words and Phrases Word or Phrase Example Numerical Expression and Simplification Quotient ofThe quotient of –24 and 3 Divided by–16 divided by –4 Ratio ofThe ratio of 2 to Multiplying and Dividing Real Numbers

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 20 Translating Words and Phrases 1.6 Multiplying and Dividing Real Numbers Example 7 Write a numerical expression for each phrase, and simplify the expression. (a) Three fourths of the difference between 8 and –2

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 21 Translating Words and Phrases 1.6 Multiplying and Dividing Real Numbers Example 7 (concluded) Write a numerical expression for each phrase, and simplify the expression. (b) 20% of the sum of 1200 and ( ) = = (1600)

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 22 Translating Words and Phrases 1.6 Multiplying and Dividing Real Numbers Example 8 Write a numerical expression for each phrase, and simplify the expression. (c) The quotient of 20 and the difference between –11 and –7

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 23 Translating Simple Sentences into Equations 1.6 Multiplying and Dividing Real Numbers Example 9 Write each sentence in symbols, using x to represent the number. (a) Five times a number is 40. 5x = 40 (b) The quotient of a number and –8 is 6.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 24 CAUTION It is important to recognize the distinction between the types of problems found in Examples 7 and 8 and those in Example 9. In Example 7 and 8, the phrases trans- late as expressions, while in Example 9, the sentences translate as equations. Remember that an expression is a phrase, while an equation is a sentence. Translating Simple Sentences into Equations 1.6 Multiplying and Dividing Real Numbers