Multiplying and Dividing Signed Numbers. Finding the Product of Two Signed Numbers. (a) (3)( 4) ( 4) ( 4) ( 4) 12 (b) (4)( 5) ( 5) ( 5) ( 5) ( 5) 20

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1 SECTION.4 Multiplying and Dividing Signed Numbers.4 OBJECTIVES 1. Multiply signed numbers 2. Use the commutative property of multiplication 3. Use the associative property of multiplication 4. Divide signed numbers 5. Use the distributive property Man s mind, once stretched by a new idea, never regains its original dimensions. Multiplication can be seen as repeated addition. We can interpret Oliver Wendell Holmes We can use this interpretation together with the work of Section.3 to find the product of two signed numbers. E x a m p l e 1 Finding the Product of Two Signed Numbers Multiply. Note that we use parentheses ( ) to indicate multiplication when negative numbers are involved. (a) (3)(4) (4) (4) (4) 12 (b) (4)(5) (5) (5) (5) (5) 2 CHECK YOURSELF 1 Find the product by writing as repeated addition. 4(3) Looking at the products we found by repeated addition in Example 1 should suggest our first rule for multiplying signed numbers. To Multiply Signed Numbers The product of two numbers with different signs is negative.

2 Section.4 Multiplying and Dividing Signed Numbers 29 The rule is easy to use. To multiply two numbers with different signs, just multiply their absolute values and attach a minus sign to the product. E x a m p l e 2 The product must have two decimal places. The product is negative. You can simplify as before in finding the product. Finding the Product of Two Signed Numbers Find each product. (5)(6) 3 (1)(12) 12 (7)(9) 63 (1.5)(.3) CHECK YOURSELF 2 Find each product. (a) (15)(5) (b) (.8)(.2) (c) The product of two negative numbers is harder to visualize. The following pattern may help you see how we can determine the sign of the product. (3)(2) 6 (2)(2) 4 (1)(2) 2 ()(2) (1)(2) 2 (2)(2) 4 Do you see that the product is increasing by 2 each time the first number decreases by 1? We already know that the product of two positive numbers is positive. This suggests that the product of two negative numbers is positive, and this is in fact the case. To extend our multiplication rule, we have the following.

3 3 Chapter The Arithmetic of Signed Numbers To Multiply Signed Numbers 2. The product of two numbers with the same sign is positive. E x a m p l e 3 Finding the Product of Two Signed Numbers Find each product. CHECK YOURSELF 3 Find each product Since the numbers have the same sign, the product is positive. (9)(6) 54 (.5)(2) 1 (a) (5)(7) (b) (8)(6) (c) (9)(6) (d) (1.5)(4) Caution! Be Careful! (8)(6) tells you to multiply. The parentheses are next to one another. The expression 8 6 tells you to subtract. The numbers are separated by the operation sign. To multiply more than two signed numbers, apply the multiplication rule repeatedly. E x a m p l e 4 Finding the Product of a Set of Signed Numbers Multiply. CHECK YOURSELF 4 Find the product. (5)(7)(3)(2) (35)(3)(2) (15)(2) 21 (5)(7) 35 (35)(3) 15 (4)(3)(2)(5)

4 Section.4 Multiplying and Dividing Signed Numbers 31 We saw in Section.3 that the commutative and associative properties for addition could be extended to signed numbers. The same is true for multiplication. What about the order in which we multiply? Look at the following examples. E x a m p l e 5 Using the Commutative Property of Multiplication Find the products. (5)(7) (7)(5) 35 (6)(7) (7)(6) 42 The order in which we multiply does not affect the product. This gives us the following rule. The Commutative Property of Multiplication The centered dot represents multiplication. This could have been written as a b b a The order in which we multiply does not change the product. Multiplication is commutative. In symbols, for any a and b, a b b a CHECK YOURSELF 5 Show that (8)(5) (5)(8). What about the way we group numbers in multiplication? Look at Example 6. E x a m p l e 6 The symbols [ ] are called brackets and are used to group numbers in the same way as parentheses. Using the Associative Property of Multiplication Multiply. [(3)(7)](2) or (3)[(7)(2)] (21)(2) (3)(14) We group the first two numbers on the left and the second two numbers on the right. Note that the product is the same in either case.

5 32 Chapter The Arithmetic of Signed Numbers The Associative Property of Multiplication The way we group the numbers does not change the product. Multiplication is associative. In symbols, for any a, b, and c, (a b) c a (b c) CHECK YOURSELF 6 Show that [(2)(6)](3) (2)[(6)(3)]. Two numbers, and 1, have special properties in multiplication. The Multiplicative Identity The product of 1 and any number is that number. We call 1 the multiplicative identity. In symbols, for any a, a 1 a E x a m p l e 7 Multiplying Signed Numbers by 1 Find the products. CHECK YOURSELF 7 Find the product. (8)(1) 8 (1)(15) 15 (1)(1) What about multiplication by? Multiplying by Zero The product of and any number is. In symbols, for any a, a

6 Section.4 Multiplying and Dividing Signed Numbers 33 E x a m p l e 8 Multiplying Signed Numbers by Zero Find the products. CHECK YOURSELF 8 Find the product. (9)() ()(23) ()(12) Another important property in mathematics is the distributive property. The distributive property involves addition and multiplication together. We can illustrate the property with an application. Remember: The area of a rectangle is the product of its length and width: A L W 1 3 Area 1 15 Area 2 We can find the total area by multiplying We can find the total area as a sum the length by the overall width, which is or of the two areas. found by adding the two widths. (Area 1) (Area 2) Length Overall Width Length Width Length Width 3 (1 15) So { 3 (1 15)

7 34 Chapter The Arithmetic of Signed Numbers This leads us to the following property. Note the pattern. a(b c) a b a c We distributed the multiplication over the addition. The Distributive Property If a, b, and c are any numbers, a(b c) a b a c and (b c)a b a c a E x a m p l e 9 Using the Distributive Property Use the distributive property to simplify (remove the parentheses in) the following. (a) 5(3 4) Note: It is also true that 5(3 4) Note: It is also true that 1 3 (9 12) 1 3 (21) 7 5(3 4) (b) 1 3 (9 12) CHECK YOURSELF 9 Use the distributive property to simplify (remove the parentheses). (a) 4(6 7) (b) 1 (1 15) 5 The distributive property applies to all signed numbers. First let us look at an example of multiplication distributed over addition. E x a m p l e 1 Distributing Multiplication over Addition Use the distributive property to remove the parentheses and simplify the following. (a) 4(2 5) 4(2) 4(5) (b) 5(3 2) (5)(3) (5)(2) 15 (1) 5 CHECK YOURSELF 1 Use the distributive property to remove the parentheses and simplify the following. (a) 7(3 5) (b) 2(6 3)

8 Section.4 Multiplying and Dividing Signed Numbers 35 The distributive property can also be used to distribute multiplication over subtraction. E x a m p l e 1 1 Distributing Multiplication over Subtraction Use the distributive property to remove the parentheses and simplify the following. (a) 4(3 6) 4(3) 4(6) (b) 7(3 2) 7(3) (7)(2) 21 (14) CHECK YOURSELF 11 Use the distributive property to remove the parentheses and simplify the following. (a) 7(3 4) (b) 2(4 3) A detailed explanation of why the product of two negative numbers must be positive concludes our discussion of multiplying signed numbers. The Product of Two Negative Numbers The following argument shows why the product of two negative numbers must be positive. From our earlier work, we know that a 5 (5) number added to its opposite is. Multiply both sides of the statement by 3. (3)[5 (5)] (3)() A number multiplied by is, so on the (3)[5 (5)] right we have. We can now use the distributive property (3)(5) (3)(5) on the left. Since we know that (3)(5) 15, the 15 (3)(5) statement becomes We now have a statement of the form 15. This asks, What number must we add to 15 to get, where is the value of (3)(5)? The answer is, of course, 15. This means that (3)(5) 15 The product must be positive. It doesn t matter what numbers we use in the argument. The product of two negative numbers will always be positive.

9 36 Chapter The Arithmetic of Signed Numbers Multiplication and division are related operations. So every division problem can be stated as an equivalent multiplication problem Since Since Since the operations are related, the rules of signs for multiplication are also true for division. To Divide Signed Numbers 1. If two numbers have the same sign, the quotient is positive. 2. If two numbers have different signs, the quotient is negative. E x a m p l e 1 2 The numbers 2 and 5 have different signs, and so the quotient is negative. Dividing Two Signed Numbers Divide 2 by 5. 2 (5) 4 Since 2 (5)(4) CHECK YOURSELF 12 Write the multiplication statement that is equivalent to 36 (4) 9 E x a m p l e 1 3 The two numbers have the same sign, and so the quotient is positive. Dividing Two Signed Numbers Divide 2 by CHECK YOURSELF 13 Find each quotient. 48 (a) (b) (5) (5) 6 Since 2 (5)(4)

10 Section.4 Multiplying and Dividing Signed Numbers 37 As you would expect, division with fractions or decimals uses the same rules for signs. Example 14 illustrates this concept. E x a m p l e 1 4 First note that the quotient is positive. Then invert the divisor and multiply. Dividing Two Signed Numbers Divide. CHECK YOURSELF 14 Find each quotient (a) (b) 4.2 (.6) Be very careful when is involved in a division problem. Remember that divided by any nonzero number is. However, division by is not allowed and will be described as undefined. E x a m p l e 1 5 A statement like 9 has no meaning. There is no answer to the problem. Just write undefined. Dividing Signed Numbers When Zero Is Involved Divide. (a) 7 (b) 4 (c) 9 is undefined. (d) 5 is undefined. CHECK YOURSELF 15 Find the quotient if possible. (a) (b) 12 7 The result of Example 15 can be confirmed on your calculator. That will be included in the next example.

11 38 Chapter The Arithmetic of Signed Numbers E x a m p l e 1 6 Dividing with a Calculator Use your calculator to find each quotient. (a) The key stroke sequence on a graphing calculator () Enter results in a Divide by error message. The calculator recognizes that it cannot divide by zero. On a scientific calculator, / results in an error message. (b) The key stroke sequence () () 4.58 Enter or / 4.58 / yields 2.4 CHECK YOURSELF 16 Find each quotient (a) (b) CHECK YOURSELF ANSWERS 1. (3) (3) (3) (3) (a) 75; (b).16; (c) (a) 35; (b) 48; (c) 54; (d) (a) 52; (b) (a) 14; (b) (a) 49; (b) (4)(9). 13. (a) 8; (b) (a) 5 ; (b) (a) ; (b) undefined. 16. (a) 4.8; (b) undefined.

12 E x e r c i s e s Multiply (6)(12) 3. (4)(3) (8)(9) 6. (8)(3) 7. (7)(6) 8. (12)(2) 9. (1)() 1. (1)(1) 11. (8)(8) 12. ()(5) 13. (2)(4) 14. (25)(8) 15. (9)(12) 16. (9)(9) 17. (2)(1) 18. (1)(3) 19. (4)(5) 2. (25)(5) 21. (1)(15) 22. (5)(6) (11)(2) () (5)(3)(8) 3. (4)(3)(5) 31. (2)(8)(5) 32. (7)(5)(2) 33. (2)(5)(3)(5) 34. (2)(5)(5)(6) 35. (4)(3)(6)(2) 36. (8)(3)(2)(5) Use the distributive property to remove parentheses and simplify the following (6 9) (5 9) 39. 8(9 15) 4. 11(8 3) 41. 5(8 6) 42. 2(7 11) 43. 4(6 3) 44. 7(2 3) Divide (3) (2)

13 4 Chapter The Arithmetic of Signed Numbers Undefined Undefined (1) (1) (12) weeks 8. $ $ $ F (9) (8) (4) Divide using a graphing calculator. Round answers to the nearest thousandth Dieting. A woman lost 42 pounds (lb). If she lost 3 lb each week, how long has she been dieting? 8. Mowing lawns. Patrick worked all day mowing lawns and was paid $9 per hour. If he had $125 at the end of a 9-hour day, how much did he have before he started working? 81. Unit pricing. A 4.5-lb can of food costs $8.91. What is the cost per pound? 82. Investment. Suppose that you and your two brothers bought equal shares of an investment for a total of $2, and sold it later for $16,232. How much did each person lose? 83. Temperature. Suppose that the temperature outside is dropping at a constant rate. At noon, the temperature is 7 F and it drops to 58 F at 5: P.M. How much did the temperature change each hour?

14 Section.4 Multiplying and Dividing Signed Numbers Test tube count. A chemist has 84 ounces (oz) of a solution. He pours the solution into test tubes. Each test tube holds 2 3 oz. How many test tubes can he fill? To evaluate an expression involving a fraction (indicating division), we evaluate the numerator and then the denominator. We then divide the numerator by the denominator as the last step. Using this approach, find the value of each of the following expressions (8) ( 5)( ( 3 12) )( 5) 9. ( 8) (3) (2 )( 4) 91. Create an example to show that the division of signed numbers is not commutative. 92. Create an example to show that the division of signed numbers is not associative. 93. Here is another conjecture to consider: ab a b for all numbers a and b. (See the discussion in Exercises.3, problem 87, concerning testing a conjecture.) Test this conjecture for various values of a and b. Use positive numbers, negative numbers, and. Summarize your results in a rule. 94. Use a calculator (or mental calculations) to compute the following: 5.1, 5. 1, 5 5 5,, In this series of problems, while the numerator is always 5, the denominator is getting smaller (and is getting closer to ). As this happens, what is happening to the value of the fraction? Write an argument that explains why 5 could not have any finite value.