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239 CH 44 MULTIPLYING AND DIVIDING SIGNED NUMBERS Division and Those Pesky Zeros O ne of the most important facts in all of mathematics is that the denominator (bottom) of a fraction can NEVER be zero. Sometimes this is phrased “Never divide by zero.” What’s the big deal? Recall from somewhere in your past that 56 8 because 8 7 = 56 7 This is the result of dividing by zero. If you prefer, we can write the division problem and its “check” like this: 8 7 56 which is checked by seeing that 8 7 = 56. Let’s put zeros in fractions and see what happens. Zero on the Top How shall we interpret the division problem 0 = ??? 7 What number times 7 yields an answer of 0? Well, 0 works; that is, 0 0 because 0 7 = 0. 7 So the answer is 0. Moreover, no other number besides 0 will work. Ch 44 Multiplying and Dividing Signed Numbers 240 Zero on the Bottom Now let’s put a zero only on the bottom and see what happens: 9 = ??? 0 Your first guess is probably 0; let’s check it out: 9 = 0 would be true only if 0 0 = 9, which it isn’t. 0 How about we try an answer of 9? 9 = 9 which is checked by seeing if 9 0 = 9. Sorry. 0 Could the answer be ? Nope; 0 = 0, not 9. In fact, any number we conjure up as the answer will have to multiply with 0 to make a product of 9. But this is impossible, since any number times 0 is always 0, never 9. In short, no number in the whole world will work as the answer to this division problem. Zero on the Top AND the Bottom Now for an even stranger problem with division and zeros: 0 = ??? 0 We can try 0: 0 = 0 which seems reasonable, since 0 0 = 0. 0 Let’s try an answer of 5: 0 = 5 , which -- amazingly -- works out too. 0 Could possibly work? 0 = -- still true, since 0 is surely 0. 0 That’s three answers for this division problem: 0, 5, and . Is there any end to this madness? Apparently not, since any number we imagine will multiply with 0 to make a product of 0. In short, every number in the whole world will work in this division problem. Ch 44 Multiplying and Dividing Signed Numbers 241 Summary: 1) 2) 3) Zero on the top of a fraction is perfectly okay, as long as the bottom is NOT zero. The answer to this kind of division problem is always zero. For example, 0 = 0 . 7 There is no answer to the division problem 9 . Clearly, we 0 can never work a problem like this. There are infinitely many answers to the division problem 0 . This may be a student’s dream come true, but in 0 mathematics we don’t want a division problem with billions and billions of answers. Each of the division problems with a zero in the denominator leads to a major conundrum, so we summarize cases 2) and 3) by stating that DIVISION BY ZERO IS UNDEFINED! More examples: 0 = 0 34 17 is undefined 0 “Black holes are where God divided by zero.” Steven Wright 0 is undefined 0 Note: The two division problems with zero in the denominator may both be undefined, but they’re undefined for totally different reasons. Your teacher may require that you understand this. Ch 44 Multiplying and Dividing Signed Numbers 242 Homework 1. Evaluate each expression, and explain your conclusion: a. 0 15 2. 3. 4. 5. 6. b. 32 0 c. 0 0 Evaluate each expression: a. 0 17 b. 0 9 c. 6 6 17 3 2 d. 3 8 1 100 e. 98 0 f. 44 0 g. 73 8 2 8 2 h. 7 40 23 23 i. 0 0 j. 2 k. 52 25 0 03 l. 4 53 2 10 3 9 9 9 10 10 0 = 0 because a. 0 is the only number that when multiplied by gives 0. b. No number times equals 0. c. Every number times equals 0. 0 is undefined because 0 a. no number times 0 equals 0. b. every number times 0 equals 0. c. any number divided by itself is 1. 7 is undefined because 0 a. 0 is the only number multiplied by 0 to get 7. b. No number times 0 equals 7. c. Every number times 0 equals 7. a. The numerator of a fraction is 0. What can you conclude? b. The denominator of a fraction is 0. What can you conclude? Ch 44 Multiplying and Dividing Signed Numbers 243 Multiplying Signed Numbers Recall that the result of multiplying two numbers is called the product of the two numbers. So the product of 3 and 5 is 15. Some ways to represent the product of 3 and 5 are 35 35 3(5) (3)5 (3)(5) (all of these products equal 15) Also recall that the order in which you multiply a pair of numbers makes no difference in the product: ab = ba for any numbers a and b. [This is called the commutative property of multiplication.] Positive Times Positive What should 6 4 be? It better be 24. In other words, we see what we learned in elementary school: A positive number times a positive number is positive. Positive Times Negative This one’s not so obvious. We’ll determine the answer by sneaking up on the problem while it’s not looking: (7)(2) 14 since a positive times a positive is positive (7)(1) 7 same rule, or, anything times 1 is itself (7)(0) 0 anything times 0 is 0 (7)( 1) ??? what should the product be? What comes next in the sequence 14, 7, 0, ? Since this sequence of numbers is decreasing by 7 at each step, the next number in the sequence must be 7, and we see that (7)(1) = 7. It appears that A positive number times a negative number is negative. Negative Times Positive To calculate (3)(4), first use the commutative propery to reverse the order of the factors: (3)(4) = (4)(3), which is now a product of a positive with a negative. By the previous rule, we know that the answer is 12. So (3)(4) = 12, and thus, A negative number times a positive number is negative. Ch 44 Multiplying and Dividing Signed Numbers 244 Negative Times Negative Now for the most interesting situation, the product of two negative numbers -- for example (5)(1). ( 5)(2) 10 since a negative times a positive is negative ( 5)(1) 5 same rule, or, anything times 1 is itself ( 5)(0) 0 anything times 0 is 0 ( 5)( 1) ??? what should the product be? What comes next in the sequence of answers 10, 5, 0 ? This sequence of numbers is growing by 5 at each step. The next number is therefore 5, giving us the following logical pattern: ( 5)(2) 10 ( 5)(1) 5 The answers are growing by 5. ( 5)(0) 0 ( 5)( 1) = 5 We’ve reached the inescapable conclusion that A negative number times a negative number is positive! See if you can deduce the two rules in the following box from the four rules stated above. Multiplying Signed Numbers : If the signs are the same, the product is positive. If the signs are different, the product is negative. Ch 44 Multiplying and Dividing Signed Numbers 245 Homework 7. 8. Find the product: a. (17)(3) b. (4)(7) c. 3(10) d. (3)(4) e. 7(2) f. 2(7) g. 1(8) h. 1(9) i. (1)(134) j. (765)(0) k. 3 4 l. 7 10 m. (18)(2) n. 7(3) o. 8(7) p. 3 9 q. (2)(99) r. (1)(7) s. (7)(10) t. (4)(5) Find the product: a. (2)(3)(4) b. 7 6 3 c. (1)(2)(3)(4) d. 4 4 4(1)(3) e. 5(1)(1)(2) f. 4(3)(2)(8) g. (2)(1)(3) h. (7)(6)(2)(1) i. (1)(1)(1)(8) j. (1)(3)(5)(7) k. 3(4)(5)2 l. (2)(2)(1)(3)(5)0 Dividing Signed Numbers The secret to the division rules for signed numbers is the fact that division is checked by multiplication. For example, 56 8 only because 8 7 = 56. 7 Positive Divided by Positive 6 3 , since (3)(2) = 6. 2 Therefore, a positive number divided by a positive number is positive. Ch 44 Multiplying and Dividing Signed Numbers 246 Negative Divided by Positive 6 3 , because (3)(2) = 6. 2 Thus, a negative number divided by a positive number is negative. Positive Divided by Negative 6 3 . This can be checked by seeing that (3)(2) = 6. 2 Conclusion: a positive number divided by a negative number is negative. Negative Divided by Negative 6 3 , which is confirmed by the fact that (3)(2) = 6. 2 We see that a negative number divided by a negative number is positive. Dividing Signed Numbers : If the signs are the same, the quotient is positive. If the signs are different, the quotient is negative. Notes: 1. 2. Do these four rules for dividing signed numbers remind you of anything? The rules for dividing signed numbers are the same as the rules for multiplying signed numbers. 10 = 10 = 10 because each of these three 2 2 2 division problems results in 5. In general, the following three fractions are equal: a = a = a b b b Ch 44 Multiplying and Dividing Signed Numbers 247 Homework 9. Find the quotient (leave answers as simplified fractions): a. 30 5 b. 60 10 c. 100 20 d. 14 2 e. 3 6 f. 18 4 g. 24 5 h. 34 24 j. 345 345 k. 10 10 l. c. 3 3 9 7 d. 18 2 2 18 g. 12 15 3 3 h. 9 1 6 5 i. 10. 11. 1 9 120 70 Evaluate each expression: 10(2) 4 a. 5 1 1 2 b. e. 1 2 3 2 4 f. 10 2 2 1 Evaluate each expression, leaving your answers as decimals rounded to the tenths place: a. (2)( 3) ( 3)(4) b. 1 2 3 2 3 4 c. 20 (16 4) 32 22 d. 1 ( 2) 1 4 e. 10 ( 2) ( 1 4) f. (1)(2)( 3)( 4) 12(2)( 1) Ch 44 Multiplying and Dividing Signed Numbers 248 Solutions 1. a. 0 = 0 since 0 15 = 0, and 0 is the only number that accomplishes 15 this. 32 is undefined because any number times 0 is 0, never 32; thus NO 0 number works. b. 0 is undefined because any number times 0 is 0; thus EVERY 0 number works. c. 2. a. 0 b. 0 c. 0 d. 0 e. Undefined f. Undefined i. Undefined j. Undefined g. Undefined h. Undefined k. Undefined l. 0 3. a. b. 6. a. You can’t conclude anything -- it depends on what’s on the bottom. If 4. 5. b. the bottom is a non-zero number (like 7), then 0 = 0 . If the bottom is 7 zero, then 0 is undefined. 0 b. This time we can conclude that the fraction is undefined, since division by 0 is undefined, no matter what’s on the top of the fraction. 28 9 21 20 7. a. 51 g. 8 m. 36 s. 70 b. h. n. t. 8. a. 24 g. 6 b. 126 h. 84 c. 30 i. 134 o. 56 d. 12 j. 0 p. 27 e. 14 k. 12 q. 198 f. 14 l. 70 r. 7 c. 24 i. 8 d. 192 j. 105 e. 10 k. 120 f. 192 l. 0 Ch 44 Multiplying and Dividing Signed Numbers 249 9. e. 1 2 a. 6 b. 6 c. 5 d. 7 f. 9 = 4 1 2 2 g. 24 = 4 4 5 5 l. 12 = 1 5 7 7 h. 17 = 1 5 12 12 i. g. 9 k. 1 10. a. 4 b. 5 c. 0 d. 1 e. 1 f. 7 11. a. 0.5 b. 0.7 c. 0 d. 1 e. 1.6 f. 1 Ch 44 Multiplying and Dividing Signed Numbers 1 9 j. 1 h. 8 250 Jean Piaget (1896-1980) Swiss cognitive psychologist Ch 44 Multiplying and Dividing Signed Numbers