4 CHAPTER 1 Critical Thinking Skills Did You Know? Solution a)51 5 5210531554 205 5 25 56 3057 3558 4059 45510 50 × = × = × = × = × = × = × = × = × = × = b) Notice that the ones digit of the products are either 5 or 0. Based on these examples, we can make the conjecture that products involving 5 and natural numbers will have a ones digit of 5 or 0. As we will discuss in Section 5.1, this statement is indeed true. 7 Now try Exercise 29 Example 4 Use Inductive Reasoning to Make a Conjecture Pick any number and multiply the number by 4. Next, add 2 to the product and divide the sum by 2. Finally, subtract 1 from the quotient. a) What is the relationship between the number you started with and the final answer? b) Repeat this procedure for several different numbers. Note the original number and the final number. c) Use inductive reasoning to make a conjecture about the relationship between the original number and the final number. Solution a) We will go through the procedure step by step. Now try Exercise 37 Pick a number: say, 5 Multiply the number by 4: 5 4 20 × = Add 2 to the product: 20 2 22 + = Divide the sum by 2: 22 2 11 ÷ = Subtract 1 from the quotient: 11 1 10 − = Notice that we started with the number 5 and ended with the number 10. b) If we repeat the procedure, this time starting with the number 6, we will end with the number 12. If we start with the number 9, we will end with the number 18. If we start with the number 15, we will end with the number 30—verify these results for yourself. c) Based on the examples from parts a) and b), we can conjecture that when we follow the given procedure, the number you end with will always be twice the original number. 7 The result reached by inductive reasoning is often correct for the specific cases studied but not correct for all cases. History has shown that not all conclusions arrived at by inductive reasoning are correct. For example, Aristotle (384–322 b.c.) reasoned inductively that heavy objects fall at a faster rate than light objects. About 2000 years later, Galileo (1564 –1642) dropped two pieces of metal—one 10 times heavier than the other—from the Leaning Tower of Pisa in Italy. He found that both hit the ground at exactly the same moment, so they must have traveled at the same rate. See the Did You Know on the left for a similar experiment carried out on the moon by astronaut David Scott. An Experiment Revisited Apollo 15 astronaut David Scott used the moon as his laboratory to show that a heavy object (a hammer) does indeed fall at the same rate as a light object (a feather). Had Galileo dropped a hammer and feather from the Tower of Pisa, the hammer would have fallen more quickly to the ground and he still would have concluded that a heavy object falls faster than a lighter one. If it is not the object’s mass that is affecting the outcome, then what is it? The answer is air resistance or friction: Earth has an atmosphere that creates friction on falling objects. The moon does not have an atmosphere; therefore, no friction is created. m David Scott on the moon JSC PAO Web Team/NASA Learning Catalytics Keyword: Angel-SOM-1.1 (See Preface for additional details.)
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