1.1 Inductive and Deductive Reasoning 3 In Examples 1 and 2, we cannot conclude that the results are true for all counting numbers. From the patterns developed, however, we can make predictions. This type of reasoning process, arriving at a general conclusion from specific observations or examples, is called inductive reasoning, or induction. Example 2 The Sum of an Odd Number and an Even Number If an odd number and an even number are added, will the sum be an odd number or an even number? Solution Let’s look at a few examples in which one number is odd and the other number is even. 3 4 7 9 6 15 23 18 41 5 12 17 5 14 19 81 32 113 + = + = + = + = + = + = All these sums are odd numbers. Therefore, we might predict that the sum of an odd number and an even number is an odd number. 7 Now try Exercise 27 Induction often involves observing a pattern and from that pattern predicting a conclusion. Imagine an endless row of dominoes. You knock down the first, which knocks down the second, which knocks down the third, and so on. Assuming the pattern will continue uninterrupted, you conclude that any one domino that you select in the row will eventually fall, even though you may not witness the event. Inductive reasoning is often used by mathematicians and scientists to develop theories and predict answers to complicated problems. For this reason, inductive reasoning is part of the scientific method. When a scientist or mathematician makes a prediction based on specific observations, it is called a hypothesis or conjecture. After looking at the products in Example 1, we might conjecture that the product of two even numbers will be an even number. After looking at the sums in Example 2, we might conjecture that the sum of an odd number and an even number is an odd number. As described in the opening paragraph of this section, the science of biometrics is used for personal identification. By studying the biometrics of millions of people, scientists have never found two people who have the exact same fingerprints, iris patterns, DNA, or voice patterns. By induction, then, a conclusion can be reached that each of these biometrics provides a unique identification. A general conclusion is reached through the observation of specific cases. Therefore, the science of biometrics makes use of inductive reasoning. Examples 3 and 4 illustrate how we arrive at a conclusion using inductive reasoning. Example 3 Products Involving 5 a) Select some natural numbers and multiply the numbers by 5. b) Observe the ones digit (the rightmost digit) of the products from part (a). Use inductive reasoning and make a conjecture regarding products involving 5 and natural numbers. Definition: Inductive Reasoning Inductive reasoning is the process of reasoning to a general conclusion through observations of specific cases.
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