Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 1.1 Inductive Reasoning

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Induction – An Introduction Complete the patterns by finding the next number (or object) in the pattern: Can you get all 7??? 1. 1, 5, 9, 13, ,1, 2, 3, 5, 8, , 6, 18, 54, , 4, 9, 16, , 2, 4, 7, 11, J, F, M, A, M,

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Natural Numbers The set of natural numbers is also called the set of counting numbers. N = {1, 2, 3, 4, 5, 6, 7, 8, …} The three dots, called an ellipsis, indicate that the numbers continue in the same manner

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Divisibility If a ÷ b has a remainder of zero, then a is divisible by b. The even counting numbers are divisible by 2. They are 2, 4, 6, 8,…. The odd counting numbers are not divisible by 2. They are 1, 3, 5, 7,…

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Inductive Reasoning Consider the following pattern: Using induction, we may conjecture that the sum of an odd and an even number is odd

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Inductive Reasoning The process of reasoning to a general conclusion through observations of specific cases (also called induction). Induction is often used by mathematicians and scientists to make predictions

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Inductive Reasoning – Another Example: In millions of tests, no two people have been found to have the same fingerprints or DNA. By induction, then, we believe that fingerprints and DNA provide a unique way to identify people. In fact, fingerprints and DNA can be used in a court of law as evidence to convict persons of crimes

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Scientific Method Inductive reasoning is a part of the scientific method. When we make a prediction based on specific observations, it is called a hypothesis or conjecture

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Counterexample In testing a conjecture, if a special case is found that satisfies the conditions of the conjecture but produces a different result, that case is called a counterexample. Only one exception is necessary to prove a conjecture false. If a counterexample cannot be found, the conjecture is neither proven nor disproven

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example of a Counterexample: Recall that a prime number is a natural number that has exactly two factors. Thus, the following numbers are prime: 3, 5, 11, 17, 31 It would appear that all prime numbers are odd. However, the number 2 is a counter- example shows that this hypothesis is false

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Deductive Reasoning Deductive reasoning, or deduction, is the process of reasoning to a specific conclusion from a general statement. Example: The sum of the angles of a triangle is 180°