1. Inductive Reasoning Section 1.2

2. Objectives: Use inductive reasoning to make conjectures.

3. Key Vocabulary Conjecture Inductive Reasoning Counterexample

4. Inductive Reasoning Inductive reasoning is the process of observing patterns and making generalizations about those patterns. It is the basis of the scientific method. Mathematicians use inductive reasoning to make discoveries; then they attempt to verify their discoveries logically. This is the type of reasoning we will use throughout this course.

5. Using Inductive Reasoning Inductive reasoning consists of 3 stages: 1. Look for a Pattern: Look at several examples or data. Use diagrams and tables to help discover a pattern. 2. Make a Conjecture. Use the pattern to make a general conjecture. Okay, what is that?  A conjecture is an unproven statement that is based on observations. It is a generalization of the pattern observed. 3. Verify the conjecture. Use logical reasoning to verify the conjecture is true IN ALL CASES. (You will do this later in the course).

6. Example 1: Making a Conjecture Complete the conjecture. Conjecture: The sum of the first n odd positive integers is ?. How to proceed: List some specific examples and look for a pattern.

7. Example 1: Making a Conjecture First odd positive integer: 1 = 1 2 1 + 3 = 4 = 2 2 1 + 3 + 5 = 9 = 3 2 1 + 3 + 5 + 7 = 16 = 4 2 The sum of the first n odd positive integers is n 2.

8. ANSWER The sum of any two odd numbers is even. Complete the conjecture. Conjecture: The sum of any two odd numbers is ____. ? SOLUTION Begin by writing several examples. 1 + 1 = 2 3 + 13 = 16 5 + 1 = 6 21 + 9 = 30 3 + 7 = 10 101 + 235 = 336 Each sum is even. You can make the following conjecture. Example 2: Make a Conjecture

9. Complete the conjecture. Conjecture: The sum of the first n odd positive integers is ____. ? 1 1 = 1 2 1 + 3 4 = 2 2 1 + 3 + 5 9 = 3 2 1 + 3 + 5 + 7 16 = 4 2 SOLUTION List some examples and look for a pattern. ANSWER The sum of the first n odd positive integers is n 2. Example 3: Make a Conjecture

10. ANSWER odd Complete the conjecture based on the pattern in the examples. 1. EXAMPLES 7 × 9 = 63 1 × 1 = 1 3 × 5 = 15 11 × 11 = 121 3 × 11 = 33 1 × 15 = 15 Conjecture: The product of any two odd numbers is ____. ? Your Turn:

11. ANSWER n2 – 1n2 – 1 Complete the conjecture based on the pattern in the examples. 2. EXAMPLES 7 · 9 = 8 2 – 1 1 · 3 = 2 2 – 1 3 · 5 = 4 2 – 1 9 · 11 = 10 2 – 1 5 · 7 = 6 2 – 1 11 · 13 = 2 2 – 1 Conjecture: The product of the numbers (n – 1) and (n + 1) is ____. ? Your Turn:

12. Counterexample To prove that a conjecture is true, you need to prove it is true in all cases. To prove that a conjecture is false, you need to provide a single counter example. A counterexample is an example that shows a conjecture is false.

13. Example 4: Finding a counterexample Show the conjecture is false by finding a counterexample. Conjecture: For all real numbers x, the expressions x 2 is greater than or equal to x.

14. Example 4: Finding a counterexample: Solution Conjecture: For all real numbers x, the expressions x 2 is greater than or equal to x. The conjecture is false. Here is a counterexample: (0.5) 2 = 0.25, and 0.25 is NOT greater than or equal to 0.5. In fact, any number between 0 and 1 is a counterexample.

15. ANSWER The conjecture is false. Show the conjecture is false by finding a counterexample. Conjecture: The sum of two numbers is always greater than the larger of the two numbers. SOLUTION Here is a counterexample. Let the two numbers be 0 and 3. The sum is 0 + 3 = 3, but 3 is not greater than 3. Example 5: Find a Counterexample

16. Example 4 ANSWER The conjecture is false. Show the conjecture is false by finding a counterexample. Conjecture: All shapes with four sides the same length are squares. SOLUTION Here are some counterexamples. These shapes have four sides the same length, but they are not squares. Example 6: Find a Counterexample

17. Show the conjecture is false by finding a counterexample. 1. If the product of two numbers is even, the numbers must be even. ANSWER Sample answer: 7 · 4 = 28, which is an even number, but 7 is not even. The conjecture is false. Your Turn:

18. Show the conjecture is false by finding a counterexample. 2. If a shape has two sides the same length, it must be a rectangle. ANSWER Sample answer: These shapes are not rectangles, so the conjecture is false. Your Turn:

19. Note: Not every conjecture is known to be true or false. Conjectures that are not known to be true or false are called unproven or undecided.

20. Example 7: Examining an Unproven Conjecture In the early 1700’s, a Prussian mathematician named Goldbach noticed that many even numbers greater than 2 can be written as the sum of two primes. Specific cases: 4 = 2 + 210 = 3 + 716 = 3 + 13 6 = 3 + 312 = 5 + 718 = 5 + 13 8 = 3 + 514 = 3 + 1120 = 3 + 17

21. Example 7: Examining an Unproven Conjecture Conjecture: Every even number greater than 2 can be written as the sum of two primes. This is called Goldbach’s Conjecture. No one has ever proven this conjecture is true or found a counterexample to show that it is false. As of the writing of this text, it is unknown if this conjecture is true or false. It is known; however, that all even numbers up to 4 x 10 14 confirm Goldbach’s Conjecture.

22. Example 8: Using Inductive Reasoning in Real-Life Moon cycles. A full moon occurs when the moon is on the opposite side of the Earth from the sun. During a full moon, the moon appears as a complete circle.

23. Example 8: Using Inductive Reasoning in Real-Life Use inductive reasoning and the information below to make a conjecture about how often a full moon occurs. Specific cases: In 2005, the first six full moons occur on January 25, February 24, March 25, April 24, May 23 and June 22.

24. Example 8: Using Inductive Reasoning in Real-Life - Solution A full moon occurs every 29 or 30 days. This conjecture is true. The moon revolves around the Earth approximately every 29.5 days. Inductive reasoning is very important to the study of mathematics. You look for a pattern in specific cases and then you write a conjecture that you think describes the general case. Remember, though, that just because something is true for several specific cases does not prove that it is true in all cases.

25. Joke Time What did E.T.’s Mom say when he got home? Where on Earth have you been? What happened when the pig pen broke? The pigs had to use a pencil.

26. Assignment 1.2 Exercises Pg. 11 – 13; #1 – 25 odd