Section 1.1 The Nature of Mathematical Reasoning Math in Our World

Section 1.1The Nature of Mathematical

Reasoning

Math in Our World

Learning Objectives

Identify two types of reasoning. Use inductive reasoning to form conjectures. Find a counterexample to disprove a

conjecture. Explain the difference between inductive and

deductive reasoning. Use deductive reasoning to prove a conjecture.

Reasoning is the process of logical thinking.

Two Types of

Reasoning

Inductive Reasoning (Induction)

Deductive Reasoning (Deduction)

Inductive Reasoning

The process of reasoning that arrives at a general conclusion based on the observation of specific examples.

It involves…•Looking for patterns•Making a Conjecture (an educated guess)

EXAMPLE 1 Using Inductive Reasoning to Find a Pattern

Use inductive reasoning to find a pattern, and then find the next three numbers by using that pattern.

1, 2, 4, 5, 7, 8, 10, 11, 13, __, __, __


SOLUTION

To find the pattern, look at the first number and see how to obtain the second number. Then look at the second number and see how to obtain the third number, etc.

1 2 4 5 7 8 10 11 13 ___ ___ __

The pattern seems to be to add 1, then add 2, then add 1, then add 2, etc. So a reasonable conjecture for the next three numbers is 14, 16, and 17.

+1 +2 +1 +1 +1+1 +1+2+2+2+2


Make a reasonable conjecture for the next figure in the sequence.


SOLUTION The flat part of the figure is up, right, down, and then left. There is a solid circle in each figure. The sequence then repeats with an open circle in each figure. So we could reasonably expect the next figure to be .

EXAMPLE 3 Using Inductive Reasoning to Make a Conjecture

When two odd numbers are added, will the result always be an even number?

Use inductive reasoning to determine your answer.

EXAMPLE 3 Using Inductive Reasoning to Make a Conjecture

Since all the answers are even, it seems reasonable to conclude that the sum of two odd numbers will be an even number.

Note: Since the sum of every pair of odd numbers hasn’t been tried, we can’t be 100% sure that the answer will always be an even number by using inductive reasoning.

SOLUTION We will try several specific examples:

3 7 10

59 14

199 28

255 30

127 28

21 3354

Counterexample

One specific example that proves the conjecture false. To find a counterexample…

•You must start with a number(s) or object(s) that matches the premise of the conjecture.

•Pick them in a way that will give you an incorrect response.

EXAMPLE 4 Finding a Counterexample

Find a counterexample that proves the conjecture below is false.

Conjecture: A number is divisible by 3 if the last two digits are divisible by 3.

EXAMPLE 4 Finding a Counterexample

SOLUTION We’ll pick a few numbers at random whose last two digits are divisible by 3, then divide them by 3, and see if there’s a remainder.

Start with 1,527:Next 11,745:At this point, you might start to suspect that the conjecture is true, but you shouldn’t! We’ve only checked two cases.

Now try 1,136:

This counterexample shows that the conjecture is false.

1,5273509

11,74533,915

1,13633782

3

Deductive Reasoning

The process of reasoning that arrives at a conclusion based on previously accepted general statements.

It does not rely on specific examples.

EXAMPLE 5 Using Deductive Reasoning to Prove a Conjecture

Consider the following problem: Think of any number. Multiply that number by 2, then add 6, and divide the result by 2. Next subtract the original number. What is the result?(a) Use inductive reasoning to make a

conjecture for the answer.(b) Use deductive reasoning to prove your

conjecture.


SOLUTION (a) Inductive reasoning will be helpful in forming a conjecture. We’ll choose a couple specific numbers at random and perform the given operations to see what the result is.

Number:Multiply by 2:

Add 6:Divide by 2:

Subtract the original number:Result:

So we might form a conjecture that the result will always be the number 3. But this doesn’t prove the conjecture, as we’ve tried only two of infinitely many possibilities.

12

122 24

24 6 30

302 15

15 12 3

3

5

52 10

106 16

162 8

8 5 3

3


SOLUTION (b) The problem with the inductive approach is that we can’t check every possible number. Instead, we’ll choose an arbitrary number and call it x. If we can show that the result is 3 in this case, that will tell us that this is the result for every number. Remember, we’ll be doing the exact same operations, just on an arbitrary number x.

Number:Multiply by 2:

Add 6:Divide by 2:

Subtract the original number:Result:

Now we proved our conjecture for all numbers.

x

x 2 2x

2x 6 2x 6

2x 62

x 3

x 3 x 3

3


Consider the following problem: Select a number. Add 50 to the number. Multiply the sum by 2. Subtract the original number from the product.What is the result?

(a) Use inductive reasoning to arrive at a general conclusion.(b) Use deductive reasoning to prove your

conclusion is true.


SOLUTION (a) Inductive reasoning will be helpful in forming a conjecture. We’ll choose a couple specific numbers at random and perform the given operations to see what the result is.

Number:Add 50:

Multiply by 2:Subtract the original number:

Result:

The conjecture is that the final answer is 100 more than the original number. But this doesn’t prove the conjecture, as we’ve tried only two of infinitely many possibilities.

1212 50 62 62 2 124 124 12 112 112

5050 50 100 100 2 200 200 50 150

150


SOLUTION (b) Now we’ll try using deduction. Remember, we’ll be doing the exact same operations, just on an arbitrary number x.

Number:Add 50:

Multiply by 2:Subtract the original number:

Result:

Our conjecture was right: the final answer is always 100 more than the original number.

x50x

2( 50) 2 100x x 2 100x x

100x

EXAMPLE 7 Comparing Inductive and Deductive Reasoning

Determine whether the type of reasoning used is inductive or deductive.

The last six times we played our archrival in football, we won, so I know we’re going to win on Saturday.

SOLUTION Inductive Reasoning!This conclusion is based on six specific occurrences, not a general rule that we know to be true.

EXAMPLE 8Comparing Inductive and Deductive Reasoning

Determine whether the type of reasoning used is inductive or deductive.

The syllabus states that any final average between 80% and 90% will result in a B. If I get a 78% on my final, my overall average will be 80.1%, so I’ll get a B.

SOLUTION Deductive Reasoning!Although we’re talking about a specific person’s grade, the conclusion that I’ll get a B is based on a general rule: all scores in the 80s earn a B.

Classwork

p. 12-15: 7, 9, 15, 17, 19, 22, 24, 28, 32, 35, 39, 41, 43, 47, 51, 67

Documents

Section 1.1 The Nature of Mathematical Reasoning Math in Our World