Math in Our World
Section 1.1
The Nature of Mathematical
Reasoning
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, __, __, __
EXAMPLE 1
Using Inductive Reasoning to
Find a Pattern
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 ___ ___ __
+1 +2 +1 +2 +1 +2
+1
+2
+1
+2
+1
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.
EXAMPLE 2
Using Inductive Reasoning to
Find a Pattern
Make a reasonable conjecture for the next
figure in the sequence.
EXAMPLE 2
Using Inductive Reasoning to
Find a Pattern
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
SOLUTION We will try several specific examples:
3

7

10
19

9

28
1

27

28
5

9

14
21

33

54
25

5

30
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.
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.
,
527

3

509
Start with 1,527: 1
11
,
745

3

3
,
915
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.

2
Now try
1,136:
1
,136

3

378
3
This counterexample shows that the conjecture is false.
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.
EXAMPLE 5
Using Deductive Reasoning to
Prove a 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.
12
5
Number:
5

2

10

2

24
Multiply by 2: 12
Add 6: 24

6

30
10

6

16
16

2

8

2

15
Divide by 2: 30



5

3

12

38
Subtract the original number: 15


3
3
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.


EXAMPLE 5
Using Deductive Reasoning to
Prove a Conjecture
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:
x
Multiply by 2: x

2

2
x
Add 6: 2
x

6

2
x

6
x6
Divide by 2: 2
x3

2
Subtract the original
number: x

3

x

3

Result:
3
Now we proved our conjecture for all numbers.


EXAMPLE 6
Using Deductive Reasoning to
Prove a Conjecture
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.
EXAMPLE 6
Using Deductive Reasoning to
Prove a 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.
50
12
Number:
0

5
0

1
0
0
2

5
06
2 5
Add 50: 1
Multiply by 2: 6
0
0

2

2
0
0
2

2

1
2
4 1
Subtract the original number: 1
2
0
0

5
0

1
5
0
2
4

1
2

1
1
2

Result: 112
150
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.
EXAMPLE 6
Using Deductive Reasoning to
Prove a Conjecture
SOLUTION
(b) Now we’ll try using deduction. Remember, we’ll be doing the
exact same operations, just on an arbitrary number x.
x
Number:
x5
0
Add 50:
Multiply by 2: 2
(5
x

0
)

2
x

1
0
0
Subtract the original number: 21
x

0
0

x

Result:
x
1
0
0
Our conjecture was right: the final answer is always 100 more than
the original number.
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 8
Comparing 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