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CHAPTER 1
Mathematical Reasoning
Section 1.1
Inductive Reasoning
There are many ways in which a person can reason. Some of them are better
suited to doing some things than others. There are two ways to reason that are
commonly used in mathematics:
•Deductive Reasoning
•Inductive Reasoning
Inductive Reasoning uses a series of examples or observations to
generate a conclusion about something more general. The examples are
sometimes referred to as a specific instance of a certain concept or idea.
Example of Inductive Reasoning
Lincoln is on a penny (Lincoln is a specific instance of a President)
Jefferson is on a nickel (A nickel is a specific instance of a coin)
Roosevelt is on a dime
Washington is on a quarter
Conclusion: All US minted coins have a President on them.
Inductive Reasoning is particularly useful for a person who is studying
mathematics to identify patterns that occur. For example her are a few patterns:
1= 1 = 12
1+3 = 4 = 22
1+3+5 = 9 = 32
1+3+5+7 = 16 = 42
What general conclusions can you draw from these specific instances?
The sum of all the odd numbers starting from 1 up to a certain odd
number is always a perfect square.
The sum of the first k odd numbers starting at 1 is k2.
Try the following for the values of x= 0, 1, 2
Pick a number x.
Find the cube of x. (Compute x3)
Subtract 3 times the square of x. (Subtract 3x2)
Add 3 times x. (Add 3x)
0  3 0  3 0  0  0  0  0
3
2
1  3 1  3 1  1  3  3  1
3
2
2  3  2  3  2  8  12  6  2
3
2
Conclusion: You will always get the number you start with.
Is this correct? Try it for the number 3
33  3  32  3  3  27  27  9  9
Inductive reasoning has both its advantages and disadvantages. Its big
advantages is it gives a way to identify patterns. The disadvantage is that it is not
always possible to test the pattern in all possible cases. The number of cases
might be infinite or too big to be practical.
In order to be able to handle conclusions about very large (or sometimes infinite)
collections of specific things we sometimes apply deductive reasoning.
Deductive reasoning uses a collection of general principles (sometimes called a
hypothesis or premise) to draw a conclusion about something.
Example of Deductive Reasoning
Hypothesis:
Dr. Daquila voted in the last election.
Only people over 18 years old can vote.
Conclusion :
Dr. Daquila is over 18 years old.
An example of how inductive and deductive reasoning can work together
Pick a number.
Add 5 to that number.
Multiply by 4.
Subtract 20.
Divide by 2.
Try this for the numbers: 0, 1, 2, 3
0  0  5  5  4  5  20  20  20  0  0  2  0
1 1  5  6  4  6  24  24  20  4  4  2  2
2  2  5  7  4  7  28  28  20  8  8  2  4
3  3  5  8  4  8  32  32  20  12 12  2  6
Conclusion: You get 2 times the number you start with.
Is this correct? We can not check all cases since there is an infinite amount of
numbers. We can use a form of deductive reasoning.
We take as our hypothesis (or premise) two things we know
Hypothesis:
x is a number
The rules of algebra
Conclusion: (Applied at each step)
Pick a number
x
Add 5
x+5
Multiply by 4
4(x+5)
Subtract 20
4(x+5)-20
Divide by 2
(4(x+5)-20)2
Apply some of the rules of algebra:
(4(x+5)-20)2=(4x+20-20)2=(4x+0)2=(4x)2=2x
We see that we always end up with 2x.