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Survey of Math - MAT 140
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Inductive Reasoning
Some warm up might be necessary, the book does not go this far but sometimes it is better to go beyond the book.
1
Number Sense
Idea #1 In nity- a never ending progression
Example 1 Is there an in nity of sand grains on the Outer Banks of North Carolina, could it be counted?
Think a bit on this, could this be done, could every grain of sand be counted?
The answer is YES, because eventually (in a few million years) every grain of sand could be counted! An in nity
assumes that once you think that you reached it, there is always one more, and one more, and .......
Idea #2 The Real Numbers-the basis of most of mathematics is Numbers and Numeration.
Natural Numbers (counting numbers)
These were used by the oldest of people and was easy to understand. To list them all would be impossible so we
use something called Set Notation to accomplish this. Using ellipses to indicate a continuing pattern to in nity
the Natural numbers would be:
N D f1; 2; 3; 4; 5; :::g
Whole Numbers Here is a huge addition to the number system, the concept of ZERO. We will nd that for
thousands of years the concept of zero was avoided. Think how you would answer the following question and
you will have an idea of why zero was a complex creation: How many oranges are in an empty bowl? Why
NONE of course.
W D f0; 1; 2; 3; 4; :::g
Integers (the essence of the idea of Negative) The idea of Negative, the absence of "stuff" is necessary to talk
about simple Accounting used in a Check Book. Deposits are .C/ and Checks written are . / to the account.
I D f:::; 3; 2; 1; 0; 1; 2; 3; :::g
p
where p&q are integers. They can also be representative
q
of decimal numbers that terminate .0:25/ or that repeat .0:122 22:::/ :
p
There is no list that can be compiled, like above with the integers. So they are just designated as any number
q
in above sentence.
Rational Numbers better known as fractions
So are these all the possible numbers? The Greeks (Pythagorean's) thought that all numbers (for them it was lengths)
could be represented as a rational number. They found an exception to this idea using the Pythagorean Theorem (the
square of the hypotenuse equals the sum of the squares of the legs on any right triangle, c2 p
D a 2 C b2 /: They found a
number (length) that could not be represented by (measured as) a rational number, mainly 2 . This type of number
is called:
Irrational Numbers are numbers that cannot be represented as a Rational Number. Thus they are decimal
numbers that do not terminate and do not repeat. One of the most famous of these irrational number is Pi
D 3: 141 592 654:::
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
Page: 2
Think: If some game show asks you p
to tell them any Real Number, what would you tell them? Most people pick a
Counting Number, very few will say " 13": Why do you think this is so?
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Pattern's with Inductive Reasoning
Another topic in most mathematical areas is the study of patterns.
Example 2 What is the sum of an even number and an odd number? Do a couple of examples, just remember that
even numbers must be divisible by 2
Example 3 What is the sum of two odd numbers?
Your conclusions above are based on Inductive Reasoning, reasoning based on observations of a nite number of
events. This is the reasoning that the brain uses when it hears rumors. This is the kind of reasoning that led to the
thought that the Sun revolved around the Earth. Which if you think about it, seems pretty reasonable. The Sun comes
up in the East and Sets in the West. It "looks" like it is going around us.
Inductive Reasoning is also a major part of the Scienti c Method:
# 1. Observe some aspect of the universe.
# 2. Invent a tentative description, called a hypothesis, that is consistent with what you have observed.
# 3. Use the hypothesis to make predictions.
# 4. Test those predictions by experiments or further observations and modify the hypothesis in the light of your
results. (borrowed from phyun5.ucr.edu/~wudka/Physics7/)
Exercise 4 Find the next three numbers in the following patterns. See if this Hypothesis holds up for all three.
1. 1; 4; 7; 10; _____; _____; _____ Here you have to nd some way of going from 1 to 4 using some operation (or
combination of operations). I usually start with addition (subtraction). There is a difference of 3.. Lets see if
this hold up for the next one 4 C 3 D 7 and the next 7 C 3 D 10: So using inductive reasoning we nish the
patter:
1; 4; 7; 10; 13; 16; 19
2. 16; 8; 4; _____; _____; _____
Here if we subtract 16
8 D 8 , but this does not hold up for the next one
16
8
8 8 D 0: So maybe this is another operation, say division.
D 8 and D 4 so we just need to keep
2
2
dividing by 2:
16; 8; 4; 2; 1; 1=2
3. 27; 18; 12; ____ Here subtraction doesn't work and neither does division. This must be a combination pattern.
27
We know that 27 is divisible by 3 and so is 18 and 12 so lets start there
D 9 now if we double that 9 .2/ D
3
18
12
D 6 and double: 6 .2/ D 12 so now:
2D8
18 which is the next element. Lets test this pattern:
3
3
27; 18; 12; 8
would nish this pattern.
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
Page: 3
Example 5 Play the following game starting with any Counting Number. I have one possible answer here
1
2
3
Pick a number
multiply by 6
subtract 8
4
divide by 2
5
add 4
7
7 6 D 42
42 8 D 34
34
D 17
2
17 C 4 D 21
7
42
34
17
21
What pattern exists between the number picked and the nal answer? My guess would be that the nal number is
three times the number that we started with. This would be my hypothesis (conclusion). Test is out to make sure this
is a good conclusion.
This is considered Inductive Reasoning. There is no way by picking all of the possible numbers so that we would be
absolutely sure that this pattern works. To test that we would have to test an in nite number of numbers, impossible
because there will always be one more to test!!!!
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Deductive Reasoning
So there must be some way in which we can be SURE that our Hypothesis is true. This type of Reasoning is the one
you encountered in HS Geometry, Deductive Reasoning
Example 6 Deduce using a Variable that our Hypothesis above is correct for all real numbers.
1
2
3
Pick a number
multiply by 6
subtract 8
4
divide by 2
5
add 4
N
6N
6N 8
6N 8
D 3N 4
2
.3N 4/ C 4 D 3N
N
6N
6N 8
3N
4
3N
Now we can say with absolute certainty that indeed the last number is three times the rst, proving our conjecture
(hypothesis) for all numbers.
3.1
Counter Example
A way to disprove a Hypothesis is the method of Counter Example, where instead of nding examples that t the
conjecture we try to nd at least one that does not.
Example 7 Conjecture: The sum of an odd and an even is prime*.
*prime numbers are only divisible by 1 and itself (two different numbers) thus the list of prime numbers start off at
f2; 3; 5; 7; 11; 13; 17; 19; 23; :::g
2C3D5
wor ks
3C4D7
wor ks
4 C 5 D 9 Here is one that does not work
Since the last one does not work we can say with absolutely certainty that our Conjecture is FALSE!
Copyright 2007 by Tom Killoran