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7) There is a conclusion.
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Irrational Numbers
Daniel Dreibelbis
MAS 4932
March 12, 2008
Abstract
This paper describes the properties of irrational numbers. We begin by defining what an
irrational number is, distinguishing it from rational numbers. Examples of commonly
used rational numbers are given. We then give a simple proof that the square root of 2 is
an irrational number. If this was a twenty page paper, I would have more things to say
here, and my abstract would list all of the major goals of my paper. As is, I only have
two goals, so I will end this abstract now.
1) Introduction
Irrational numbers have been used throughout mathematics. They are really special.
Legend has it that Pythagoras and his followers were the first mathematicians to
recognize the existence of irrational numbers. This was quite a problem for Pythagoras,
because he and his school believed that all distances would be rational multiples of each
other. In fact, the Pythagorean Theorem was used to prove the existence of an irrational
number, and it destroyed their entire system of belief.
In Section 2, we give the definitions of rational and irrational numbers, and we
give some examples of each. In Section 3, we give a proof that
2 is irrational.
2) Definitions
Throughout this paper, we only consider real numbers, as opposed to complex numbers.
We first need to define a rational number:
Definition 2.1. [1] A rational number is a number that can be written as p/q,
where p and q are integers.
Example 2.2. The following numbers are all rational: 2, 0, -1, 2/3, 100/102,
234.45, -4.123423523, etc.
As the name implies, irrational numbers will the numbers that are not rational:
Definition 2.4. [2] An irrational number is a number that cannot be written as p/q
with p and q integers.
Example 2.5. [2] The following numbers are all irrational: π, e,
2,
3.
3) Square root of 2
There is a very nice proof showing that
Theorem 3.1 [3] The number
Proof: Assume that
2 is irrational.
2 is irrational.
2 is not irrational, i.e., assume that
there exists relatively prime integers p and q such that
2 is rational. The
2  p / q . Rearranging and
squaring both sides, our equation becomes
2q2 = p2
Since the left-hand side is even, that implies that the right-hand side is even, which
implies that p is even. But if p is even, then p = 2k for some integer k. Plugging this into
our equation, we get
2q2 = 4k2
So q2 = 2k2, which implies q2 is even, which implies q is even. Hence 2 divides both p
and q, which is a contradiction to the assumption that p and q are relatively prime. Hence
2 must be irrational.
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4) Conclusion
We have barely scratched the surface of all the wonderful applications of irrational
numbers. It is truly amazing how often they show up in mathematics. While our proof
about
2 was relatively simple, proving that other numbers are irrational (like e and π)
take quite a bit of ingenuity. Irrational numbers are our friend, and we cannot escape
them, even if we wanted to.
References
(1) Weisstein, Eric W. "Rational Number." From MathWorld--A Wolfram Web
Resource. http://mathworld.wolfram.com/RationalNumber.html
(2) Weisstein, Eric W. "Irrational Number." From MathWorld--A Wolfram Web
Resource. http://mathworld.wolfram.com/IrrationalNumber.html
(3) Smith, D., Eggen, M, and St. Andre, R. A Transition to Advanced Mathematics.
Third Edition, Brooks/Cole, 1990.