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### Rational and Irrational Numbers

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Date: 2/2/96 at 0:4:38
From: Anonymous
Subject: Rational and Irrational numbers

What is the difference between rational and irrational numbers?
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Date: 2/2/96 at 14:34:37
From: Doctor Syd
Subject: Re: Rational and Irrational numbers

Hello!

Before we talk about rational and irrational numbers, let's make clear
one other definition.  An INTEGER is in the set:

{...-3, -2, -1, 0, 1, 2, 3, ...}

It is just a positive or negative whole number.  Thus 454564 is an
integer, but 1/2 isn't.

Now, a rational number is any number that can be written as a ratio of
two integers (hence the name!).  In other words, a number is rational if
we can write it as a fraction where the numerator and denominator are
both integers.  Now then, every integer is a rational number, since
each integer n can be written in the form n/1.  For example 5 = 5/1 -
thus 5 is a rational number.  However, numbers like 1/2,
45454737/2424242, and -3/7 are also rational since they are fractions
where the numerator and denominator are integers.

An irrational number is any real number that is not rational.  By "real"
number I mean, loosely, a number that we can conceive of in this world,
one with no square roots of negative numbers (numbers where square roots
of negative numbers are involved are called complex, and there is lots
of neat stuff there, if you are curious).  A real number is a number
that is somewhere on your number line.  So, any number on the number
line that isn't a rational number is irrational.  For example, the
square root of 2 is an irrational number because it can't be written as
a ratio of two integers.

How would you imagine we would show something like that?  The proof
is a proof by contradiction.  We assume that the square root of 2 CAN
be written as p/q for some integers, p and q, and we get a contradiction.
The proof has a little trick to it, but if you're curious about it, write back
and I can tell you more!

Other irrational numbers include:
square root of 3, the square root of 5, pi, e, ....

I hope this answers your question.  There are lots of neat properties of
rational numbers, irrational numbers and real numbers.  For instance, it
turns out that if you were to try to gauge how many rational numbers,
irrational numbers, and real numbers there are between 0 and 1, you
would find that while there are infinitely many of each kind of number,
there are many, many more irrational numbers than rational numbers.  The
sizes of the infinities involved are somehow a little different.  Another
property is that between any two rational numbers on the number line
there is an irrational number; also, between any two irrational numbers
there is a rational number.

So, these are some things for you to ponder!  Write back if you have any
more questions!

-Doctor Syd,  The Math Forum

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