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### Numbers between 0 and 9999

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Date: 7/15/96 at 21:48:43
From: Lucas W Tolbert
Subject: How Many Numbers....

My Dad and I were arguing last night.  I said there were only
10000 whole numbers counting zero between 0 and 9999 but he said there
are an infinite number.  I know I am right but he wants me to ask what
you think.
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Date: 7/15/96 at 22:18:32
From: Doctor Sydney
Subject: Re: How Many Numbers....

Dear Lucas,

You were right!  Whole numbers are the numbers we count with:  1, 2,
3, ... One way to define a whole number is that if you were to write a
whole number in the form of a fraction, where it is reduced as far as
possible (meaning there are no common divisors in the numerator and
the denominator), there would be a 1 in the denominator.  So, the only
whole numbers between 0 and 9999 inclusive are: 0, 1, 2, 3, ..., 9998,
9999.

However, there ARE an infinite number of RATIONAL numbers and REAL
numbers between 0 and 9999.  A rational number is a number that can be
written as a fraction where the numerator and denominator are positive
or negative whole numbers, and a real number is any number that you
can find on the number line, whether it be a rational number like 1/2
or an irrational number like pi.

Infinity has some strange, interesting properties.  For instance, it
turns out that the number of real numbers between 0 and 9999 is the
same as the number of real numbers between 0 and 1!  What's more,
mathematicians have defined several different "sizes" of infinity.
So, the infinity associated with the rational numbers between 0 and 1
is "smaller" than the infinity associated with the real numbers
between 0 and 1.  However, the total number of whole numbers ({1, 2,
3,..., 9999, 10000, ...}) is the same "size" as the number of rational
numbers between 0 and 1.  As you can see, problems dealing with
infinity can be tricky and surprising, but they can turn out to be
write back!  We love infinity (or at least I do)!

Hope I was able to help in the dispute and hope it has a peaceful
resolution!

:)

-Doctor Sydney,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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