Numbers between 0 and 9999

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.

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 
pretty fun!  If you want to learn more about the different "sizes" of 
infinity or any other questions you have about infinity, please do 
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/