Subsets of Real Numbers and Infinity

Date: 08/22/2001 at 17:26:39
From: Kevin Kelley
Subject: Subsets of Real Numbers and Infinity

Today in my Algebra II class we had a rather vehement argument, to 
which I have not been able to find a definitive answer.

Both the Whole Number set and the Integer set have an infinite number 
of entries, yet some of my classmates persisted that there were more 
numbers in the Integer set because it includes negatives as well.

Am I correct in saying that they both have an infinite number of 
numbers within them, and therefore are of the same size, or are my 
classmates correct in saying that the Integer set has more numbers in 
it?

Date: 08/23/2001 at 10:34:22
From: Doctor Peterson
Subject: Re: Subsets of Real Numbers and Infinity

Hi, Kevin.

You are right. Although there are in fact different infinities (for 
example, there are more real numbers than integers), the whole numbers 
and the integers have the same number of elements. (More surprisingly, 
so does the set of rational numbers.)

You can see this by matching the two sets up:

    0  1  2  3  4  5  6  7  8 ...
    0  1 -1  2 -2  3 -3  4 -4 ...

Since I can count the integers by matching each one up to a whole 
number, the two sets are the same size.

You can read more about these ideas here:

   Large Numbers and Infinity - Dr. Math FAQ
   http://mathforum.org/dr.math/faq/faq.large.numbers.html   

   Counting Rationals and Integers - Dr. Math archives
   http://mathforum.org/dr.math/problems/kelley.10.6.99.html   

   Sets Containing an Infinite Number of Members - Dr. Math archives
   http://mathforum.org/dr.math/problems/kate2.3.98.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/