Infinity minus Infinity

Date: 10/26/95 at 19:38:3
From: Dominic Parkinson
Subject: Infinity

Dr Math,

INFINITY - INFINITY = ?

Thanks.

Date: 11/5/95 at 16:17:2
From: Doctor Jeremy
Subject: Re: Infinity

There really isn't an answer to this one.  Infinity isn't really a 
number, so you can't do the same things with it.  Infinity is "how many" 
of something that cannot be counted because there is no end to it.

Since infinity isn't a number, what "minus" means isn't really clear.  
If we let "infinity - infinity = ?" mean "If you have infinitely things 
and take away infinitely many of them, how many do you have left?", 
there are a lot of things the answer could be.

The answer could be zero.  If you have all the numbers (there are
infinitely many of them) and take them all away (again infinite) you 
have none left.

The answer could be some number.  For example, you can have all the 
whole numbers greater than zero (i.e. 1, 2, 3, 4 and so on).  You can 
remove all the numbers greater than four.  Then the only ones you've got 
left are 1, 2, 3, and 4.  So in that case the answer is four.

The answer could be infinite.  If you have all numbers, and you remove
all even numbers, you still have infinitely many left.

This definition I've used doesn't allow for  numbers that are negative 
or aren't whole numbers.  If you allow those, there will be even more 
possibilities.

In short, the notion of subtracting with infinity isn't really defined.
You can't really do it, and if you could, the answer wouldn't mean 
much.

-Doctor Jeremy,  The Geometry Forum