Infinity times Infinity

Date: 10/15/98 at 21:00:39
From: jeff
Subject: Infinity

What is infinity multiplied by infinity?

Date: 10/22/98 at 01:19:57
From: Doctor Andrewg
Subject: Re: Infinity

Hi Jeff,

Well, that's a pretty interesting question that you have asked. I've 
always found the idea of infinity to be really interesting, and 
sometimes even a bit scary! Imagine if you had to do an infinite 
amount of homework - how long would that take? Would you ever be 
finished?

Let's think about multiplying numbers that are not infinity first, so I 
can think a bit more about your question.  

If we want to see how much a number times another number is (say, 5 
times 3) we can imagine that we have the first number of piles, each 
containing the second number of things, and see how many things we have 
overall. In the case of 5 times 3, we can imagine 5 piles of 3 things, 
for a total of 15 things. Or we could imagine 3 piles of 5 things for a 
total of 15 things. The things we are counting can be anything you like 
- apples, books, counters, anything at all.

Can we do this with infinity times infinity?

How many piles do we need to have an infinite number of piles? And how 
many things so in each pile so that there is any infinity of them? If 
we have a number, say a billion piles, then we will still not have an 
infinite number of piles. And no matter how many piles we add, there 
still won't be an infinite number of piles! At the same time, no matter 
how many things we put in each pile there won't be an infinite number 
in that pile either.  

So I infinity can't be a number, and neither can infinity times 
infinity. 

The problem is that infinity (sometimes shown using a symbol that looks 
like an eight on its side) isn't really a number at all. It represents 
a symbol for a value that is so large that we can't imagine it - no 
matter what number you give me, it will always be less than infinity.  

One way of looking at this is that if you had an infinite number of 
piles, each containing an infinite number of something, then you would 
still have an infinite number of things overall.

So infinity times infinity is still infinity.

In higher-level mathematics we can deal with different types of 
infinity - perhaps you will study these one day. In the meantime keep 
up the great questions.

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