Finite Numbers

Date: 05/13/2003 at 18:03:51
From: Bola
Subject: Finite numbers

Dr. Math, 

I was wondering if you could tell me about finite numbers.

Date: 05/14/2003 at 14:20:03
From: Doctor Ian
Subject: Re: Finite numbers

Hi Bola, 

This has been an interesting question to think about. Informally, the
definitions of 'finite' and 'infinite' are somewhat circular: 
Something is infinite if it's not finite, and it's finite if it's not
infinite. But that doesn't shed a lot of light on the subject, does 
it? 

My dictionary has this definition:

3. Mathematics. 
  a. Bounded in an interval. Said of a quantity defined
     in an interval. 
  b. Incapable of being put into one-to-one correspondence
     with a part of itself. Said of a set.
  c. Real or complex, as distinguished from ideal. Said
     of a number. 

But it's not clear that these terms are going to be meaningful to you.
There are a lot of subtleties to understanding this. 

I guess I'd define a finite number this way: If it's possible to reach 
a number by doing arithmetic operations (+, -, x, /) starting with 1, 
in a sequence that will come to an end, the number is finite. 

So this handles any integer that you can think of, since you can get 
to any integer by starting from 0 and adding or subtracting 1 some
number of times. (You can get there in fewer steps if you use
multiplication.)  You may need a _lot_ of steps, e.g., if the number
you've selected is a million billion trillion; but eventually you
reach a final step, and that's what makes even a huge number finite.

You can get to any rational number by making two integers and dividing
one by the other (so long as you don't divide by zero). 

Unfortunately, this definition doesn't cover irrational numbers, like
pi or e. However, for any irrational number, we can always find a pair 
of rational numbers that surround it. 

So we can say that a number is finite if we don't have to do an 
endless sequence of operations to generate it; or if it lies between
two finite numbers. 

There are other subtleties, like how to deal with 'numbers' that have
components, like complex numbers or vectors. But if you ignore those,
does this definition make sense?  

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