Equations with Infinity

Date: 07/29/98 at 03:37:47
From: Robert Rundle
Subject: Infinity

Given that infinity is a concept which has not been proven, how is it 
possible to use it in maths equations? 

Is it just used as a concept number, and does that mean that any 
equation which uses infinity is also a concept and cannot be proved 
or disproved?

Date: 07/29/98 at 15:08:46
From: Doctor Rob
Subject: Re: Infinity

Equations such as 1/infinity = 0 are not equations in the real number
(or complex number) system. They are shorthand for some equations
involving limits. The "infinity" in the above expression is short for:

       lim    g(x)
   x->infinity

where g(x) is any function which grows without bound as x does. In this 
case, the equation is shorthand for:

   1/[    lim    g(x)] =    lim    1/g(x) = 0
      x->infinity       x->infinity

This limit also uses the symbol infinity, but the limit has a precise 
definition:

       lim    f(x) = L
   x->infinity

is defined to mean that for any e > 0, no matter how small, there
exists a B (depending on e and f), such that for all x > B,
|f(x)-L| < e.  In other words, as x grows without bound, f(x) gets
arbitrarily close to L.

Such equations involving limits are, indeed, provable (or disprovable), 
and so the shorthand versions are provable (or disprovable).

One must, however, be very careful when writing down such shorthand
"equations" to be sure the full version is true. For example,
infinity - infinity = 0 is not true, and this is one of the ways you 
can see that infinity does not behave like a real number. To see the 
correctness of this last statement, write:

   infinity - infinity = lim (x+c) - lim x = lim (x+c-x) = lim c = c

where c is any real number, and the limits are taken as x->infinity, or 
write:

   infinity - infinity = lim (2*x) - lim x = lim (2*x-x) = lim x 
                       = infinity

This means that infinity - infinity is not a meaningful shorthand for
any limit equation, since it has no particular fixed value. Such
expressions are called "indeterminate" expressions, because you cannot
determine their values.

- Doctor Rob, The Math Forum
Check out our web site! http://mathforum.org/dr.math/