Infinity in Undefined (Divide by 0) Situations

Date: 9/5/96 at 21:18:23
From: Jonathan White
Subject: Infinity in Undefined (Divide by 0) Situations

In economics, there is something called elasticity. Elasticity is
simply an inverse slope. If a graph of elasticity is a horizontal 
line, it is referred to as infinitely elastic. Wouldn't this really 
be undefined because it is a divide by zero error? Is there some 
relationship between infinity and dividing by zero that I don't 
understand or is it simply mislabled for simplicity?

Date: 9/6/96 at 19:34:11
From: Doctor Tom
Subject: Re: Infinity in Undefined (Divide by 0) Situations

Hi Jonathan,

Nowhere is it written in stone what mathematical objects should
be studied.  Obviously, objects that are interesting and useful
have received the most attention.

For example, the real numbers are very important, but there's
no law of the universe that says we couldn't think about a system
that includes the real numbers AND another thing called "infinity."

The trouble is, for most applications it's very tough to figure
out how to integrate this new "infinity" with the other numbers
in a useful way.

Suppose we add this new infinity, and call it "I" (I call it "I"
since I can't type the usual 8-on-its-side symbol here).

At first, we think, "great -- I can now say that 1/0 = I".  OK,
but what's I - I?  Is it zero?  If so, we no longer have the
associative law, since (1 + I) - I = I - I = 0, but
1 + (I - I) = 1 + 0 = 1.  Believe me, it would be a real pain
to try to do algebra without the associative law.

So rather than stuff an "infinity" into the system, mathematicians
usually use the word "infinity" in a very controlled way that's
defined in terms of actual numbers.  For example, if I say,
"What is the limit, as x approaches infinity, of (x+1)/x?"
Without going into the details, this definition translates into
looking at what happens to that expression with very large
values of x.

Similarly, we can say what we mean for an expression to "approach
infinity."  It just means that whatever actual number you name,
it eventually gets bigger than that.  And so on.

But with these careful definitions, you'll find that you can
almost always get away with a little sloppiness in your thinking,
and still usually get the right answer, so some people let this
sloppiness creep into what they say.  I can define, for example,
a vertical line to have a slope of infinity, but I can't operate
on it (legally, at least) with actual mathematical operations.

But it makes sense if you think of a vertical line as a limit of
lines that are closer and closer to vertical.  Those lines have
slopes that get larger than any number you care to name, so in
a mathematical sense, the slopes of those lines tend to infinity.

It's a little dangerous to say, "the vertical line has a slope
of infinity," but many people do.  As long as you understand this
to mean that you can't actually use the value of infinity in
calculations, you're okay.

I hope this helps.

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