Introduction

The rational numbers -- the set of all fractions -- can be defined into a field, that is a set together with two operations -- + and . -- such that we may interpret + as addition and . as multiplication and get something that behaves the way we're used to. In other words, there should be inverses to all elements under + and ., there should be neutral elements -- a one for . and a zero for + -- and we should be able to do as much as possible with it.

So far, I've just stated all that you learn in school, even if I've done it in a terse and somewhat quirky way. But you learn a lot of other things in school, don't you? You may not take the square root of negative numbers... You may not divide by zero... and so on.

Some mathematicians don't like this kind of limitations, and so they define the things you may not do, and see what it leads to.

Expanding the field of rational numbers

One 'classical' way of expanding this is to form the Riemann sphere (it's just a circle until you mix in the complex numbers, but let's not...). You take the ordinary line of numbers, and wrap it up into a circle. Now you have 0 at the bottom of the circle, the positive numbers distributed counterclockwise half a lap up, and the negative numbers clockwise on the other half. At the top, we place an element that we call infinity (oo).

Now, we may define some rules for counting with infinity:

Some care must be taken though... Once your in infinity, you've a hard time getting out of there, and quite some operations won't work like you want them to...

This expansion is quite often labelled Q^oo

But there still are some things that are not yet defined. oo+oo for an example, or oo*0 or 0/0. So, what do we do? We amend the field with a new element!

If we define 0/0