This article is classified "Real"
In some circumstances, infinity (to be algebraically represented in this
article by the letter "I") can be used as a number:
* in order to engineer a formula,
* to extract and display some theory, or
* to fill some empty space on a text book.
However, in most cases, it makes a very bad number, and is mostly used as a
concept. The reason for this is that it isn't really a number.
Take the following rules:
1) Infinity divided by a finite number is infinite (I / f = I);
2) Any finite number divided by infinity is a number
infinitesimally larger than, but never equal to, zero
(f / I = 1 / I);
3) Infinity divided by infinity is one (I / I = 1),
or in fact any other positive number (I / I = and so on...);
4) Infinity multiplied by zero (no infinity) is zero (I * 0 = 0);
5) Infinity divided by a positive finite number is infinity
(I / +f = I);
6) Infinity divided by a negative finite number is minus
infinity (I / -f = -I);
7) Infinity divided by zero is not possible;
8) Infinity plus infinity is infinity (I + I = I);
9) Zero divided by infinity (nothing divided into infinity)
equals zero (0 / I = 0);
10) Infinity plus a finite number is infinity (I + f = I);
11) Infinity minus a finite number is infinity (I - f = I); but
12) Infinity minus infinity, due to the nature of infinity, can be
zero, infinity, or minus infinity (I - I = -I, 0, I).
In some of these respects it appears to be a number; it shares some of the
same properties. Unfortunately, when you start using it as a number, you
experience a few problems. You might be thinking, so why isn't it a
number? Well, if it is, then it is each and every imaginable and
unimaginable number, while being none of them. If you can define a number,
it isn't infinity. Therefore, infinity is not a number.
Perhaps then, it is lots of numbers. Perhaps a variable? Well, this is an
interesting point to consider, but again, any number you might apply to
infinity, is quite obviously not infinity. It doesn't matter how many
numbers you apply to it, if they are numerable, they are not infinity.
Can you calculate it? Well, considering the nature of infinity, no. As
soon as you apply a value to infinity, you find that it isn't. Pick a
number; you can always add one more.
So what can you use it for? In most cases, people use infinity to replace
a number that is so large, they can't be bothered to work it out. Infinity
is usually applied to the boundless limits of space, and time. The theory
being that if you go far enough in space, you can go further [1], and time
is just as bad.
The closest way to experience infinity is to buy an "auto-reverse" tape
recorder, fix it to a reliable power source, and never turn it off. Of
course, you will never really experience infinity this way, as you will
eventually die, the tape will break, the machine will break, or the power
source will die [2].
Curiosities
===========
Here follows a couple of examples of how one might happily misuse infinity
for their own amusement, cheerfully putting a few rules to one side:
If I + I = I and I + 2 = I
Then I + I = I + 2
So I = I + 2 - I
Then I = 2
But in the same spirit you could also say that:
Since I + 2 = I and I - 4 = I
Then I + 2 = I - 4
So 2 + 4 = 0
Then 2 = -4
Which is completely absurd, and is therefore the basis of most of the most
advanced forms of modern creative accounting. In fact, all this proves is
that infinity is most definitely not a constant value, and trying to use it
as one can cause all manner of problems. Trying to fiddle with the rules of
infinity can be hazardous to your health.
[1] And further [1].
[2] Especially if you forget to pay the bill.