Infinity, The Truth About (Real)

What Is It?

Author: Aaron Rice
Date: 1997/02/10
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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.

See also

• Infinity
• Infinity, The Trouble With
• Insomnia
• Time Travel