# Undefined + Undefined = Undefined

We know that infinity divided by infinity is undefined:

∞ — ∞ |
= Undefined |
---|

If you don't believe me, here is the proof.

Most people would immediately stop working on an equation if they had undefined anywhere in the equation... but not me. So, I wondered what would happen if we tried breaking down the equation above, and I think I discovered something that I've never seen before, yet I believe it is intuitively obvious if you think about it.

First, we know that this is true:

If we substitute this equation into the numerator of the first equation, then we would get:

∞ + ∞ ——— ∞ |
= Undefined |
---|

This fraction can then be split into two fractions like this:

∞ — ∞ |
+ | ∞ — ∞ |
= Undefined |
---|

If we substitute the very first equation twice in this new equation, then we would get this equation:

As I said before, this seems obvious, yet I don't know of anyone who bothered proving it.

by Phil for Humanity

on 07/10/2011