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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:

Undefined + Undefined = Undefined

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

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