# What does Infinity Divided by Infinity Equal?

At first, you may think that infinity divided by infinity equals one. After all, any number divided by itself is equal to one, however infinity is not a real or rational number. I am going to prove what infinity divided by infinity really equals, and you may not like the answer.

First, I am going to define this axiom (assumption) that infinity divided by infinity is equal to one:

∞ — ∞ |
= 1 |
---|

Since

**∞ = ∞ + ∞**, then we are going to substitute the first infinity in our axiom:

∞ + ∞ ——— ∞ |
= 1 |
---|

The next step is to split this fraction into two fractions:

∞ — ∞ |
+ | ∞ — ∞ |
= 1 |
---|

Next, substitute the axiom twice into the equation, we get:

1 | + | 1 | = 1 |
---|

Finally, this can be rewritten as:

2 | = 1 |
---|

This equation is obvious incorrect. Therefore, infinity divided by infinity is

**NOT**equal to one. Instead we can get any real number to equal to one when we assume infinity divided by infinity is equal to one, so

**infinity divided by infinity is undefined**.

If you still do not believe me, let's try proving this another way. Let's start with our axiom again.

∞ — ∞ |
= 1 |
---|

Next, split this fraction into two parts.

∞ * | 1 — ∞ |
= 1 |
---|

Solve the fraction of 1 divided by ∞. At first, you would think 1 divided by ∞ is equal to 0, however that is not correct because that would mean 0 times ∞ would equal to 1. And that is proven not true here. However, 1 divided by ∞ does equal a limit approaching 0. In other words, 1 divided by ∞ does not equal a number or is undefined.

∞ * | undefined | = 1 |
---|

As a result, we reached a dead end.

**Therefore, infinity divided by infinity is still undefined.**

by Phil B.