# 0^0 is Undefined

What does 0^0 equal?

Some people think 0^0 equals 1, while other people think 0^0 is undefined. Here is a simple proof to answer this mystery.

Assume that:

Take the natural log or ln of both sides of the equation:

Since ln (a^b) = b * ln a, then this is true:

When x is greater than or less than zero (x != 0), then (ln x) is a number (n), so this is true:

Thus:

If we apply Euler's number (e) to both sides of the equation, this would be the result:

Simplified, this is:

More simply:

Therefore, if we substitute this equation into the original assumption., then we would have proved:

Thus, any non-zero number to the zeroth power equals 1.

However, when x is equal to zero (x=0), then (ln x) is (ln 0), and this equation is undefined.

Since we had this equation above:

We can substitute to this equation when x is equal to 0.

This equals:

Therefore, we must stop working on this equation. As a result:

or

by Phil for Humanity

on 09/14/2011