What is 0 to the Power of 0?

We know that a nonzero number to the power of zero is one and zero to the power of a nonzero number is zero. So what is zero to the power of zero? Is it One? Zero? Undefined? It turns out that there is no single accepted answer — and this illustrates that mathematical definitions are not objective.

We know that a³ is equal to a multiplied by itself 3 times, or a•a•a. Then, 0³ is 0 multiplied by itself 3, which is still 0. That gives us the pattern:

This suggests that 0⁰ = 0, but it is not the whole story. What happens when we start with negative exponents and work our way up? We know that 0^-3 = 1/0³ = 1/0, which is undefined. That gives us the pattern

Now things are not so clear. The positive exponents tell us that 0⁰ = 1 and the negative exponents tells us that 0⁰ = undefined. Let’s look at this another way. We know that 3⁰ = 1, so we have the pattern

So now there are three possibilities: zero, undefined, and one. How do we choose? The answer is that we can do just that: choose. But we should choose carefully.

It was mathematicians who chose to call 1/0 undefined and warn us against dividing by zero, and this is for a good reason. If we say 1/0 is equal to some number x, then we are really saying that 0•x = 1 If we follow the rule that zero multiplied by any number is zero, then 0 = 1. So, if we choose to divide by zero we need to accept that 0 = 1.

The 1/0 example illustrates how mathematical definitions are constructed. In the real world, zero squirrels is not the same as one squirrel, so it makes sense to avoid saying that 0 = 1. To do this we must define 1/0 as undefined. The question of zero to the power of zero is not as clear as dividing by zero.

The most commonly accepted answer is 0⁰ = 1, but some mathematicians use 0⁰ = undefined. This choice depends on which is most useful and consistent given the context.

One reason why 0⁰ = 1 is generally preferred is that it’s consistent with the combinatorial definition of the exponent for integers, which says that a^b is the number of ways to form a set of b elements from a set of a elements. There is one way to form a set of zero elements from a set of zero elements, so this means that 0⁰ = 1.

The ambiguity of zero to the power of zero reveals that math is not a set of pre-ordained rules. Mathematicians intentionally design definitions and axioms — statements like 1/0 undefined that are assumed to be true–and the rest of math is what logically follows.

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This article was edited by Simon Thill and George Morgulis.