Why is 0 0 is not defined

What is 0 to the power of 0?

Reading time: 4 min

The question about the result of “zero to the power of zero” () cannot be answered unequivocally.

Here are some considerations about this issue:

version 1

because for every number the following applies: (see also principle of permanence)

Variant 2

because for zero to the power of a number, the result is zero:

Variation 3

Not defined, because both variants 1 and 2 create a contradiction, i.e. there is no uniqueness, which is problematic in mathematics. Incidentally, this will also be the reason why many pocket calculators input or output at.

Conclusion

The answer is often found:

Use what makes sense for the math problem at hand.
It is often useful to use.

There is a variety of literature on this and, as I said, you encounter different approaches. In computer science, for example, prevailed. For fun you can test it yourself and enter 0 ^ 0 in Google.

Another approach

If we write and use the power law, the result is:

It must therefore apply:

So the question arises, which number multiplied by itself results in itself? Here we can think of two numbers for which this is possible:

and

Solved in general:

So we see that there are two solutions with and.

Basically, however, is preferable. If we chose, new problems would arise in higher mathematics.

Approach via limit values

The consequence has the limit, there

The consequence has the limit value.

The sequence also has the limit value for each zero sequence.

The above is not proof of the correctness of the definition, but shows essential cases for which applies.

goes against, even though and each time against.

miscellaneous

Wolframalpha defines zero to the power of zero as "undefined".