### Zero to the power of zero

Calculations usually give a correct result if you take 0^{0} ≝ 1, but you must pay attention, and sometimes you have to check the calculation carefully.

##### Explanation

You can multiply every number with 0. The result always remains the same, because

0 × 7 = 0

0 × 6958 = 0

You can also multiply 0 with itself

For every power *n* ≠ 0 applies 0^{n} = 0. Why don't we know this for it not for *n* = 0 ? Let us have a look at it. We know that

what you can also write as

The exponent may become very small, the result remains zero. Therefore we now take Δ*x* and write very cheeky

because infinitely small is not zero. *Is this correct or not?* Well, this shows that there is a difference between infinitely small and zero. And therefore 0^{0} ≝ 1 must be determined *by definition*, because you cannot calculate it.

##### Additional information

The American computer scientist Donald Knuth contends strongly: **0 ^{0}**

**has to be 1**.