How many zeros are there in zero

Freistetter formula world: How much is zero to the power of zero?

Actually, the confusing question just revolves around a simple exponentiation. So about repeated multiplication. If we want to do 2 to the power of 3, all we have to do is multiply the number 2 by itself three times. 5 to the power of 7 is nothing more than 5 multiplied by itself seven times. The calculation is not a problem, and you immediately get the results 8 and 78 125. Admittedly, in the second case I used the calculator. But if I want to calculate the following expression, it becomes difficult:

x = 00

This article is featured in Spectrum - The Week, 47/2020

You have to multiply the number 0 by yourself. And exactly zero times. How do you do that? My calculator obviously knows how to do it, because it immediately gives the result x = 1. But a closer look reveals the hidden depths of the question. Anyone who remembers the arithmetic rules of exponentiation may also know that with an exponent of zero, the result is always 1.

On the other hand, the result of an exponentiation is always zero if the base is zero. In our case, however, the base and exponent are zero. So is the result 0 or 1? It seems obvious, 00 = 1 to be calculated. After all, that's what the calculator did too. However, you have to take into account that the rule “No matter what,› to the power of zero ‹is always equal to 1”, strictly speaking, only applies if the base is not equal to zero. So it cannot be applied here.

Even with simple reshaping, the problem cannot be solved. You could try such a calculation, for example: 0x = 01 + x-1 = 01 · 0x-1 = 0 · 0x-1, and since every multiplication by zero gives the result 0, it must also be 00 be equal to 0. Only then one would have ignored that 0x-1 for x = 0 corresponds to the expression 1/0. And division by 0 is not allowed.

To date there is no clear answer

You have to use more complex methods if you want to get to the bottom of the matter. For example, by looking at what the expression xx happens when the number x gets smaller and smaller: it gets closer and closer to the number 1; exactly the same thing happens if you use ever larger values ​​from negative numbers and thus approach x = 0.