X² is X times X.
X¹ is X.
In X^Y Y is the factor that decides how many Xs are getting multiplied.
So why is X⁰ 1? It comes out of nowhere. Zero Xs are getting multiplied, so there’s no numbers being multiplied, meaning there’s nothing.
In: Mathematics
Yes, zero X’s are being multiplied. So we do nothing, correct. But nothing to what is the question, and what would do nothing. What is the baseline value? What is the baseline “identity” (the term used) for multiplication that does nothing?
Let start with addition. 2+3 = what? Well, 5 you say. Okay, but why? Why not 6? You can add 2 and then add 3 and end up at 6. It just means you started at one, the first natural number. Why is this wrong, or more importantly why is this not helpful way to define this operation? Well, because doing no additon would leave you with one. One does something to addition. Adding one to anything, well, changes the number. So what do we want to use as the starting point for addition? Zero, because adding zero to anything does nothing. So we start at zero, and doing no additon leaves us at zero. The identity value for addition is zero.
Now, you’re assuming zero should hold for multiplication, and hence exponentiation as well. Zero is nothing after all? Well, no, not for the multiplication operation. Zero does a lot, it turns everything to zero. If our base, our start, for multiplication was zero, every operation would be zero. The start point for multiplication, the value that does nothing, the identity, is one. X^0, doing no multiplication, is one. The identity value for multiplication is one. Not zero.
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