why X⁰ is equal to 1

Question

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

Answer 1

Because 1 is the neutral element of multiplication.

Let’s look at how multiplication is defined by addition first. Multiplication says how many additions are performed:

* 2X = X+X
* 1X = X
* 0X =

Now that’s weird too, right? What does it mean that the right side is “empty”? To avoid that. we can expand it by adding 0 to the right side without changing the result. The fact that adding 0 doesn’t change the value is called 0 being the neutral element of addition.

* 2X = 0+X+X 
* 1X = 0+X
* 0X = 0

Better, right?

Now we can do the same thing with powers and multiplication. However, we have to think what is the neutral element of multiplication? Can’t be 0 because multiplying something with zero definitely changes its value. It has to be 1. 1 is the only number you can multiply some other value with, that doesn’t change the value.

So, let’s do the same thing with powers but expand the right side by the neutral element of multiplication:

* X^2 = 1XX
* X^1 = 1X
* X^0 = 1

There we have it!