why is everything to the power of 0 1?
In: 3
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X^1 = X
X^2 = X times X
So you multiply it by itself each time you add to the power. Inversely, you divide it by itself each time you subtract from the power.
So X^0 = X/X = 1
Everything (I’ll use *n* to mean any normal number) to the power of 1 is *n*. *n* to the power of -1 is 1/*n*. Multiplying 2 exponents adds the number in them, so *n*^1 * *n*^-1 is the same as *n*^1-1 or *n*^0 which translates to *n* * 1/*n* = *n*/*n* = 1.
Why does it work? Largely because 1 is the crossover point between dividing and multiplying, which is what happens when you go from negative powers to positive powers. So the crossover point between the two will be related; negative and positive powers at 0, while dividing and multiplying numbers at 1.
The true reason is that humans invented math and they decided that’s how they wanted it to work!
But let me explain why they chose that by showing a pattern.
3^3 = 27
3^2 = 9
3^1 = 3
As we subtract 1 from the exponent, we divide the result by 3.
So what’s should 3^0 be? If we follow the pattern, 3^0 should be 1.
Another way to look at this is by thinking about mathematical “identities.” An identity in math is a special number that leaves anything it touches unchanged. When adding or subtracting, the identity is 0. If you add 0 or subtract 0, you don’t change anything. But for multiplying and dividing, the identity is 1.
Remember earlier we said that when we subtract from the exponent, we divide from the final result? Well now it makes sense. When we have the subtracting identity in the exponent, we should have the dividing identity in the final result. That’s why anything to the 0 power becomes 1. Although I should point out these are normally called the additive identity and multiplicative identity, even though they work for subtracting and dividing too.
When you add together no elements, you end up with 0, right? Well, what’s special about 0 with respect to addition? The answer is that 0 is the “do nothing” number with respect to addition, 0 + x = x for any number x. Another way to see this is if you want to come up with a procedure to add together a list of numbers. You start by setting your counter to 0 and then go through the list adding each number to the counter. If the list is empty then you just end up with 0.
Now the question of x^(0). We’re essentially asking “what happens if you take 0 copies of x, put them in a list, and multiply them together?” Well the procedure for multiplying a list of numbers is that you start with the “do nothing” number – which in the case of multiplication is 1, since 1*x = x for any number x. Then you multiply that number by each number in the list. If your list happens to be empty then you just get 1, which is why x^(0) = 1.
(This is true even for x=0, which is one reason why in most contexts where mathematicians choose to define 0^(0) then they define it to be 1.)
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