It kinda doesn’t make sense.
x¹= x
x² = x*x
x³= x*x*x
etc…
and even with negative numbers you’re still multiplying the number by itself
like (x)-² = 1/x² = 1/(x*x)
In: 1797
Well, a simple reason is that we want x^a times x^b to be x^(a+b).
So: x^1 is x. x^(-1) is 1/x. What is x times 1/x? It’s 1. But that’s also x^1 times x^(-1) = x^(1 + -1) = x^(0).
A somewhat more formal approach is to think of x^0 as an empty product. You’re not multiplying anything, which is the same as multiplying by 1. Or to extend your logic from the OP:
> It kinda doesn’t make sense
> x*1 = x
> x*2 = x + x
> x*3 = x + x + x
So in this case, x*0 is the empty sum, which is the same as not adding anything, which is the same as adding 0. (And of course, x * 0 is in fact 0.)
Let’s look at powers of 2:
2¹=2
2²=4
2³=8
2⁴=16
2⁵=32
So to get the next power of 2, you just multiply by 2 (2×2=4, 4×2=8, 8×2=16, …). Which means to get the previous power, you need to *divide* by 2. So 2⁰ should be 2¹/2=2/2=1.
You get square of x by multiplying x by x. The cube of x by multiplying the square by x and so in. So what would you need to multiply x with, to get x
Think of it as
x^n =x^(n-1) *x
So
x^(n-1)=(x^n) /x
Now replace n by 1.
Think of it like its describing a space. X is a line. X^2 is a 2d grid. X^3 is a 3d cube. X^4 is a hypercube. You can make each x the length of an axis describing a space.
Now go backwards. X^4 describes a hypercube (tesseract). X^3 describes a cube. X^2 a plane. X^1 a line. So x^0? It must be a point. Or a single unit of space.
Both can happen, it depends on what kind of thing is x and in what context. Sometimes x^0 =1 and sometimes x^0 is undefined.
There are some context where x^0 doesn’t “make sense”, and in that case it might be better to leave x^0 undefined.
But when it does make sense, why not define it? The more possible input the operation can accept, the more manipulation you can do. There are generally no harm in defining the operation to work on extra input. The only possible downside is that if the extra input is useful, then it’s not worth the effort of defining it.
When x is a number (in many sense of “number”) and 0 is supposed to be a natural number 0 or an integer 0, then x^0 =1. Why? Think about sum. If x*0 is x add to itself 0 times, and you know x*0 =0, right? To perform a sum, you start with 0, and keep adding, so if there are nothing to add, you get 0 back. Same here. x^0 mean x times itself 0. To perform a product, you start with 1, and keep multiplying. If you have nothing to multiply, you get back 1.
This convention is called “empty product equal 1” convention. This is applicable to all forms of product. If someone say “what’s the product of all prime numbers strictly less than n” and n happen to be 2, then the answer is 1, because there are no prime numbers strictly less than 2.
Since others have already explained the WHY, I thought I might give a different explanation as to HOW X⁰=1 makes sense.
Try multiplying 2², then 2¹, 2½, 2⅓, 2¼, you get the idea. You will begin to notice the closer you get to 2⁰, the closer the result equals 1.
Imagine, instead of it being just X, X^(1) is actually X/1. If you multiply by any value equivalent to one, the original value remains unchanged, so multiplying by 1/1 is perfectly fine, the main purpose is that I want to *express* positive powers of X by X^(y) /1. If you have one x, it’s just one on the top. X^(2) same just with two, etc. Why I’m doing this will be clear momentarily.
So with negative powers, X^(-y) for instance, it’s instead 1/X^(y). For every negative power you go, there’s another x multiplied by the on the *bottom* this time. So for positive powers of X they go on the top, and for negative they go on the bottom.
What happens if you have no X’s in either direction, positive or negative? Nothing on the top, nothing on the bottom, you get 1!
This is a very good video explanation I found a while ago.
I’ve had it explained this way, maybe it’ll help. You are correct that x^2 = x*x, but it is also 1*x*x, since they’re the same thing.
X^1 = 1*x, and so continuing the pattern we get
X^0 = 1
Think of powers as how many times you multiply 1 by the number ie 2^(0) = 1 : 2^(1) = 1 * 2 : 2^(2) = 1 * 2 * 2
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