eli5: why is x⁰ = 1 instead of non-existent?

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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.)

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.

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.

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.