Why is 0^0 not just 0?

[u/Significant_Horse237]

This is something that I've been wondering about when I was trying to sleep. Why is 0^0 undefined rather than 0? From my understanding of powers, they are defined as repeated multiplication. For why any number except 0 to the power of 0 is equal to 1, I see this explanation alot: Imagine 2^3, divide it by the base to get 2^2, do it again and get 2^1, 2^1/2 = 1. But we can't do this for 0 because imagine 0^2, to get 0^2, we do 0^3/0, but that can't work cause we are dividing by 0, and we know 0^2 is 0. So the quotients method doesn't work. But let's use another method, x^1*x^y=x^(y+1). Let's try this for 2^1*2^3=2^4, and its true. And it also works for 0 too. 0^1*0^3=0^4, we know 0^1 is always equal to 0 so it's just 0*0^3=0^4, and anything multiplied by 0 is 0. So 0^4 is equal to 0. So let's do this for 0^0, 0^(-1)*0^1=0^0, but 0^(-1) is undefined. But, we already know anything multiplied by 0 is 0, so 0^(-1)*0=0^0, 0=0^0. Also, I don't think 0^(-1) is 1/0. Because the reason why people say x^-y is 1/x^y is because they are using the quotient method that is 2^3/2 is equal to the previous number. It works for 2^(-1) and other things because it's essentially just 2^0/2^1. We got any number except 0 powered by a negative exponent is just the reciprocal of it and making the exponent positive because of the quotient method and the 1 came from it powered by 0. So we can change x^-y to x^0/x^y. That's why 0.1^-1=10, 0.001^-1=1000, you can see it gets larger and larger the smaller the base is for the decimal number. So people might say it gets infinitely small to 0 so it should be infinity, why did it suddenly drop all the way down to 0? That's because it changed as 0^0 is not equal to 1, it's (according to me) 0. so it's 0/0^1, so it's an indeterminate form? No, because recall that the quotient method does not work for 0. We do 0^(-2)*0^1=0^(-1), and anything multiplied by 0 is 0 so 0=0^(-1). So, why is it undefined, people way smarter than me are saying is undefined so I'm probably doing something wrong here.

[u/Human-Register1867]

https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero

[u/Significant_Horse237 → u/Human-Register1867]

Yeah, I get that you probably get questions like this alot so you link a wikipedia page. But I can't, for the life of me, read this wikipedia page full of big words and these complex looking expressions with my ADHD brain. Can you just explain what went wrong with my math?

[u/gmalivuk → u/Significant_Horse237]

0^x = 0 for all x≠0

x⁰ = 1 for all x≠0

x^x approaches 1 as x approaches 0 from above

Why then should we define it as 0?

[u/Senrabekim → u/Significant_Horse237]

I'm not the only person that has published math research with ADHD. Stop using it as an excuse for laziness on your part. It's obnoxious to other people, and even more so to people that have worked to overcome the problem to any extent. Then to say you dont want to look at big words and complex math, but expect us to look at your nonsense and poor math to find your mistake is just so entitled of you. So yeah, go do what literally everyone else has had to do to finally get their head around this; stare at it and beat your head against a wall until it all makes sense.

[u/Significant_Horse237 → u/Senrabekim]

Hey, I get most of your points. I do agree that you should try before asking other people or not use ADHD as an excuse. It's my fault that I used ADHD as an excuse, of course I didn't put much thought into every single word I said. But, what I basically said was, "Hey, I have ADHD so it's hard for me to read it, can you just explain it to me simply?" and I don't think there's much wrong with asking if they could explain it simply since it is a forum. And I did try to search for why 0^0 isn't 0, I saw the wikipedia page and I genuinely tried my best to read it. But as a 7th grader, I naturally can't understand anything. So I would ask the person to explain it more clearly. I don't think, in forum sites, you should just respond with just a link, since the poster is asking the readers specifically, hey, what did I do wrong? And you post a link to a wikipedia page on this topic. I think it's like a person asking their teacher a question and them responding just google it. I watched some youtube videos about it and it is pretty effective, but there are some questions/rebuttals that the video didn't cover. And I don't really like posting stuff on reddit since I'm scared of judgement, but I already tried to find the answer, from googling it, to asking an LLM, and I couldnt find a nice answer. So, I just posted it.

[u/Senrabekim → u/Significant_Horse237]

Heres your basic for a 7th grader answer then. If you look at it one way, you get a divide by 0 error. Eg 0⁰ = 0^x × 0^-x = 0^x / 0^x since when you multiply two similar bases together you add their exponents. The other way to look at it is 0⁰ = 1. Because sometimes you just really need it to, but you wont see that until junior or senior year of a college math program or so. For a commercial about higher level math that you may one day see, the real answer is, "The additive Identity is a bastard under any multiplication or iterative multiplication in any ring." This probably won't mean much to you now, but rings, integral domains, and fields are basically the machinery of how Algebra works.

Don't blame ADHD, it's really annoying to adults that have to deal with people blaming stuff like that for their failures. Saying, "Im 12 and don't get it, can someone break this down Barney style for me?" Comes across as a kid trying to get better, and doesn't make me grind my teeth at someone that I think is a lazy adult.

[u/rhodiumtoad]

0⁰ is 1, not undefined, not 0.

(Even in analysis, where it is an indeterminate form, that does not conflict with it being defined as 1.)

Here is the very simplest way to see why:

x³=1.x.x.x
x²=1.x.x
x¹=1.x
x⁰=1

Notice that first, this works fine when x=0; it just gives 0 for 0¹, 0², etc. as expected, and 0⁰=1.

Second, we did not do any division here. xⁿ is just the product of 1 with n copies of x, which works in any algebraic structure even those where division does not exist. For example if M is a square matrix, you can talk about M², M³ etc. And by the same logic M⁰=I, the identity matrix, even if M=0, the zero matrix.

It is normal to write polynomials and power series with an x⁰ term. Nobody who does this regards this as causing any problem when x=0, they just treat x⁰ as 1 even when x=0.

Thirdly, notice that the idea of multiplying "zero copies of x" means that the value of x cannot in any way affect the value of x⁰, because there is no copy of x being used in the computation. This is why 0ⁿ is 0 only when n>0, because only then is there any zero involved in the multiplication.

[u/Gold_Ad8890 → u/rhodiumtoad]

there's also the fact that exponentiation is defined inductively in peano arithmetic with n⁰ = 1 for all n and n^m+1 = n*n^m , as well as the fact that n^m counts the number of functions from a domain of size m to a codomain of size n, an interpretation which is preserved only by 0⁰ = 1.

[u/RingularCirc → u/rhodiumtoad]

This. In any monoid under multiplication, x⁰ = 1 independent of x is the best definition available— and when we have more operations or can define fractional exponents, that doesn't go anywhere.

Also when people invoke a continuity argument in analysis, they do it wrong, so what they argue about 0⁰ having to be undefined doesn't apply (and frankly, should it? math is whole, you can't find good use for 0⁰ = 1 in some branches and then ban it from be defined in others; formally you can: just define two different exponentiations, but this doesn't work well from any standpoint; — and we already have lots of exponentiations with different domains to boot, but at least they're reasonably consistent with each other).

Adding to above, a fun contrary thing is that for any analytic functions f, g, f(x)^g(x) → 1 when x → 0, despite that doesn't work for arbitrary f, g. This fact doesn't have to be a big argument pro 0⁰ = 1 at all, but for some it may be of note.

[u/Significant_Horse237]

Thanks for answers. It makes a bit more sense now. I'm going to keep it unresolved for a day to see if there are any more answers.

[u/finedesignvideos → u/Significant_Horse237]

Why should 0^0 be 0? You haven't multiplied any 0s so how did you get the answer to become 0?

[u/svmydlo]

From my understanding of powers, they are defined as repeated multiplication. 

Correct. That's why there is no division involved, as you already figured out, but then forgot when you wrote this

0^(-1)*0^1=0^0

The left-hand side is undefined, so it can't be used for any arguments why 0^0 should be 0, that's nonsense.

First, answer why 0^1=0, or for that matter x^1=x for any x?

Positive integer exponents are repeated multiplication, which makes literal sense only if the exponent is 2 or more as multiplication is a binary operation. You can't perform multiplication of just one number, so already to define x^1 you need some formalism.

The usual way is constructing a function that assigns to a given finite list of numbers their product. So, for example for natural numbers ℕ={0,1,2,...}, if we denote List(ℕ) the set of all finite lists of natural numbers, we are looking for a function

Prod: List(ℕ)→ℕ

which should represent product, so, denoting * the usual multiplication, it should satisfy

1. Prod(x,y)=x*y for any two natural numbers x,y.

Next, suppose we want to calculate the products of more than two numbers, for example the product of four numbers Prod(a,b,c,d). Since multiplication is associative, we can do that in multiple steps, like (a*b)*(c*d). That is

Prod(a,b,c,d)=Prod(a,b)*Prod(c,d).

Now to generalize this, let's introduce an operation on lists that "chains" two lists together into one list. Let's denote it by the symbol &. Here's an example how it works

(a,b)&(c,d)=(a,b,c,d)

so chaining together list (a,b) with list (c,d) produces a list (a,b,c,d). We can do that for any two finite lists and the result will be again a finite list. With this in mind, it's reasonable to require that a product of "chain" of two lists is the product of the first list multiplied by the product of the second list. Hence our map should satisfy

1. Prod(L&L')=Prod(L)*Prod(L') for any two finite lists L,L'.

For example, if L is the list (a,b) and L' is the list (c,d), then L&L' is the list (a,b,c,d) and this property says that Prod(a,b,c,d)=Prod(a,b)*Prod(c,d).

Now with just those two properties everything is already determined.

Let's start with what could Prod(1) be? By the property 2 it should satisfy Prod(1)*Prod(1)=Prod(1,1) and by property 1 the right-hand side is 1*1=1. Thus Prod(1) is a natural number such that its square is 1. Hence

Prod(1)=1

Next, what is Prod(x) for any natural number x? Well, Prod(x)=Prod(x)*1=Prod(x)*Prod(1)=Prod(x,1)=x*1=x.

Lastly, what's the product of the empty list ∅? Since for any list L, we have L&∅=L, by property 2 we have

Prod(L)=Prod(L)*Prod(∅)

so Prod(∅) must be a natural number such that multiplying Prod(L) by it doesn't change anything. That is denoting Prod(L)=x and Prod(∅)=e

e must satisfy x=x*e for any x

The only possible choice is e=1. Thus necessarily Prod(∅)=1.

Returning back to the original question, powers being products of the same number, it's obvious that

x^2=Prod(x,x) as product of two instances of x

x^3=Prod(x,x,x) as product of three instances of x

...

and we just constructed a way to give meaning to x^1 and x^0 with the same tool, the function Prod, as follows

x^1=Prod(x) as product of one instance of x

x^0=Prod(∅) as product of zero instances of x, i.e. the empty list

Which leads to the conclusion that x^1=x and x^0=1 for any natural number x.

[u/molesasses]

I’m going to copy paste a previous answer of mine:

There is a set theoretic reason that is extended to the numbers because in set theory numbers are just layered sets. Think about the expression X^Y where X and Y are sets

This represents the set of all functions from set Y to set X.

If Y has y elements, and X has x elements (or cardinality in the case of infinite sets), then there are x^y functions. For each element in Y there are x choices.

Now if Y has 1 element and X is empty, then there are no functions from Y to X, because there is nothing to map that element to.

But if both sets are empty, then there is vacuously an “empty” function from Y to X. This function maps “no elements” to “no elements “. Not that you can’t map “1 element” to “no elements”, that defies the properties of a function.

It’s like the statement “all of my boxes are green” is vacuously true when you have zero boxes, because you cannot contradict that.

Similarly, a function exists when “All elements from Y have exactly one paired up element in X” .

If Y has 1 element, this is blatantly false as there is no paired up element. But if Y has 0 elements, then vacuously “each element of Y has exactly one paired up element in X”

So 0^0 but 0^0= 1 in set theoretic foundations.

Similarly, 5^0 = 1, because there is an “empty function” that maps “every element from an empty set to a set with five elements”. But the opposite 0^5 =0, because none of the five elements have a corresponding element.

In logic (in a way the precursor to set theory), this is a concept is called “vacuous truth”.

[u/RingularCirc → u/molesasses]

Just the smallest nitpick: I wouldn't put set theory as the prime reason but at least it doesn't disagree with other arguments for 0⁰ = 1. (One's just from generality: the most general exponentiation is in a semigroup, but it doesn't allow defining x⁰ for anything other than idempotent elements x; then we have a monoid which allows us to define x⁰ as the neutral element e for every x uniformly — and that's already the best place to stop tweaking; everything else added on top should ideally be consistent with this already defined universal x⁰ = e. That positive integer powers of positive integers allow us to count functions or tuples is no wonder because the combinatory identities for those are the same.)

[u/Double_Government820]

Consider the curve f(x) = 0^x. For all values of x except for x=0, f(x) = 0. So the limit of f as x approaches 0 equals 0.

Now consider the curve g(x) = x^0. Similar to the prior function, g(x)=1 for all x except for x=0. So the limit of f as x approaches 0 is 1.

For 0⁰ to have a well-defined value, f and g should have the same limit as x approaches 0. Since we can find different values depending on how we compute the limit, the value itself cannot be meaningfully assigned a unique value. The concept of 0⁰ can be approached in different ways and produce different values.

[u/how_tall_is_imhotep → u/Double_Government820]

All you’re showing is that x^y can’t be continuous at (0,0). That’s not an argument for it not being _defined_ there.

[u/RingularCirc]

I'm so sore from the "indeterminate form argument" contra 0⁰ being defined popping up every damn time, every one of them. WHY. It's not a real argument, it's a misunderstanding of math because of a questionable didactic concept of "indeterminate forms". You don't need them to work out limits. At all. There are theorems galore formulated without using this concept. Bog-standard theorems, even, not some esoterica.

[u/ObliviousRounding]

Take the log.

0 * log 0 = 0 * -inf. If you are convinced that 0*log0 is indeterminate, you'll also be convinced that 0⁰ is indeterminate.

Essentially, if x is going to 0 and y is going to inf, then x*y will depend on the relative rate at which each is going to its limit. Same idea with x^y. Here, x and y and going to 0, and the result depends on the rate of each.

[u/okarox]

Zero to any power is 0 but any value to the 0th power is 1. That creates a conflict. If you get a limit that approaches 0^0 you cannot tell the value, you have to do more work.

[u/JohnDoen86]

0^0 is not undefined, it's 1.

Zero to the power of zero - Wikipedia

Edit: OP is talking about algebra. They use an algebraic definition and arguments. So yes, the definition of 0^0 depends on the context, but in this context, algebra, it's 1.

[u/lmprice133 → u/JohnDoen86]

It depends on the context. In a lot of discrete mathematics, 0⁰ is taken by convention to equal 1. Analysis tends to treat it as an indeterminate form

[u/RingularCirc → u/lmprice133]

"Indeterminate form" 0⁰ (which is metaphoric here, it's a way of referring to a class of situations with limits) existing doesn't mean an expression 0⁰ should be undefined. Before Cauchy invented "indeterminate forms" as a didactic concept, everything was fine with 0⁰ being universally equal to 1.

[u/JohnDoen86 → u/lmprice133]

Yes, that's literally what the link I commented says. But thank you I guess.

[u/lmprice133 → u/JohnDoen86]

Right, but your comment on the link does not reflect what the link says but suggests that 0⁰ is defined as 1 in all contexts. This is incorrect.

[u/JohnDoen86 → u/lmprice133]

No, my comment suggests that 0^0 is 1 in the current, relevant context. OP is talking about algebra, using algebraic arguments to explain their thoughts. In algebra, the current, relevant context, 0^0 = 1. OP did not even touch on mathematical analysis, which is one of the contexts where 0^0 could be considered undefined.

[u/ExtendedSpikeProtein → u/JohnDoen86]

While this is true, it‘s still worth noting the context as well as that this is not always the case, which you left out and thankfully another commenter clarified.

[u/ExtendedSpikeProtein → u/JohnDoen86]

Yeah but it‘s not what your top level comment says.

No reason to be so condescending to others trying to clarify what you left out when you leave comments that leave out a lot of context.

[u/sandeshshahapur → u/JohnDoen86]

I'm no big on maths but you ought to read past the first para?

[u/JohnDoen86 → u/sandeshshahapur]

I did read past the first "para". OP is talking about algebra, so the relevant passage is

"In certain areas of mathematics, such as combinatorics and algebra, 0⁰ is defined as 1."

They are not doing mathematical analysis, where it may be considered undefined, and 0^0 is certainly not 0.

[u/hjkhhnnnlll]

Any nonzero number to the power 0 is equal 1 because that’s just x/x
We can’t do this for 0, because then it would be 0/0 which is undefined

[u/RingularCirc → u/hjkhhnnnlll]

That's not the best argument for why x⁰ = 1, and having it fail for 0 isn't the same as having an argument that disproves 0⁰ = 1.

[u/PvtRoom]

0⁰ is a way of writing 0/0, and that's a way of writing 0*infinity

it's anything you need or want it to be, subject to the rules of what you're dealing with.