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I have been a witness to many a discussion about numbers to the power of zero, but I have never really been sold on any claims or explanations. This is a three part question, the parts are as follows...

1) Why does $n^{0}=1$ when $n\neq 0$? How does that get defined?

2) What is $0^{0}$? Is it undefined? If so, why does it not equal 1?

3) What is the equation that defines exponents? I can easily write a small program to do it (see below), but what about in equation format?

I just want a little discussion about numbers to the power of zero, for some clarification.

Code for Exponents: (pseudo-code/Ruby)

def int find_exp (int x, int n){
    int total = 1;
    return total;
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Actually, $0^0=1$, too. – Thomas Andrews Nov 11 '12 at 19:22
@ThomasAndrews $0^0$ is an indeterminate form. As the limit of $x^x$ from the right, yes it limits to $1$. But as the limit of $0^x$, it tends toward 0. – Shaun Ault Nov 11 '12 at 19:25
Being an indeterminate form does not imply that something is undefined. $0^0$ is defined and equal to $1$. @ShaunAult – Thomas Andrews Nov 11 '12 at 19:26
Sure, you can define your code that way. On the other hand, there are also many other good reasons to define $n^0 = 1$. Notably, it is consistent with $1 = n^p / n^p = n^{p-p} = n^0$. – Shaun Ault Nov 11 '12 at 19:27
@ShaunAult, the value of the expression $0^0$ is not necessarily the limit of a different expression $x^y$ as $x$ and $y$ tend to zero. The notion of "NaN" is a red herring in any case; it is a value defined by the IEEE 754 floating-point standard, not a concept in mathematics. – Rahul Nov 11 '12 at 20:10
up vote 11 down vote accepted

It's basically just a matter of what you define the notation to mean. You can define things to mean whatever you want -- except that if you choose a definition that leads to different results than everyone else's definitions give, then you're responsible for any confusion brought about by your using a familiar notation to mean something nonstandard.

Most commonly we define $x^0$ to mean $1$ for any $x$. What you find in discussions elsewhere are argument that this is a useful definition, not arguments that it is correct. (Definitions are correct because we choose them, not for any other reason. That's why they are definitions).

Some people choose (for certain purposes) to explicitly refrain from defining $0^0$ to mean anything. That choice is (supposedly) useful because then the map $x,y\mapsto x^y$ is continuous in the entire subset of $\mathbb R\times\mathbb R$ it is defined on. But it's an equally valid choice to define $0^0$ to mean $1$ and then just remember that $x,y\mapsto x^y$ is not continuous at $(0,0)$.

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I agree with most of your answer, except the definition we choose is not $x^0$ in this case, it is another definition which implies the value of $x^0$. See – Wok Nov 12 '12 at 17:00
But when considering the lone expression $0^0$ without the context of the function $x^y$, I've always seen the choice to refrain from defining $0^0$ to be firstly about two competing basics of arithmetic: that usually, $x^0=1$, and usually $0^x=0$. These two things are at odds with $0^0$. – alex.jordan Nov 16 '12 at 20:16

To 1): We define the exponents of a nonzero integer $a$ such that they satisfy the relation $a^ba^c=a^{b+c}$ for ay integers $b,c$, with $a^1=a$. In order for exponents to be well defined, we thus need $a^0=1$.

To 2): It depends on how you define it. If you define it via the limits $\lim_{x\rightarrow 0} x^0$ or $\lim_{x\rightarrow 0} x^x$, then $0^0=1$. If you define it as $\lim_{x\rightarrow 0} 0^x$, then $0^0=0$.

To 3): Exponents are defined simply by $a^n=\underbrace{a\cdot a\cdot \,...\, \cdot a}_{n}$.

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For 1), so some mathematician once said that $a^{0} = 1$? For 2), Which is then correct? Or is it both? For 3), I can kinda wrap my head around that, but I have yet to come across that particular way of denoting a range... – fr00ty_l00ps Nov 11 '12 at 19:29
No, we never define $0^0=0$. Just because it is a limit doesn't mean we define it that way. It is either $1$ or undefined. – Thomas Andrews Nov 11 '12 at 19:33
1) is needed to make the notion of exponent well defined. For 2), there is no "correct" one. It is a matter of definition. – Espen Nielsen Nov 11 '12 at 19:35

It is for various reasons convenient to define $0^0$ as being equal to $1$. For one thing, consider the Binomial Theorem, or power series. It is useful to be able to write $$(1+x)^n =\sum_{k=0}^n \binom{n}{k}x^k,$$ or $$e^x=\sum_{k=0}^\infty \frac{x^k}{k!}.$$ In each of these equations, if we want the expression on the right to give the correct answer when $x=0$, we need to set $0^0=1$.

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Ah, that makes sense, thanks ^v^ – fr00ty_l00ps Nov 11 '12 at 20:02

From definition of division of powers with the same base we have that $$\frac{a^n}{a^m}=a^{n-m}$$ Assuming that $n=m$ from left side we get $$\frac{a^n}{a^n}=1$$ and from right side we get $$\frac{a^n}{a^n}=a^{n-n}=a^0$$ comparing the last two equations we have that $$a^0=1$$

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(1) For intuition, if $k\ge0$ is an integer, take $x^k$ to mean "$1$ multiplied $k$ times by $x$", and $x^{-k}$ with $(x\neq 0)$ to mean "$1$ divided $k$ times by $x$." For integers $n\geq 0$, we may define $n!$ as the number of distinct ways to line up $n$ distinct objects--the only way to line up $0$ objects is to not line up any objects.

(2) We often define $0^0$ to be $1$, which accords with the intuitive definition above--if we multiply $1$ by $0$ not at all, then we still just have $1$. If we don't use the intuitive definition, we can define $0^0$ as the limit of $x^x$ as $x$ approaches $0$ from above, which gives us $1$, as well. Now, some prefer not to define $0^0$ at all, because they'd like $y\mapsto 0^y$ to be continuous on its domain of definition, and not have a jump discontinuity at $y=0$.

(3) We can extend integer powers to rational powers as follows: We say $y=x^{\frac1m}$ for some integer $m>0$ if $x=y^m$. If $m$ is odd, there will be a unique solution $y$ to the equation $x=y^m$. If $m$ is even and $x<0$, there will be no real solution $y$; if $m$ is even and $x\ge0$, then there is at least one real solution $y$, and we will take $x^{\frac1m}$ to be the nonnegative solution. At that point, given integers $k,m$ with $m>0$ and $\frac k m$ in lowest terms, we define $x^{\frac k m}:=\left(x^{\frac1m}\right)^k$ for such $x$ as this is possible. Finally, we can extend from the rationals to the rest of the reals with continuity arguments.

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The invention of numbers was one of the biggest breakthroughs in the history of math. It marked the realization that this sack of pebbles $$\{ \blacktriangle\;\blacktriangle\;\blacktriangle\;\blacktriangle\;\blacktriangle \}$$ this string of knots $$-\bullet-\bullet-\bullet-\bullet-\bullet-$$ and this bone full of tally marks $$/\,/\,/\,/\,/$$ are all incarnations of a single thing, the abstract quantity five. That leap of abstraction has become so prosaic for us that it almost feels weird to do arithmetic by actually counting things. In some cases, though, it can be illuminating to go back to the basics—back to the days when we didn't have numbers, and we did all our arithmetic by counting things. Your question is one of those cases.

In what follows, I'll use a capital letter like $X$ to stand for a finite set of things, like a herd of goats or a pile of beads, and I'll use the symbol $|X|$ to stand for the number of things in the set.

Exponentiation is a tricky operation, as you've clearly noticed, so let's warm up with something simpler. If you have two piles of beads, $A$ and $B$, the simplest thing you can do with them is shove them together to make a bigger pile, which is often written $A \sqcup B$. You should easily be able to convince yourself that, on the level of numbers, $|A \sqcup B| = |A| + |B|$. In other words, the concrete operation of shoving two piles together corresponds to the abstract operation of adding two numbers. Addition of whole numbers is often defined in this way.

Here's a slightly tougher warm-up. If you have a bunch of shirts, $H$, and a bunch of skirts, $K$, you might wonder how many different outfits you can make by pairing a shirt with a skirt. The set of outfits is usually written $H \times K$. You should be able to convince yourself that $|H \times K| = |H| \cdot |K|$. In other words, the concrete operation of counting pairs corresponds to the abstract operation of multiplication. Multiplication of whole numbers is often defined in this way.

Now that we're warmed up, suppose you have a set of paints, $C$, and a bag of beads, $X$. You might wonder how many different ways there are to color each bead with one of the paints. The set of ways to color the beads is usually written $C^X$. If you try a few examples, you'll see that $|C^X| = |C|^{|X|}$. Exponentiation of whole numbers is often defined this way.

Finally, we can get to your question. Suppose you have a bunch of paints, but the bag of beads is empty. Is it possible for you to paint all the beads? Sure: you just don't do anything! In fact, not doing anything is the only way to paint all the beads in the bag, since there are no beads. So, when the set $C$ has a bunch of paints, but the bag $X$ is empty, $|C^X| = 1$. If you define exponentiation by counting colorings, that means $|C|^0 = 1$ for any positive number $|C|$.

To make matters worse, suppose you have no paints and no beads. Happily, you can still paint all the beads: once again, you just don't do anything. Like before, not doing anything is the only way to paint all the beads, so $|C^X| = 1$ even when both $C$ and $X$ are empty. If you define exponentiation by counting colorings, that means $0^0 = 1$.

On the other hand, suppose you don't have any paints, but you do have some beads. In this case, you can't paint all the beads, because you have no paints! There are just no ways to paint all the beads. In other words, when $C$ is empty but $X$ is not, $|C^X| = 0$. If you define exponentiation by counting colorings, that means $0^{|X|} = 0$ for any positive number $|X|$, just like you'd expect.

Here's a bonus. André Nicolas argued that $0^0$ should be $1$ in order to make the binomial theorem true. Even those weird-looking numbers $\binom{n}{k}$ can be defined using finite sets: if you have $N$ toys and $K$ kids, $\binom{N}{K}$ is the set of ways you can pick out enough toys to have one for each kid. (Note that you don't give each toy to a particular kid: you just want the numbers of kids and toys to be the same.) If you get out your set of paints $C$ and another set of paints $D$ and start painting various numbers of kids and handing out toys based on how many colors of kids there are, you should somehow be able to convince yourself that the binomial theorem is true, even when $C$ doesn't have any paints in it. That's why André Nicolas came up with the same rules for zero exponentials as we just did.

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Another approach...

It can be shown that there exists infinitely many, what I call "exponent-like functions" that can be defined on the set of natural numbers $N$. By an exponent-like function $f$ on $N$, I mean $f$ such that:

  1. $f: N\times N\to N$

  2. $f(x,0)=1$ for $x\ne 0$

  3. $f(x,y+1)=f(x,y)\cdot x$

For all $x_0\in N$, there exists a unique exponent-like function $f$ such that $f(0,0)=x_0$.

It can be shown that, except for the value of $f(0,0)$, all exponent-like functions, as defined here, are identical.

From each such exponent-like function, we can derive the usual Laws of Exponents for non-zero bases corresponding to:

  1. $x^{y+z}=x^y\cdot x^z$

  2. $(x^y)^z= x^{y\cdot z}$

  3. $(x\cdot y)^z=x^z\cdot y^z$

If we define $0^0=1$, then these Laws of Exponents are true for all bases, including $0$. It might then be argued that we must have $0^0=1$. But the same is true for $0^0=0$ (but no other values), and we are no further ahead! If we are to look at exponentiation on $N$ as simply repeated multiplication, then $0^0$ is inherently ambiguous. We can formally define exponentiation on $N$ as follows:

  1. $\forall x,y\in N: x^y\in N$
  2. $\forall x\in N: (x\ne 0 \implies x^0 = 1)$
  3. $\forall x,y\in N: x^{y+1}=x^y\cdot x$

Here, $0^0$ is a natural number, but no value has been assigned to it.

See formal proofs, etc. in "Oh, the ambiguity!" at my math blog.

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To define x^0, we just cannot use the definition of repeated factors in multiplication. You have to understand how the laws of exponentiation work. We can define x^0 to be: x^0=x^(n-n)=x^n/x^n. Now, let us assume that x^n=a. It would then be simplified as x^n/x^n=a/a=1. So that's why x^0=1 for any number x.

Now, you were asking what does 0^0 mean. Well, let us the example above: 0^0=0^(n-n)=0/0. Here is where it gets confusing. It is more likely to say that 0/0 equals either 0 or 1, but it turns out that 0/0 has infinitely many solutions. Therefore, it is indeterminate. Because we mathematicians want to define it as some exact value, which is not possible because there are many values, we just say that is undefined.

NOTE: 0^0 still follows the rule of x^0=1. So it is correct to say that x^0=1 for ANY value of x.

I hope this clarify all your doubts.

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Oh sorry, you probably don't know why 0/0 has infinitely many solutions. Well here's the thing: 0/0=x, where is x is allegadly undefined/unknown. Undo multiplication and you obtain the equivalent equation 0x=0. Every number (including complex numbers) make this equation true. So we can assume that 0/0 equal all numbers at the same time. This kind of values are called "indeterminate", meaning that there cannot be a specific value assigned for it. Call it 0^0=0/0 by the example shown above, 0^0 is just as indeterminate as 0/0. – Angel Nov 16 '13 at 23:12

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