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Could someone provide me with a good explanation of why $0^0=1$?
My train of thought: $$x>0\\ 0^x=0^{x-0}=\frac{0^x}{0^0}$$ so $$0^0=\frac{0^x}{0^x}=\,?$$ Possible answers:
PS. I've read the explanation on mathforum.org, but it isn't clear to me.
In general, there is no good answer as to what $0^0$ "should" be, so it is usually left undefined.
Basically, if you consider $x^y$ as a function of two variables, then there is no limit as $(x,y)\to(0,0)$ (with $x\geq 0$): if you approach along the line $y=0$, then you get $\lim\limits_{x\to 0^+} x^0 = \lim\limits_{x\to 0^+} 1 = 1$; so perhaps we should define $0^0=1$? Well, the problem is that if you approach along the line $x=0$, then you get $\lim\limits_{y\to 0^+}0^y = \lim\limits_{y\to 0^+} 0 = 0$. So should we define it $0^0=0$?
Well, if you approach along other curves, you'll get other answers. Since $x^y = e^{y\ln(x)}$, if you approach along the curve $y=\frac{1}{\ln(x)}$, then you'll get a limit of $e$; if you approach along the curve $y=\frac{\ln(7)}{\ln(x)}$, then you get a limit of $7$. And so on. There is just no good answer from the analytic point of view. So, for calculus and algebra, we just don't want to give it any value, we just declare it undefined.
However, from a set-theory point of view, there actually is one and only one sensible answer to what $0^0$ should be! In set theory, $A^B$ is the set of all functions from $B$ to $A$; and when $A$ and $B$ denote "size" (cardinalities), then the "$A^B$" is defined to be the size of the set of all functions from $A$ to $B$. In this context, $0$ is the empty set, so $0^0$ is the collection of all functions from the empty set to the empty set. And, as it turns out, there is one (and only one) function from the empty set to the empty set: the empty function. So the set $0^0$ has one and only one element, and therefore we must define $0^0$ as $1$. So if we are talking about cardinal exponentiation, then the only possible definition is $0^0=1$, and we define it that way, period.
Added 2: the same holds in Discrete Mathematics, when we are mostly interested in "counting" things. In Discrete Mathematics, $n^m$ represents the number of ways in which you can make $m$ selections out of $n$ possibilities, when repetitions are allowed and the order matters. (This is really the same thing as "maps from $\{1,2,\ldots,m\}$ to $\\{1,2,\ldots,n\\}$" when interpreted appropriately, so it is again the same thing as in set theory).
So what should $0^0$ be? It should be the number of ways in which you can make no selections when you have no things to choose from. Well, there is exactly one way of doing that: just sit and do nothing! So we make $0^0$ equal to $1$, because that is the correct number of ways in which we can do the thing that $0^0$ represents. (This, as opposed to $0^1$, say, where you are required to make $1$ choice with nothing to choose from; in that case, you cannot do it, so the answer is that $0^1=0$).
Your "train of thoughts" don't really work: If $x\neq 0$, then $0^x$ means "the number of ways to make $x$ choices from $0$ possibilities". This number is $0$. So for any number $k$, you have $k\cdot 0^x = 0 = 0^x$, hence you cannot say that the equation $0^0\cdot 0^x = 0^x$ suggests that $0^0$ "should" be $1$. The second argument also doesn't work because you cannot divide by $0$, which is what you get with $0^x$ when $x\neq 0$. So it really comes down to what you want $a^b$ to mean, and in discrete mathematics, when $a$ and $b$ are nonnegative integers, it's a count: it's the number of distinct ways in which you can do a certain thing (described above), and that leads necessarily to the definition that makes $0^0$ equal to $1$: because $1$ is the number of ways of making no selections from no choices.
Coda. In the end, it is a matter of definition and utility. In Calculus and algebra, there is no reasonable definition (the closest you can come up with is trying to justify it via the binomial theorem or via power series, which I personally think is a bit weak), and it is far more useful to leave it undefined or indeterminate, since otherwise it would lead to all sorts of exceptions when dealing with the limit laws. In set theory, in discrete mathematics, etc., the definition $0^0=1$ is both useful and natural, so we define it that way in that context. For other contexts (such as the one mentioned in mathforum, when you are dealing exclusively with analytic functions where the problems with limits do not arise) there may be both natural and useful definitions.
We basically define it (or fail to define it) in whichever way it is most useful and natural to do so for the context in question. For Discrete Mathematics, there is no question what that "useful and natural" way should be, so we define it that way.
This is merely a definition, and can't be proved via standard algebra. However, two examples of places where it is convenient to assume this:
1) The binomial formula: $(x+y)^n=\sum_{k=0}^n {n\choose k}x^ky^{n-k}$. When you set $y=0$ (or $x=0$) you'll get a term of $0^0$ in the sum, which should be equal to 1 for the formula to work.
2) If $A,B$ are finite sets, then the set of all functions from $B$ to $A$, denoted $A^B$, is of cardinality $|A|^{|B|}$. When both $A$ and $B$ are the empty sets, there is still one function from $B$ to $A$, namely the empty function (a function is a collection of pairs satisfying some conditions; an empty collection is a legal function if the domain $B$ is empty).
$0^{0}$ is just one instance of an empty product, which means it is the multiplicative identity 1.
I'm surprised that no one has mentioned the IEEE standard for $0^0$. Many computer programs will give $0^0=1$ because of this. This isn't a mathematical answer per se, but it's worth pointing out because of the increasingly computational nature of modern mathematics, so that one doesn't run afoul of anything.
The use of positive integer exponents appears in arithmetic as a shorthand notation for repeated multiplication. The notation is then extended in algebra to the case of zero exponent. The justification for such an extension is algebraic. Furthermore, in abstract algebra, if $G$ is a multiplicative monoid with identity $e$, and $x$ is an element of $G$, then $x^0$ is defined to be $e$. Now, the set of real numbers with multiplication is precisely such a monoid with $e=1$. Therefore, in the most abstract algebraic setting, $0^0=1$.
Continuity of $x^y$ is irrelevant. While there are theorems that state that if $x_n \to x$ and $y_n \to y,$ then $(x_n + y_n) \to x+y$ and $(x_n)(y_n) \to xy$, there is no corresponding theorem that states that $(x_n)^{(y_n)} \to x^y$. I don't know why people keep beating this straw man to conclude that $0^0$ can't or shouldn't be defined.
Maybe it's a good idea to put the problem into a broader perspective.
The minimum we need to define a power is a multiplicative semigroup, which basically means we have a set with an associative operation which we write as multiplication. If $S$ is the semigroup, the power function is defined as $$S\times \mathbb Z^+ \to S, (x,n) \mapsto x^n = \begin{cases} x & n=1\\ x x^{n-1} & n>1 \end{cases}$$ This power function has the fundamental properties $$x^mx^n = x^{m+n},\quad (x^m)^n = x^{mn}\tag{*}$$ Note that up to now, we have used absolutely nothing about the elements of $S$ except for the fact that we can multiply them.
Now it is a natural question whether we can extend that definition from the positive integers to the non-negative integers, in other words, whether we can define $x^0$. Of course we would want to define it in a way that the power laws (*) still hold. This especially implies the following relations: $$(x^0)^2 = x^0 \tag{I}$$ (that is, $x^0$ is idempotent) and $$x^0 x = x x^0 = x.\tag{II}$$ Note that this does not imply that for any $x\ne y$, $x^0=y^0$.
Now for a given $S$ and a given $x\in S$, there are three possible cases:
There exist more than one element $z_i\in S$ fulfilling those equations. This is the case for $0^0$ because both $0$ and $1$ fulfil the equations. In that case, you have several choices:
Now one particularly interesting case is if you have a neutral element, that is, an element $e\in S$ so that $ex=xe=x$ for all $x\in S$. A semigroup with such an element is called a monoid.
It is easy to check that in this case, $e$ fulfils both (I) and (II) for any element $x\in S$. Therefore you always get a valid power function by defining $x^0=e$. Indeed, by doing so, you get the general form $$x^n = \begin{cases}e & n=0\\ x x^{n-1} & n>0\end{cases}$$ which means that in a monoid, $x^0=e$ clearly is a distinguished, and therefore preferable choice of the power function, even for $x$ where other choices would be possible. Note that the real numbers form a monoid under multiplication, with $e=1$. Therefore this is a first hint that $0^0=1$ is a good definition.
The definition $0^0=1$ turns out to be useful also in other areas, for example when considering the linear structure (that is, the distributive law with addition), as it makes sure that e.g. the binomial formula $$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^k b^{n-k}$$ also holds if any of $a$, $b$ or $a-b$ is $0$.
There's however one structure that doesn't favour $0^0=1$, and that is continuity: $x^y$ is discontinuous at $(0,0)$, and every value can be obtained as limit of a suitably chosen sequence of arguments approaching the origin. Therefore from the point of continuity, $0^0$ can be considered unspecified. However, since $x^y$ is discontinuous at the origin whether or not one defines it there, and independent of the value one chooses, that is not really an argument against choosing $0^0=1$, but more an argument against using that definition blindly.
I'm not aware of a context where the definition $0^0=0$ would be more useful, but I wouldn't want to exclude that it exists. Any other value of $0^0$ would violate condition (I) above, and therefore likely not be useful at all.
So in summary, the definition $0^0=1$ is the most useful in most situations, and not harmful in situations where one would otherwise leave $0^0$ unspecified, and if any context exists where another definition would be more useful, it's arather unusual context. Therefore the definition $0^0=1$ is the most reasonable one.
It's pretty straight forward to show that multiplying something by $x$ zero times leaves the number unchanged, regardless of the value of $x$, and thus $x^0$ is the identity element for all $x$, and thus equal to one.
For the same reason, the sum of any empty list is zero, and the product is one. This is when a product or sum of an empty list is applied to a number, it leaves it unchanged. Thus if the product $\Pi()$ = 1, then we immediately see why $0! = 0^0 = 1$.
Without this property, one could prove that $2=3$, by the ruse that there are zero zeros in the product on the left (zero is after all, a legitimate count), and thus $2*0^0$, and since $0^0$ as indeterminate, could be 1.5, and thus $2=3$. I think not.
The approaches to $0^0$ by looking at $x^y$ from different directions, fails to realise that for even lines close to $x=0$, the line sharply sweeps up to 1 as it approaches $y=0$, and that the case for $x=0$, it may just be a case of not seeing it sweep up. On the other hand, looking from the other side, even in a diagonal line (ie $(ax)^x$), all do rapidly rise to 1, as x approaches 0. It's only when one approaches it from $0^x$ that you can't see it rising. So the evidence from the graph of $x^y$ is that $0^0$ is definitely 1, except when approached from $y=0$, when it appears to be zero.
Knuth's answer is at least as good as any answer you're going to get here: http://arxiv.org/pdf/math/9205211v1.pdf See pp. 4-6, starting at the bottom of p. 4.
"Everybody knows" that $$ e^z = \sum_{n=0}^\infty \frac{z^n}{n!}, $$ and when $z=0$ then the first term is $\dfrac{0^0}{0!}$, so of course $0^0$ is $1$ since it's an empty product.
But it's also an indeterminate form because $\displaystyle\lim_{x\to a} f(x)^{g(x)}$ can be any positive number, or $0$ or $\infty$, depending on which functions $f$ and $g$ are, if $f(x)$ and $g(x)$ both approach $0$ as $x\to a$.
Another reason to define $0^0=1$ comes from probability, in particular Bernoulli trials.
These are independent repeated trials of an experiment, whose outcome is either positive or negative. So let $p$ be the probability of a success for each trial. Then the probability of exactly $k$ successes out of $n$ trials is $$p_k = {n \choose k}p^k(1-p)^{n-k}.\tag{B}$$ Now, suppose $p=1$. $(B)$ yields $p_n=0^0$. However, we already know that each trial will certainly occur, resulting in $p_k=0$ for all $k<n$ and $p_n=1$, whence $$0^0=1.$$
$0^0$ is undefined. It is an Indeterminate form.
You might want to look at this post.
Why is $1^{\infty}$ considered to be an indeterminate form
As you said, $0^0$ has many possible interpretations and hence it is an indeterminate form.
For instance,
$\displaystyle \lim_{x \rightarrow 0^{+}} x^{x} = 1$.
$\displaystyle \lim_{x \rightarrow 0^{+}} 0^{x} = 0$.
$\displaystyle \lim_{x \rightarrow 0^{-}} 0^{x} = $ not defined.
$\displaystyle \lim_{x \rightarrow 0} x^{0} = 1$.
A clear and intuitive answer can be provided by ZFC Set-Theory. As described in Enderton's 'Elements of Set Theory (available free for viewing here; see pdf-page 151): http://sistemas.fciencias.unam.mx/~lokylog/images/stories/Alexandria/Teoria%20de%20Conjuntos%20Basicos/Enderton%20H.B_Elements%20of%20Set%20Theory.pdf, the set of all functions from the empty set to the empty set consists merely of the empty function which is 1 function. Hence $0^0$ = 1.
Below is copied verbatim from The Math Forum, except MathJax formatting has been applied.
What is $0$ to the $0$ power?
This answer is adapted from an entry in the sci.math Frequently Asked Questions file, which is Copyright (c) 1994 Hans de Vreught ([email protected]).
According to some Calculus textbooks, $0^0$ is an "indeterminate form." What mathematicians mean by "indeterminate form" is that in some cases we think about it as having one value, and in other cases we think about it as having another.
When evaluating a limit of the form $0^0$, you need to know that limits of that form are "indeterminate forms," and that you need to use a special technique such as L'Hopital's rule to evaluate them. For instance, when evaluating the limit $\sin(x)^x$ (which is $1$ as $x$ goes to $0$), we say it is equal to $x^x$ (since $\sin(x)$ and $x$ go to $0$ at the same rate, i.e. limit as $x\to0$ of $\sin(x)/x$ is $1$). Then we can see from the graph of $x^x$ that its limit is $1$.
Other than the times when we want it to be indeterminate, $0^0 = 1$ seems to be the most useful choice for $0^0$. This convention allows us to extend definitions in different areas of mathematics that would otherwise require treating $0$ as a special case. Notice that $0^0$ is a discontinuity of the function $f(x,y) = x^y$, because no matter what number you assign to $0^0$, you can't make $x^y$ continuous at $(0,0)$, since the limit along the line $x=0$ is $0$, and the limit along the line $y=0$ is $1$.
This means that depending on the context where $0^0$ occurs, you might wish to substitute it with $1$, indeterminate or undefined/nonexistent.
Some people feel that giving a value to a function with an essential discontinuity at a point, such as $x^y$ at $(0,0)$, is an inelegant patch and should not be done. Others point out correctly that in mathematics, usefulness and consistency are very important, and that under these parameters $0^0 = 1$ is the natural choice.
The following is a list of reasons why $0^0$ should be $1$.
Rotando & Korn show that if $f$ and $g$ are real functions that vanish at the origin and are analytic at $0$ (infinitely differentiable is not sufficient), then $f(x)^{g(x)}$ approaches $1$ as $x$ approaches $0$ from the right.
From Concrete Mathematics p.162 (R. Graham, D. Knuth, O. Patashnik):
Some textbooks leave the quantity 0^0 undefined, because the functions $0^x$ and $x^0$ have different limiting values when $x$ decreases to $0$. But this is a mistake. We must define $x^0=1$ for all $x$, if the binomial theorem is to be valid when $x=0$, $y=0$, and/or $x=-y$. The theorem is too important to be arbitrarily restricted! By contrast, the function $0^x$ is quite unimportant.
Published by Addison-Wesley, 2nd printing Dec, 1988.
As a rule of thumb, one can say that $0^0 = 1$, but $0.0^{0.0}$ is undefined, meaning that when approaching from a different direction there is no clearly predetermined value to assign to $0.0^{0.0}$ ; but Kahan has argued that $0.0^{0.0}$ should be $1$, because if $f(x)$, $g(x)$ $\to0$ as $x$ approaches some limit, and $f(x)$ and $g(x)$ are analytic functions, then $f(x)^{g(x)}\to1$.
The discussion of $0^0$ is very old. Euler argues for $0^0 = 1$ since $a^0 = 1$ for a not equal to $0$. The controversy raged throughout the nineteenth century, but was mainly conducted in the pages of the lesser journals: Grunert's Archiv and Schlomilch's Zeitshrift. Consensus has recently been built around setting the value of $0^0 = 1$.
References
Knuth. Two notes on notation. (AMM 99 no. 5 (May 1992), 403-422).
H. E. Vaughan. The expression '$0^0$'. Mathematics Teacher 63 (1970), pp.111-112.
Louis M. Rotando and Henry Korn. The Indeterminate Form $0^0$. Mathematics Magazine, Vol. 50, No. 1 (January 1977), pp. 41-42.
L. J. Paige. A note on indeterminate forms. American Mathematical Monthly, 61 (1954), 189-190; reprinted in the Mathematical Association of America's 1969 volume, Selected Papers on Calculus, pp. 210-211.
Baxley & Hayashi. A note on indeterminate forms. American Mathematical Monthly, 85 (1978), pp. 484-486.
Robert S. Fouch. On the definability of zero to the power zero. School Science and Mathematics 53, No. 9 (December 1953), pp. 693-696.
It depends on whether the 0 in the exponent is the real number 0, or the integer 0. These are two different objects and while the distinction is not often important, it is in this case.
Exponentiation by an integer has a universal specific meaning: Positive exponent is repeated multiplication, negative exponent is the inverse of repeated multiplication, zero exponent is the empty product, equal to the multiplicative identity denoted by 1. So if the exponent is the integer 0, then $0^0=1$ - the fact that this is an empty product with no terms trumps the fact that the terms which are not there are 0's (because they're not there).
Whereas exponentiation by a real or complex number is a messier concept, inspired by limits and continuity. So $0^0$ with a real 0 in the exponent is indeteriminate, because you get different results by taking the limit in different ways.
Note that all of the standard examples where it's "convenient" to have $0^0=1$ (e.g. power series) are all cases where the exponent is an integer.
Note that as long as the exponent is the integer 0, it doesn't matter if the base is the integer 0, the real 0, or pretty much any mathematical object on the planet for which multiplication is defined.
There are so many fantastic answers already, but no pictures which is sad. $0^0$ is undefined - is it the limited case of $x^0$, $0^x$ or $x^x$? The plots for $x^0$ (a line at 1) and $0^x$ (a line at zero) are boring, but $x^x$ is
Author/Site: Wolfram|Alpha
Publisher: Wolfram Alpha LLC
URL: https://www.wolframalpha.com/input/?i=plot+x%5Ex
Retrieval date: 6/2/2015
I don't always define $0^0$, but when I do, I define it as $1$.
Another approach and yet another result !
We define an exponential function on $\mathbb{R}$ as a function $E:\mathbb{R}\rightarrow \mathbb{R}$ such that $$ E(x+y)=E(x)E(y) \qquad \forall x,y \in \mathbb{R} $$ That's a very general and powerful definition, and I love it because captures the link between Lie algebras and Lie groups.
From this definition we have immediately that if $\exists a\in \mathbb{R}$ such that $E(a)=0$ then $E(x)=0 \; \forall x \in \mathbb{R}$. Nor continuity nor other topological properties are needed for this. In exponential fields theory such a function is called a trivial exponential function.
If $E$ is not this trivial function, it's easy to see that we must have $E(0)=1$ and, if $E(x)=1$ for some $x \ne 0$ than $E(x)=1 \forall x \in \mathbb{R}$, i.e. it is a constant (another trivial exp. function).
If $E$ is not trivial and $E(1)=a$ we have one particular funtion $E_a(n)$ and it is easy to see that $E_a(-1)=\dfrac{1}{a}$ and $E_a(n)=a^n$ (and all other properties of integer or fractional exponents), so it is natural to write $E_a(x)=a^x$ .
Now we see that $0^x$ is the null trivial exponential function, and, since this function must be always null, we have $0^0=0$.
Source: Understanding Exponents: Why does 0^0 = 1? (BetterExplained article)
A useful analogy to explain the exponent operator of the form $a \cdot b^c$ is to make $a$ grow at the rate $b$ for time $c$.
Expanding on that analogy, $0^0$ can be interpreted as $1\cdot0^0$ which is to say: grow $1$ at the rate of $0$ for time $0$. Since there is no growth (time is $0$), there is no change in the $1$ and the answer is $0^0=1$
Of course, this is just to grok and get an intuition or a feel for it. Science is provisional and so is math in certain areas. 0^0=1 is not always the most useful or relevant value at all times.
Using limits or calculus or binomial theorems doesn't really give you an intuition of why this is so, but I hope this post made you understand why it is so and make you feel it from your spleen.
A paper for general public is published on Scribd : " Zero to the Zero-th Power" (pp.7-11) : http://www.scribd.com/JJacquelin/documents
Some indeterminates forms $0^{0}, \displaystyle\frac{0}{0}, 1^{\infty}, \infty − \infty, \displaystyle\frac{\infty}{\infty}, 0 × \infty, $ and $\infty^{0}$
Futhermore,
$$\lim_{ x \rightarrow 0+ }x^{0}=1$$ and
$$\lim_{ x \rightarrow 0+ }0^{x}=0$$
Take a look at WolframMathWorld's [1] discussion.
See if this gives you any clarification.
[1] Weisstein, Eric W. "Indeterminate." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Indeterminate.html
The longstanding practice of leaving $0^0$ undefined is usually justified with arguments based on path dependent limits on the real numbers. As we see here, however, it is also possible to justify this practice based on purely discrete methods.
If the intuition of exponentiation on $N$ (where $0\in N$) is to be repeated multiplication such that $x^2=x\cdot x$, then we can formally justify the following definition:
$\forall x,y\in N: x^y\in N$ (a binary function on $N$)
$\forall x\in N:(x\ne 0\implies x^0=1)$
$\forall x,y\in N:x^{y+1}=x^y\cdot x$
Here, $0^0$ is assumed to be a natural number, but no specific value is assigned to it.
From this definition, we can derive the usual Laws of Exopnents:
$\forall x,y,z\in N: (x\ne 0 \implies x^y \cdot x^z = x^{x+y})$
$\forall x,y,z\in N: (x\ne 0 \implies (x^y)^z = x^{y\cdot z})$
$\forall x,y,z\in N: (x,y\ne 0 \implies (x\cdot y)^z = x^z\cdot y^z)$
For a detailed development based on formal proofs, see "Oh, the Ambiguity!" at my math blog.
Follow-up
A better approach, I have since found, is to construct a partial function on $N\times N$ with domain of definition $N\times N \setminus (0,0)$ such that:
$\forall x,y:[x,y\in N \land \neg [x=0\land y=0] \implies x^y\in N$
$\forall x:[x\in N\land x\ne 0\implies x^0=1]$
$0^1=0$
$\forall x,y:[x,y\in N \land \neg [x=0\land y=0] \implies x^{y+1}=x^y\cdot x]$
See my revised blog posting at above link (originally dated October 9, 2013).
only if the sets involved are not empty ... to every axiom!"
$\endgroup$
Let $$p=\lim\limits_{x\to 0} x^x$$
Then $$ln(p)=\lim\limits_{x\to 0} x\ln(x)$$
$$ln(p)=\lim\limits_{x\to 0} \frac{\ln(x)}{x^{-1}}$$
We can use L'Hopital on RHS as both numerator and denominator is tending to $\infty$
$$ln(p)=\lim\limits_{x\to 0} -\frac{1/x}{x^{-2}}$$
This can be simplified by cancelling some x terms out.
$$ln(p)=\lim\limits_{x\to 0} -x$$
Thereafter the limit can be applied
$$ln(p)=0$$
$$ p = e^0$$
Therefore $$p=1$$
$$ x^n = \underbrace{x \cdot x \cdot x \cdots x}_{n \text{ terms}} = 1 \cdot \underbrace{x \cdot x \cdot x \cdots x}_{n \text{ terms}} \implies 0^0 = 1 \cdot \underbrace ⬚_{\text{0 terms}} = 1 \\ 0^n = \underbrace{0 \cdot 0 \cdot 0 \cdots 0}_{n \text{ terms}} = k \cdot \underbrace{0 \cdot 0 \cdot 0 \cdots 0}_{n \text{ terms}} \implies 0^0 = k \cdot \underbrace ⬚_{\text{0 terms}} = k $$
