Q: What does 0^0 (zero raised to the zeroth power) equal? Why do mathematicians and high school teachers disagree?

Posted on December 4, 2010 by The Mathematician

Clever student:

I know!

$x^{0}$ = $x^{1-1}$ = $x^{1} x^{-1}$ = $\frac{x}{x}$ = $1$ .

Now we just plug in x=0, and we see that zero to the zero is one!

Cleverer student:

No, you’re wrong! You’re not allowed to divide by zero, which you did in the last step. This is how to do it:

$0^{x}$ = $0^{1+x-1}$ = $0^{1} \times 0^{x-1}$ = $0 \times 0^{x-1}$ = $0$

which is true since anything times 0 is 0. That means that

$0^{0}$ = $0$ .

Cleverest student :

That doesn’t work either, because if $x=0$ then

$0^{x-1}$ is $0^{-1} = \frac{1}{0}$

so your third step also involves dividing by zero which isn’t allowed! Instead, we can think about the function $x^{x}$ and see what happens as x>0 gets small. We have:

$\lim_{x \to 0^{+}} x^{x}$ = $\lim_{x \to 0^{+}} \exp(\log(x^{x}))$

= $\lim_{x \to 0^{+}} \exp(x \log(x))$

= $\exp( \lim_{x \to 0^{+} } x \log(x) )$

= $\exp( \lim_{x \to 0^{+} } \frac{\log(x)}{ x^{-1} } )$

= $\exp( \lim_{x \to 0^{+} } \frac{ \frac{d}{dx} \log(x) }{ \frac{d}{dx} x^{-1} } )$

= $\exp( \lim_{x \to 0^{+} } \frac{x^{-1}}{- x^{-2}} )$

= $\exp( \lim_{x \to 0^{+} } -x )$

= $\exp( 0)$

= $1$

So, since $\lim_{x \to 0^{+}} x^{x}$ = 1, that means that $0^{0}$ = 1.

High School Teacher:

Showing that $x^{x}$ approaches 1 as the positive value x gets arbitrarily close to zero does not prove that $0^{0} = 1$ . The variable x having a value close to zero is different than it having a value of exactly zero. It turns out that $0^{0}$ is undefined. $0^{0}$ does not have a value.

Calculus Teacher:

For all $x>0$ , we have

$0^{x} = 0$ .

Hence,

$\lim_{x \to 0^{+}} 0^{x} = 0$

That is, as x gets arbitrarily close to $0$ (but remains positive), $0^{x}$ stays at $0$ .

On the other hand, for real numbers y such that $y \ne 0$ , we have that

$y^{0} = 1$ .

Hence,

$\lim_{y \to 0} y^{0} = 1$

That is, as y gets arbitrarily close to $0$ , $y^{0}$ stays at $1$ .

Therefore, we see that the function $f(x,y) = y^{x}$ has a discontinuity at the point $(x,y) = (0,0)$ . In particular, when we approach (0,0) along the line with x=0 we get

$\lim_{y \to 0} f(0,y) = 1$

but when we approach (0,0) along the line segment with y=0 and x>0 we get

$\lim_{x \to 0^{+}} f(x,0) = 0$ .

Therefore, the value of $\lim_{(x,y) \to (0,0)} y^{x}$ is going to depend on the direction that we take the limit. This means that there is no way to define $0^{0}$ that will make the function $y^{x}$ continuous at the point $(x,y) = (0,0)$ .

Mathematician: Zero raised to the zero power is one. Why? Because mathematicians said so. No really, it’s true.

Let’s consider the problem of defining the function $f(x,y) = y^x$ for positive integers y and x. There are a number of definitions that all give identical results. For example, one idea is to use for our definition:

$y^x$ := $1 \times y \times y \cdots \times y$

where the y is repeated x times. In that case, when x is one, the y is repeated just one time, so we get

$y^{x}$ = $1 \times y$ .

However, this definition extends quite naturally from the positive integers to the non-negative integers, so that when x is zero, y is repeated zero times, giving

$y^{0}$ = $1$

which holds for any y. Hence, when y is zero, we have

$0^0 = 1$ .

Look, we’ve just proved that $0^0 = 1$ ! But this is only for one possible definition of $y^x$ . What if we used another definition? For example, suppose that we decide to define $y^x$ as

$y^x$ := $\lim_{z \to x^{+}} y^{z}$ .

In words, that means that the value of $y^x$ is whatever $y^z$ approaches as the real number z gets smaller and smaller approaching the value x arbitrarily closely.

[Clarification: a reader asked how it is possible that we can use $y^z$ in our definition of $y^x$ , which seems to be recursive. The reason it is okay is because we are working here only with $z>0$ , and everyone agrees about what $y^z$ equals in this case. Essentially, we are using the known cases to construct a function that has a value for the more difficult x=0 and y=0 case.]

Interestingly, using this definition, we would have

$0^0$ = $\lim_{x \to 0^{+}} 0^{x}$ = $\lim_{x \to 0^{+}} 0$ = $0$

Hence, we would find that $0^0 = 0$ rather than $0^0 = 1$ . Granted, this definition we’ve just used feels rather unnatural, but it does agree with the common sense notion of what $y^x$ means for all positive real numbers x and y, and it does preserve continuity of the function as we approach x=0 and y=0 along a certain line.

So which of these two definitions (if either of them) is right? What is $0^0$ really? Well, for x>0 and y>0 we know what we mean by $y^x$ . But when x=0 and y=0, the formula doesn’t have an obvious meaning. The value of $y^x$ is going to depend on our preferred choice of definition for what we mean by that statement, and our intuition about what $y^x$ means for positive values is not enough to conclude what it means for zero values.

But if this is the case, then how can mathematicians claim that $0^0=1$ ? Well, merely because it is useful to do so. Some very important formulas become less elegant to write down if we instead use $0^0=0$ or if we say that $0^0$ is undefined. For example, consider the binomial theorem, which says that:

$(a+b)^x$ = $\sum_{k=0}^{\infty} \binom{x}{k} a^k b^{x-k}$

where $\binom{x}{k}$ means the binomial coefficients.

Now, setting a=0 on both sides and assuming $b \ne 0$ we get

$b^x$

= $(0+b)^x$ = $\sum_{k=0}^{\infty} \binom{x}{k} 0^k b^{x-k}$

= $\binom{x}{0} 0^0 b^{x} + \binom{x}{1} 0^1 b^{x-1} + \binom{x}{2} 0^2 b^{x-2} + \hdots$

= $\binom{x}{0} 0^0 b^{x}$

= $0^0 b^{x}$

where, I’ve used that $0^k = 0$ for k>0, and that $\binom{x}{0} = 1$ . Now, it so happens that the right hand side has the magical factor $0^0$ . Hence, if we do not use $0^0 = 1$ then the binomial theorem (as written) does not hold when a=0 because then $b^x$ does not equal $0^0 b^{x}$ .

If mathematicians were to use $0^0 = 0$ , or to say that $0^0$ is undefined, then the binomial theorem would continue to hold (in some form), though not as written above. In that case though the theorem would be more complicated because it would have to handle the special case of the term corresponding to k=0. We gain elegance and simplicity by using $0^0 = 1$ .

There are some further reasons why using $0^0 = 1$ is preferable, but they boil down to that choice being more useful than the alternative choices, leading to simpler theorems, or feeling more “natural” to mathematicians. The choice is not “right”, it is merely nice.

This entry was posted in -- By the Mathematician, Math, Philosophical. Bookmark the permalink.

1,178 Responses to Q: What does 0^0 (zero raised to the zeroth power) equal? Why do mathematicians and high school teachers disagree?

Hmmm says:

January 24, 2011 at 1:59 am

And yet there is still hunger in the world. I actually do appreciate the amount of thought in this. It is far beyond me. I’m just wondering how this is applicable to real life situations. I’m actually asking.
Duh says:

January 24, 2011 at 2:39 am

Zero is not a number. It is a symbolic representation of the lack thereof. How short-sighted and contrived and trivial can you possibly be?
Now there is a question?
Error: Unable to create directory uploads/2025/05. Is its parent directory writable by the server? Physicist says:

January 24, 2011 at 8:56 am

@ Hmmm: Before you can advance the field of mathematics it’s important to understand, in detail, the fundamentals, and what can go wrong. In particular, this problem is a useful case study in “limits”, what can go wrong and how to deal with it.
Less specifically, asking “what good is math?” is a question on par with “What good is literacy?” and “What good are hands?”. If only a few people had hands, only a few people would realize how useful hands are.
Deep stuff.
Jeff says:

January 24, 2011 at 9:13 am

create14all your kinda right that 0 is not a number but in your world where 0 is inappropriately used in the number 10 doesn’t make sense. From what you are saying multi-digits don’t exist; there is no such thing as 10 O’clock instead you would be saying “one 0 O’clock” (haha that sounded so funny in my head). Another point is people would die at the ripe old age of 9. Maybe if you said you were posting from some parallel universe were the concept of base 10 was never created and people compute in binary, 10 would really be 1100 or maybe base 9 where 10 would be 11 =0 my mind would have been blown ha. but this page is awesome reminds me of a class I took were the professor was awesome.
anya says:

January 24, 2011 at 10:38 am

@hmmm yes there is still hunger in the world and had the amount of thought that went into explaining this problem been spent elsewhere, there would still be hunger in the world.
But it may make you feel better to know that mathematics has been used throughout the ages to help alleviate hunger and further agriculture
Jerry says:

January 24, 2011 at 11:59 am

Mathematics are a extremely powerful tool that has helped us from calculating the exact sugar values of your coke to the trajectories of satellites in space, from the chemical reactions in your lip gloss to the age of the universe, from calculating mass growings to creating cheaper food to feed that mass, from programming iphones to making artifitial limbs, eyes, hearts, kidneys, lungs, etc etc…Seriously guys, its ok if you dont like math or its ok if you dont understand math, but this 0 thingy goes deep into the structure of calculations to allow us to use that in our advantage. So YES, it works, it is not just “short-sighted and contrived and trivial”, its questions like these that have keep some of us on top of evolutionary development, question like these sent human to limits that no one ever dreamed of. So, please, learn some math, and dont stop questioning everything.

Gabriel P.
Physics Engieneering student for the Metropolitan Autonomous University, Mexico, D.F.
Ron says:

January 24, 2011 at 6:59 pm

From a practical point of view, zero is useless because there is no point in representing nothing. Let me illustrate…

A lawyer appears before a judge. The judge says, “Where is your client?” The lawyer says, “I have no client, but I’m ready to argue a case.”
Error: Unable to create directory uploads/2025/05. Is its parent directory writable by the server? Physicist says:

January 24, 2011 at 7:07 pm

It useful to know when you have 0 clients.
Alex says:

January 25, 2011 at 12:10 am

Go tell an engineer or an economist or an accountant that 0 is useless. 0 is most definitely a number and is absolutely integral in a lot of important and practical mathematics.
David says:

January 25, 2011 at 7:52 pm

Well, the physicist would say that 1/0, 10/0, 0.0001/0 are all very useful ratios that tell us some significant information about some physical process or phenomenon. If, as you study some nugget of physics you come across or calculate something that results in a number divided by zero, it usually means whatever it is becoming very VERY large and is fast approaching infinity. For anyone to say that zero is useless is to say that infinity is useless. Either position is narrow-minded and ignorant.

But the English teacher would say that it is an improper use of grammar for anyone to say they have zero of anything. You can have nothing, or you can be without anything, but you can’t really have zero of anything. Have a nice day. 🙂
Isaiah says:

January 26, 2011 at 8:19 pm

Incorrect “Duh”. Zero is defined as the additive identity (zero plus stuff equals stuff).
Nick says:

January 27, 2011 at 12:55 pm

Mr. Hmmm….. To solve world hunger, technologies must be developed. To solve the issue of improving technology, lots of mathematics must be precisely defined and understood. Here we are working on solving world hunger my friend.
Paul says:

January 27, 2011 at 4:35 pm

I’m actually kind of surprised that there isn’t yet any inquiry or objection about raising a number to a non-positive-integer degree, other than zeroth degree. Though it’s been fun to read about these philosophical or real world utilitarian approach to 0^0.
Umm says:

January 28, 2011 at 12:54 am

I’m just wondering how posting here applicable to real life situations. I’m actually asking.
Brian says:

January 28, 2011 at 9:20 pm

@hmmmm

Really though it doesn’t matter. It’s just us trying to explain more complicated mechanics in our fun physic world. But if it doesn’t make any sense and we can’t use it. It doesn’t matter.
Pete says:

January 29, 2011 at 6:00 am

@Hmmm Dividing by zero or more appropriately the thought behind it leads to knowledge of how we work out rates of change and hence calculus this knowledge allows us to model weather systems, build structures such as huge bridges, helps with the electronics you are using for your laptop and myriad many other applications.
Jim Ward says:

February 1, 2011 at 3:52 pm

The C language bible, Kernighan and Ritchie, and my C Reference Manual both say pow(0,0) should be a NaN (Not a Number). However, IEEE says it’s 1, and the ANSI C standard goes even further:

pow(x, ±0) returns 1 for any x, even a NaN.

Google for “What Every Computer Scientist Should Know About Floating Point” by David Goldberg for the pros and cons.
Darcy says:

February 1, 2011 at 8:36 pm

Hi, I’m definitely not a math person by any means, but as I was looking through your proofs and saw 0^0, I started wondering if that was the same thing as 0^1? Maybe that’s a bad question, but I’m genuinely interested. Thanks.
Error: Unable to create directory uploads/2025/05. Is its parent directory writable by the server? The Physicist says:

February 1, 2011 at 10:31 pm

0^1 is definitely equal to zero. 0^0 is a weird special case.
Mayor West says:

February 2, 2011 at 9:16 pm

The zeroes are taking over!!!!
Realist says:

February 2, 2011 at 10:57 pm

You can think of it this way: you have nothing = 0, then you multiply nothing by nothing so you still end up with nothing. I think that math people should rethink it they try too hard to explain something when you can just step back and realize that you can’t multiply something that has nothing by nothing. This is a good site to make people think though.
Chris says:

February 4, 2011 at 4:54 am

To all of you trying to prove that 0^0=0:

0^0 is not that same as multiplying nothing by nothing. Although I will admit that 0 is nothing, interpreting 0^0 as multiplication is simply foolish.

Here’s the logic:

0^2 = 0x0 = 0 – fair enough
0^3 = 0x0x0 = 0 – that’s reasonable
0^x = 0x0x0x….. (x amount of times) – of course

So what does it mean for x to equal zero when 0^x.
It does not mean multiplying 0 by 0 or nothing by nothing.
In fact it means not multiplying anything at all.

Since x represents the number of times we multiplied 0 by itself, if x=0, then we are multiplying 0 not at all. We are multiplying zero by itself no times.

Well, it still seems like that might still be nothing, because multiplying nothing no times at all seems like it should lead to nothingness.

Logically, however, we can see that multiplying nothing no times at all has to leave us with something, since we have no amount of nothing.

In other words, 0^0 means there are no zeroes.

But then there is the problems of 1^0 equaling 1. Maybe we just need a different definition whenever anything equals 0.

Damn, I thought I had worked my way out of this maze…
Weez says:

February 4, 2011 at 5:21 am

To myself, the statement “x^0=1”, still does not make sense. If you take into account that x^y is equivalent to multiplying x by itself y times, then x^0 should = x. In a real world sense, if you have two mice and you breed (multiply) them zero times, you would still have two mice, not one. By my logic x^0=x, as stated above. Furthermore, when you define x as 0, you would have the same result. 0 multiplied by anything still equals 0, even when multiplied by itself even as y approaches infinity. However, this is all theoretical, I’m not a mathematician, and I have no support for my theory outside of my own logic.
The Usher of Tides says:

February 4, 2011 at 11:18 am

x and y cannot be 0, nor can they be 1. 0 has a value of nothing… until brought into an equation, then it is the ‘dark matter’ of math. It doesn’t exist until we insist that it has to, and that because we’re using it, it must be something. The opposite of something is nothing, but the opposite of nothing isn’t something.
Michael Dickens says:

February 4, 2011 at 2:05 pm

I say 0^0 = 1, because otherwise a Taylor series wouldn’t work.
S. N. Balasubrahmanyam says:

February 5, 2011 at 7:21 am

I am from the country that accorded a position to 0 while inventing the decimal place-value system of representing numbers.
I tend to agree with the view that there is a discontinuity at 0 when we approach the limit x –> 0 in y^x . In an expression like 0^0 the power symbol represents the number of operations you carry out, in this case, just the operation of writing the expression 0^0 and that is equal to 1 (Obviously, heh!).
At all other values of x the meaning of the power term changes to the “normal” – the number of times you multiply y by itself.

Signing off with {;-\) ( = tongue firmly in cheek)
john fake says:

February 5, 2011 at 5:16 pm

simple. zero is representative of non existence. so non existent to the power of non existent makes non existent. 0^0=0.
Todd says:

February 5, 2011 at 9:36 pm

Dividing by zero, raising anything to zero and many other manipulations of zero are all perfectly fine, depending on your understanding of mathematics. First off, 0^0 is NOT zero…to understand 0^0 (or some constant “c” divided by zero ie: 1/0 500/0) you have to have an understanding of limits. When you take high school/college algebra they say that they are “undefined” because you do not have the tools to understand them.

Think of a problem like Lim x->0 1/x and try some numbers that approach zero and you will see how limits work.

1/1=1
1/.01=10
1/.0001=10000

You can see how, in a sense, 1/0=infinity

I also disagree that zero is just a representative of non-existence…zero is a much more important, useful, and complicated number than just nothing.
The Usher of Tides says:

February 6, 2011 at 11:03 am

say your looking for cats in rooms…. x is the number of cats in a room, and y is the number of rooms you look in. If y = 0, then you did not look in any rooms for cats and cannot determine whether or not x = 0. You are not going to say there is or isn’t a cat if you haven’t looked. To say 0 to the 0th power is incorrect. X can have no value if Y is 0.
IT Admin/CS Student says:

February 9, 2011 at 12:23 am

A viewpoint that I haven’t seen, and am surprised no one has taken on 0^0=1 is that there are cases where 0 is not zero, but rather false. So taken in this light, it would be a false false which gives you true, or a 1 as true is represented in programming, as well as several other systems.

So the short and simple explanation:
false=0
0^0=1
The Usher of Tides says:

February 9, 2011 at 6:16 pm

so maybe then, there was nothing. 0. And it was an immense nothing, 0 to the 0th. Then there was one?
Ryan Palmiter says:

February 9, 2011 at 8:12 pm

Whats 9.0^10+29, what is the name when not in scientific notation?
Error: Unable to create directory uploads/2025/05. Is its parent directory writable by the server? The Physicist says:

February 9, 2011 at 8:24 pm

3,486,784,430
Prashan P. (BA. Sociology) says:

February 11, 2011 at 5:05 pm

@Duh: 0 has been proven to be a number. http://mathforum.org/library/drmath/view/63315.html for the simple proof that it is a number. It lies on the line between -1 and 1 … did a black hole suddenly appear between those numbers? And i’m an arts major … common…

But back to the discussion of 0^0. I had always thought and learnt in my culc classes that if x^0=1 or for that matter anything to the power of 0 is equal to 1. If that statement is true then 0^0 should also be 1, as 1^0=1, which was also confusing to me at the time as 1^1=1. In fact, i’m now thinking that its not a “hard answer”, but it should be an “approaching answer”, in the sense that it should be greater then or equal to 1 (>=1). That way it states that its very close to one but not quite there. Any ideas on this issue or is it just simpler to say that anything to the power of 0 =1?
Freshman says:

February 13, 2011 at 2:57 am

In science class, a student asked why we didn’t put units on the measure of an object at rest. The teachers reply was that one can have 0 giraffes and 0 cows, but it’s still the same.

Nothing multiplied no times is nothing.

Why must humans tend to over complicate the vast majority of the aspects of life here on this tiny, insignificant planet?
SaggingPeach says:

February 14, 2011 at 3:23 am

This reminds me of quantum mechanics,
Where stuff can simultaneously exist and not exist at the same time.
If you calculate it one way it is nothing, if you calculate it the other way it’s something.
So how bout perceiving something makes the universe calculate it one way, and not perceiving something makes the universe calculate it another?
11 dimensions, i do think they exist. says:

February 15, 2011 at 2:30 pm

The concept of 0 and ∞ is fascinating. But is it truly a concept? The idea of 0 and ∞ is fascinating. But is it merely an idea? I believe we, in our current dimension of being, would never be able to grasp this ever.

Solve for 0^0, the number lies in another dimension.
Solve for ∞, the number lies in another dimension.
dd says:

February 15, 2011 at 7:46 pm

0^0
=e^(0*ln(0))
=e^(0*-inf)

Now think about complex numbers for a second…. real part, imaginary part… what if there was an infinite part as well? We could continue solving and it would look like this (ignoring the imaginary part since there is no imaginary component in either of these numbers).

=e^([0,0] * [0,-1])
=e^([0*0 – 0*-1 , 0*0+0*-1])
=e^([0,0])
=e^(0)
= 1

I will now use more fiction to solve world hunger…. “Energize”
Error: Unable to create directory uploads/2025/05. Is its parent directory writable by the server? The Physicist says:

February 15, 2011 at 10:54 pm

That’s a really cool idea! I don’t think it would work, but it’s a really solid idea!
Scientist in the making says:

February 16, 2011 at 5:39 am

Some of this stuff is way over my head, but I did see some logic in the argument that 0 can stand in for both of nothing and infinity. In which case 0 is not a specific value really, but more of a representation of a value that cannot be given value. I’m not sure that this furthers the argument of 0^0 one way or the other, but it has given me something new to ponder over.
Michael says:

February 17, 2011 at 9:50 am

In my greatest opinion I think what it all comes down to is a real life setting.

If you don’t have anything to begin with than you have nothing to power (multiply).
Ben says:

February 18, 2011 at 8:15 pm

Who cares what 0^0 is? l’Hopital’s rule makes it so we really don’t need to worry.
Devyn says:

February 18, 2011 at 9:08 pm

Answer = 1.
because it is.
Arjun Gehlot says:

February 19, 2011 at 3:55 am

Well… Goes to show you. Just when you thought you had nothing. You’ve got something.
Math Student says:

February 19, 2011 at 8:38 pm

Just as Mathematician says, x^0 is defined as 1. This is an axiom, something of a basic building block not derived from logic but designed to help us further describe any higher mathematics. Just as any axiom, it cannot be proved (look up the incompleteness theorems for some further light on axioms).
Ellis says:

February 20, 2011 at 3:04 am

Ya’know, all this talk about 0 being nothing and how you can’t have nothing no amount of times got me thinking about nothing…ironically. I guess it could be argued that the nature of nothing (and thus, 0) itSELF is also paradoxical, considering if there WAS nothing, then nothing is all that there would BE. Therefore it COULDN’T be nothing…could it? And would the same also apply to zero? Fun ideas, but I dunno how applicable they are to the actual theory.
But then again, what do I know?
Next to nothing.
🙂
Henry says:

February 20, 2011 at 5:07 am

Maybe it’s like schroedinger’s cat.
0^0 equals both One and Zero.
Luke says:

February 24, 2011 at 5:01 pm

Saying that Zero is useless is like saying some one would rather be paid $1 an hour instead of $10, or even $100 an hour.
“From a practical point of view, zero is useless because there is no point in representing nothing.” (Ron)
John says:

February 25, 2011 at 11:11 am

For all of you that don’t have a firm grasp on calculus, this concept is not one you will be able to understand. Most people have taken high school/college algebra, and in these classes it is not possible for you to divide by zero. This is not because I can’t divide by zero, because I have the math tools to be able to do this. It simply means that you don’t have the tools to fully understand this problem.

Look at this from a different angle.

Would you be able to diagram this exact sentence if you didn’t know what a verb was? No, because you don’t have the English tools to do such a thing. This is no different than the confusion here about 0^0. Without the proper skills, this is just simply not something you can understand. And, as with most things in math, it is this way simply because that is how it is. Why is a dangerous question in the math world.
Jeff Jyzler says:

February 25, 2011 at 10:01 pm

Two words.
Double Negative.
If you have nothing and multiply it by the power of nothing then nothing is not there so there must be something.

Q: What does 0^0 (zero raised to the zeroth power) equal? Why do mathematicians and high school teachers disagree?

1,178 Responses to Q: What does 0^0 (zero raised to the zeroth power) equal? Why do mathematicians and high school teachers disagree?

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