Thursday, February 24, 2011

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

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 helpful.

  1. lazer says:
    In middle you say we can define y^x as
    \lim_{z \to x^{+}} y^{z}.
    This is basically saying “let’s define y^x so that it’s always continuous.” Is it not?
  2. The Mathematician Mathematician says:
    As a function of two variables (x and y), y^x is still not continuous even with this definition. The problem is that along the line x=0 we have that \lim_{z \to 0^{+}} y^{z} = 1 for y>0, and \lim_{z \to 0^{+}} y^{z} = 0 for y = 0.
  3. David says:
    Could there be a practical calculation whose result is 0^0 ? In that case, you couldn’t claim the answer is just a matter of convenience…
  4. Mahmood says:
    I think there is some problem with the second definition in the mathematician’s answer.
    It says:
    “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.”
    I might be misunderstanding it, but isn’t that a bit like begging the question ?
    I mean, it tries to define the function of y^x in terms of the function itself, as in the right hand side, it uses the term “y^z”, which is an instance of the function that is being defined!. And even if we consider it to be a “Recursive Definition”, the definition provides no base case.
    This might be more philosophical than mathematical (It’s like Define “Define”), but I wish you can clarify it.
    Sorry of the late comment.
  5. squints says:
    I’m going on a sidetrack here.
    We all know anything divided by zero is impossible. We usually call this particular answer “undefined”. So, 2/0 = undefined.
    However, there’s an even MORE screwed up problem. What is 0/0? Well, we know n/n= 1, 0/n = 0, and furthermore, n/0=und.. There’s many different answers to this problem, including an und. one. We call these kinds of problems “indeterminate”. Think of it as an an extra step up in the mindf**k scale.
    (For you calculus geeks out there, we can define these as: an undefined limit can still approach a “nice” value, [i.e. a hole at the limit or an asymptote], whereas an indeterminate limit does not approach a “nice” value, [i.e. you can't do anything with it].)
    tl;dr 0/0 is indeterminate. There is no answer, although in some cases, we just give it an arbitrary 0 or 1 or und. or something
  6. Dart says:
    Actually no one talks about the imaginary number (i). Then division by zero would be possible.
  7. The Mathematician Mathematician says:
    Hi, Mahmood, thanks for your comment. I should have explained this better (in fact, maybe I’ll add a clarification to the post). The reason that definition works is because we all know what we mean by y^z for y>0 and z>0. The ambiguity crops up when z=0 and y=0. Hence, it’s okay to use y^z to define y^x because the y^z in the definition only ever uses values of z>0, whereas the y^x that it is defining allows for x=0. The idea is that we are extending y^z to handle the case of z=0 by using limits.
  8. The Physicist Physicist says:
    Complex numbers give you to answers some questions about the roots of polynomials that would otherwise be unattainable. But it doesn’t help out much here.
  9. Michael says:
    Google says 0^0 = 1
    You don’t question google.
  10. Doesn’t defining 0^0 as anything other than 1 create problems elsewhere? If you look at it from the perspective of exponentiation as antilogarithm, x^y=z is the same as log_x(z)=y, which is the same as ln(z)/ln(x) = y. We say y=0 and x=0, so ln(z)/ln(0)=0. We know that ln(0) = 1 because we can calculate it as a integral using limits. Therefore ln(z)/1 = 0, for which to hold ln(z) equal zero. If we solve ln (z) = 0 for z we get z=1.
    Doesn’t saying 0^0 is undefined lead to logical inconsistencies or problems because of this? If 0^0 ≠ 1 then something in the above proof has to also change, otherwise you arrive at a contradiction.
  11. The Mathematician Mathematician says:
    Hello Old Fashioned Whisky, thanks for your comment. If your analysis does lead to some insight, it is not the insight that you imagine. The problem is that if ln(0) is to be assigned a value at all, then it is -infinity, not 1. Hence, when you write ln(z)/ln(0)=0 you actually have ln(z)/(-infinity) = 0. It is generally considered to be the case that a / (-infinity) = 0 for any real number a. Hence, the formula ln(z)/ln(0)=0 tells nothing about the value of z.
  12. Chris says:
    You guys view this all wrong, book smarts can only get you so smart…
    0 to the 0 power=0
    0/0 =0
    However you look at it, it equals 0, BECAUSE!….
    Numbers do not actualy mean anything specific or equal anything physical, numbers are a figment of our imagination that everyone agrees to and uses to understand “how much” of “something”(something physically real) there is or was or may be…THEREFORE!….
    0 is “nothing”
    0 represents “literally nothing” (Aka the absence of something)
    So when you look at it for what it really is (not what a book or a mathematician sais it is) you see that you are trying to play with nothing…
      0/0= nothing, not 1
     (it doesn’t equal a number because you have to use logic or reality also I guess you could say).
    When you use just numbers and forget what they are representing you sort of stop using reality or logic, because numbers don’t exist, you always have to keep in mind what youre really “working with”…. And when it comes to the number 0… 0 represents “nothing”!!! It always represents nothing
    So now, because this is a math problem and in math we use numbers to represent values so everything has to be represented by a number…. When you look at 0/0 or 0 to the 0 power, you have to use 0 as the answer because nothing x nothing = nothing…
    And “0″ represets “nothing”!!!!!!!
    Whoow!!! I’ve been preaching the good news for years, please pass on the truth brothers and sisters, let the world know that our schooling systems that focus on just regurgitating repetitive general things written on paper and never tapping into real reasoning and understanding is messing up how we (our brains) process and view information (life)
    Wake up people! There’s more to life than what you learned in a school study book!!
    Numbers are used because of real things that exist, so we must keep this in mind when we use numbers, don’t just get in the number auto. Pilot mode, numbers have no intelligence to correct these kind of mistakes, this is were our brains are supposed to do that work
  13. Meta says:
    Do not question Wolfram|Alpha
  14. Phil E. Drifter says:
    0 x anything = 0. 0×0=0. 0x0x0=0.
  15. james says:
    Why is it that the higher in ranking person you ask the question to longer the explanation gets?
    What was so wrong with the clever students answer…
  16. Feels dumber now says:
    Wow I feel so retarded after reading this…
  17. Evil says:
    i thought it was 3
  18. so what says:
    And while all you folk are debating this, the chances of you getting a girlfriend or even a shag, is 0
  19. Enigma says:
    Dear god, tell me Chris was joking…
  20. JohnB says:
    I think there are two issues being confused here, ‘sign’ and ‘value’, they are different.
    All numbers have a sign and a value, 0 has neither, it is neither positive nor negative.
    Therefore using limits to assess its value is wrong because, for example, a small positive value will ALWAYS be positive as it approaches zero. There may be a continuous change in ‘value’ as it gets smaller, but it will have discontinuous change in ‘sign’ if it reaches zero.
    Its value is 0. Zero cannot be made larger or smaller, it is zero.
  21. The Mathematician Mathematician says:
    Hello James. The problem with the “clever student” answer is that it is… well… wrong. You cannot take x/x and plug in 0, as 0/0 is really undefined.
  22. jjg says:
    enigma, don’t worry, Chris had nothing to say.
  23. Geniuuus says:
  24. Zaphod B. says:
    I can’t believe all these people arguing over nothing, I need a Pan Galatic Gargleblaster and a patio.
  25. Awesome Astarte says:
    Seriously, life is far too short for this stuff. Please go out and just have a really good laugh. I’m begging you guys!
  26. Patrick Stewart's Clone says:
    wouldn’t you have to be able to measure 0^0 somehow to prove it?? I don’t believe convenient answers will help very much in the way of scientific advances, unless they lead to something productive. I’m no math expert, and I certainly don’t want to end up sounding like Chris, but I don’t see how just saying the answer is 1 or 0 because its mathematically convenient. I mean, actually measuring an answer to this problem could lead to some impressive scientific discoveries (how the universe began maybe, for example.) right?? Am I missing something???
  27. seismologist says:
    I love the “get a life” and “you guys will never get laid” comments. These people do not understand that this thread is intellectual mental jousting, just plain goddam fun. Some people play call of duty others fool around with abstractions like numbers and enjoy seeing what kind of cloth one can weave with them. Cheers!
  28. Kanye West says:
    Yo, Clever Student. I’m really happy for you. I’m gonna let you finish, but Einstein had the best equations of all times. One of the best equations of all time!
  29. Yahweh says:
    None of this makes any sense to me. Doesn’t
    0=nothing? So how can we raise nothing to the nothing power and get anything other than nothing? I obviously skipped a lot of math and cheated my way through the rest because this is not even remotely “clicking” in my brain.
    I will reluctantly accept that 0^0=1, however this raises an important question:
    If dividing by zero causes space-time to collapse in on itself and create a black hole, does raising zero to the zeroeth power create a universe? Food for thought…
  30. Meteo says:
    I started off intrigued, but somehow along the way I lost interest.
  31. right here, right now says:
    @Awesome Astarte, I agree!
    We already know that there is nothing provable, according to whatever the lastest asro-physical discovery or whatever that I read somewhere. Can we just accept life has unaccepted curves and bends, so let’s enjoy the ride?
    Even if you exist in another universe, you’re *here*, not *there*, so why worry about the *there*?
  32. stupid guy says:
    ha i dont care the world is domed anyways
  33. Wow says:
    I love these little kiddies that act as if math isn’t important.
    I’m currently an Engineering major, and let me tell you. Math makes everything in the world work.
    Just because you dicked around in High School, causing you to be fairly dumb don’t say this doesn’t matter.
  34. aynadan says:
  35. smartass says:
    if you take 3 apples and divide them among nobody, you have divided by zero
  36. jjg says:
    Allow the old statistitian to add 0! = 1 to the mix.
  37. Glitch Daracova says:
    It seems ridiculously simple to me, and I really don’t see how nobody else thought of it before. Look at it this way:
    4^3 is essentially 4x4x4. So 0^0 is essentially 0x….. My solution just fell apart. If I ever run into this on a math test, I’m just going to say it is “undefined”.
  38. ben says:
    0 x 0 = 0, 0 x (any int) = 0, 0^0 = 0.
  39. Matthew says:
    The statement that because this has no meaning whatsoever scientifically is silly. Look at the history of science. You will see that math that seems “pure” and not able to be applied is often used years down the road. Likewise, we see many examples of physics that stalled because we ran out of math, then someone invented/discovered it and away those projects went (sometimes even the physicist himself/herself was the person to invent it!)
    At some point, certain things are defined the way they are defined because it is useful to do it that way, and likewise not harmful. Consider 0!=1… If it weren’t that way, many a taylor series would totally fall apart. In many instances these ambiguous definitions have proofs on all sides of the answer.
    Also I’m curious about the problem that occurs when defining the limit as x approaches zero of y^x that occurs when we limit ourselves to approaching zero from the positive, as y^x is ALWAYS positive if y is positive. Graphing f(x)=y^x for any y will never go below the x axis.
    Finally, does L’Hopital have anything to say about this?
  40. MrBigDog2U says:
    Does it help to think about it this way?
    Multiplication is a binary operation – that is, you must have two operands to complete the operation. In order to evaluate y^1, you must have something to multiply y by in order to arrive at the answer. We are all in agreement that y^1 = y and the multiplicative identity is 1. Therefore, y^1 represents 1 * y (1 multiplied by y one time).
    Furthermore, y^x represents 1 * y * y * y * …. however many times are specified by the exponent (x). Thus, y^0 represents 1 multiplied by y zero times.
    This gives us:
    2^1 = 1 * 2 = 2
    3^2 = 1 * 3 * 3 = 9
    4^0 = 1 (multiplied by 4 zero times) = 1
    0^3 = 1 * 0 * 0 * 0 = 0
    0^0 = 1 (multiplied by 0 zero times) = 1
  41. The Mathematician Mathematician says:
    Yes, this construction is used in the article. Unfortunately, it is not the only way to define y^x, and in particular, using other definitions (which all are equivalent when y>0 and x>0) can lead to different conclusions about 0^0, as the article points out.
  42. High school teacher says:
    Great article!
    I felt dumb when I asked myself “why does he only considers the positive limit as x approaches 0 of f(x,0)=y^x?”
    0^x does not exist when x0, 0^-a :=1/(0^a) =1/0.
    Graphing f(x)=y^x may never go below the x axis, but f(x)=0^x is what we’re concerned with. He does consider both left and right limits for f(y)=y^0.
  43. Jeff says:
    This English prof says: The 0 represents societies inability to properly empathize with the most disadvantaged peoples. Done!
  44. Tom says:
    Just wanted to point out that 0 does not necessarily equal 0:
    Take for example these two functions:
    a) x/sin x
    b) x^2/sin x
    What are their values at x=0?
    Well – it turns out that the former evaluates to 1 whereas the latter equals 0 – although in both cases we have an expression of the kind 0/0.
    (If you don’t believe me – look up L’Hôpital’s Rule)
    My point is, that one has to know where the expression 0^0 came from to be able to evaluate it.
  45. GOD says:
    There is no Zero.
  46. Zach says:
    I stumbled upon this slightly drunk. I don’t think I woulder understand it sober anyway.
  47. sam says:
    actually Chris is not entirely wrong, numbers were ‘invented’, not discovered, to signify real objects and communicate with other people…arguing about nothing raised to the power nothing can go on forever, whatever majority people (or the clever ones) decide (through mathematical proof’ or otherwise) at this particular time, is what it is… all the equations will be treated accordingly…
  48. Caz says:
    0^0 is undefined.
    0^0=undefined because you cannot multiply nothing by nothing.
    simple as that.
    =0* (0/0)
    Therefore, 0^0 is undefined
  49. Caz says:
    To Tom:
    here’s where 0^0 comes from.
    f’(x) x^x=(x)*(x^-1)
    now plug in 0 for x
    0^-1= 0/0= undefined
    That’s where 0^0 derives from
  50. create14all says:
    0 is not a number. It is the absence of a number. 0 means that the objects being counted are not there. It is used inappropriately in mathematics, physics … (all the sciences) often. Another example is 10. 10 is the 10 th digit which uses 0 in an inappropriate way.

  1. Hmmm says:
    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.
  2. Duh says:
    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?
  3. The Physicist Physicist says:
    @ 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.
  4. Jeff says:
    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.
  5. anya says:
    @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
  6. Jerry says:
    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.
  7. Ron says:
    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.”
  8. The Physicist Physicist says:
    It useful to know when you have 0 clients.
  9. Alex says:
    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.
  10. David says:
    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. :-)
  11. Isaiah says:
    Incorrect “Duh”. Zero is defined as the additive identity (zero plus stuff equals stuff).
  12. Nick says:
    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.
  13. Paul says:
    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.
  14. Umm says:
    I’m just wondering how posting here applicable to real life situations. I’m actually asking.
  15. Brian says:
    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.
  16. Pete says:
    @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.
  17. Jim Ward says:
    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.
  18. Darcy says:
    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.
  19. The Physicist The Physicist says:
    0^1 is definitely equal to zero. 0^0 is a weird special case.
  20. Mayor West says:
    The zeroes are taking over!!!!
  21. Realist says:
    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.
  22. Chris says:
    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 = 0×0 = 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…
  23. Weez says:
    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.
  24. The Usher of Tides says:
    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.
  25. I say 0^0 = 1, because otherwise a Taylor series wouldn’t work.
  26. S. N. Balasubrahmanyam says:
    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)
  27. john fake says:
    simple. zero is representative of non existence. so non existent to the power of non existent makes non existent. 0^0=0.
  28. Todd says:
    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.
    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.
  29. The Usher of Tides says:
    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.
  30. IT Admin/CS Student says:
    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:
  31. The Usher of Tides says:
    so maybe then, there was nothing. 0. And it was an immense nothing, 0 to the 0th. Then there was one?
  32. Ryan Palmiter says:
    Whats 9.0^10+29, what is the name when not in scientific notation?
  33. The Physicist The Physicist says:
  34. Prashan P. (BA. Sociology) says:
    @Duh: 0 has been proven to be a number. 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?
  35. Freshman says:
    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?
  36. SaggingPeach says:
    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?
  37. 11 dimensions, i do think they exist. says:
    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.
  38. dd says:
    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])
    = 1
    I will now use more fiction to solve world hunger…. “Energize”
  39. The Physicist The Physicist says:
    That’s a really cool idea! I don’t think it would work, but it’s a really solid idea!
  40. Scientist in the making says:
    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.
  41. Michael says:
    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).
  42. Ben says:
    Who cares what 0^0 is? l’Hopital’s rule makes it so we really don’t need to worry.
  43. Devyn says:
    Answer = 1.
    because it is.
  44. Arjun Gehlot says:
    Well… Goes to show you. Just when you thought you had nothing. You’ve got something.
  45. Math Student says:
    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).
  46. Ellis says:
    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.
  47. Henry says:
    Maybe it’s like schroedinger’s cat.
    0^0 equals both One and Zero.