If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

The problem with dividing zero by zero

One can argue that 0/0 is ​0, because 0 divided by anything is 0. Another one can argue that 0/0 is ​1, because anything divided by itself is 1. And that's exactly the problem! Whatever we say 0/0 equals to, we contradict one crucial property of numbers or another. To avoid "breaking math," we simply say that 0/0 is undetermined. Created by Sal Khan.

Want to join the conversation?

  • piceratops ultimate style avatar for user Wilbur Donovan
    Is 0/0 = 1? Or is it undefined?

    Couldnt you say that the numbers are like a dimension where you could join the infinitys'?

    Like for example, if you wrote a number line on a piece of paper, with positive infinity at the top and negative infinity at the bottom, couldn't you fold the paper so that negative and positive infinity are the same?
    (39 votes)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user Paul Moore
      0/0 is undefined. If substituting a value into an expression gives 0/0, there is a chance that the expression has an actual finite value, but it is undefined by this method. We use limits (calculus) to determine this finite value. But we can't just substitute and get an answer.

      Example: the limit as x approaches 0 of x/x = 1, but the expression x/x is undefined at x = 0.

      In geometry, we do not consider lines to meet at infinity. We consider them to go on forever in opposite directions, never meeting. In physics, of course, space is curved, and a real "line" might, in fact, be a closed figure (which is somewhat like "meeting at infinity")... but not in geometry.

      Your thinking is great - keep it up.
      (50 votes)
  • piceratops ultimate style avatar for user Paulius Eidukas
    In my opinion, it seems that 0/0 could be equal not only to 0 and/or 1, but actually to any number.

    Let's say that 0/0 = x
    Following the principles of division and multiplication, we can re-arrange the equation like this:
    0x = 0
    From here it becomes obvious that this equation is true for any x, because 0 multiplied by anything is still equal to 0:
    0 * 0 = 0; 0/0 = 0
    0 * 1 = 0; 0/0 = 1
    0 * 2 = 0; 0/0 = 2
    ... and so on

    Is this also a valid way to prove that 0/0 is indeterminate?
    (30 votes)
    Default Khan Academy avatar avatar for user
    • hopper cool style avatar for user David
      I'm just stating what Sal said in the video, but some people say that 0/0 is obviously 0, since 0/4, for example, is zero. They say zero divided by anything is zero. However, some say anything divided by zero is undefined, since 4/0 and 5/0 are and so on. Others say that 0/0 is obviously one because anything divided by itself, just like 20/20 is 1. All of these points of view are logical and reasonable, yet they contradict each other. These statements are impossible and don't work, since a mathematical expression consisting of only constants (numbers like 1, not variables like x and y) can only have one value. These thoughts can not merge, as 0 is not 1 or undefined, and vice versa. So 0/0 must be undefined.

      Also, if you think about it more closely, (Sal also says this in the next video.) division must be able to be undone by multiplication. For example, 6 divided by 2 is 3, and it can be undone by multiplying 2 by 3 to get 6. If 0/0 is 1, then 1 times 0 is , so it is correct. If 0/0 is 0, then 0 times 0 is 0, so it is also correct. If 0/0 is undefined, then you can't multiply back. The first two can not be proved false using this method, nor can the latter, since it is not exactly defined as division anyway.
      (16 votes)
  • leaf green style avatar for user Christine Moreda
    If we have something and we divide it by 2, then don't we separate it into two pieces?
    And if we have something and divide it by 1, then don't we "separate" it into one piece?
    So if we have something and divide it by 0, wouldn't that be like making it disappear?
    Wouldn't that be impossible, rather than undefined?
    To take a thing and turn into nothing?
    (So, my first hypothesis is that dividing by zero is not undefined, but is defined as impossible.)

    (Now, to look at what we mean by "divide" or if we actually mean more than one thing.)
    Divide (per somewhere online, I read), means to calculate how many times we could add one number to itself to get another number.
    (For example, 10/5 = 2. We add 5 to itself 2 times to get 10.)

    So with two divided by zero, we could continue adding zero to itself forever, but still not get to two.
    Doesn't that make 2/0 impossible, not undefined?

    1/3 says you have 1 number of thirds.
    So with 3/0, would you have 3 number of "0ths"...
    I've never heard of a "0th" and I don't think they exist.
    0 (zero) of a pie is 0.
    1/3 of a pie is 1/3.
    1/2 of a pie is 1/2.
    1/1 of a pie is a whole.
    1/0 of a pie?

    Since dividing by zero (i.e. having a "zeroth" as a denominator), I think, is impossible, then there is no such thing as a zeroth of a pie.

    On a similar note, if you say you have zero halves (0/2),
    Or you say you have zero thirds (0/3)
    Then you have zero of those things.
    But you could also read those as zero divided by two (0/2)
    Or zero divided by three (0/3)

    Divide (per some online definition) also means calculate how many times one number goes into another.
    With 10/5, 5 goes into 10 two times. So 10/5 = 2.
    But with 0/2, how does 2 go into 0?
    (So, now it seems that 0/2 is undefined!) (Though, using a different meaning of division, we know we can have zero halves and that that simply equals 0!)
    (So that is part of the reason why it seems to me that we are not being consistent in our meaning and usage of division, and that is affecting how we think about dividing by zero.)

    (So, to restate my main point...)
    I think that dividing by zero, regardless of what you mean by "divide," is impossible.
    So next would be why this classic example meant to show that we can't divide by zero is actually flawed:
    a/0 = b

    Each side is multiplied by 0 in order to prepare to cancel out the zeros, like this:
    (a/0) x 0 = b x 0

    The problem with this is that a/0 is impossible, so when the zeros are "cancelled," what's really getting cancelled (on the left side) (along with the zero we added) is a part of an impossible number. If the number is impossible, then who are we to take a part of it out (the zero), and assume the remaining part is valid?
    It is the entire number a/0 that is impossible, not just the 0 at the bottom.

    So, if a/0 = b, and a/0 is impossible, then it seems that b is impossible too.

    I've yet to finish a review of imaginary numbers and how they are used, while keeping the above in mind.
    But I wonder if this is normal practice -- to pick and choose which parts of impossible things we use and which we don't. It seems that would really mess up our math.

    I'm curious how this applies to the study of more advanced math, science, and astronomy.
    Any thoughts on dividing by zero, or any of this?

    4/10/17 I corrected this post and would love to hear/read anyone's thoughts on it!
    (17 votes)
    Default Khan Academy avatar avatar for user
    • aqualine ultimate style avatar for user Jubjub Bird
      1. Separating into n pieces means that you need n new pieces to get the original piece. If you separate into 0 pieces means that you need 0 of the new piece/s to get to the original 1. In other words, you're trying to find- meaning define- a number that when multiplied by zero you get 1. Which is impossible.
      Note that I didn't call p/0 impossible. I called it impossible to define, which is why it's undefined.

      2. (No 2, because in the question it's just a definition)

      3. When you say "forever", it means that you would have to add an infinite amount of 0's to get to 2. But doing that would only give you 0, meaning you'll need to add a few more zeros- another infinite amount- and you'll stay at zero. And you keep going, and staying at zero. So yeah, it is impossible- impossible to define it's answer.

      4-5. Interesting way of looking at it. But that doesn't prove that it's not undefined.

      6. 0/p is 0, just like you say.

      7. 2 does go into 0, just like it goes into 1 and into 0.3. in fact, it goes into 0 better than into them! Into zero it goes exactly 0 times (a whole number), but into 1 it goes only a half time and into 0.3 it goes only 0.15 times (both are fractions).

      8-11. The flaw in your logic is that you aren't allowed to multiply by 0 when solving equations.

      12. I don't know what to say about this, because I don't know anything about imaginary numbers.

      13. I also don't know much about how this relates to more advanced math, science, and astronomy.

      4/26/17 I re-answered to the corrected post, and I still say the following:
      Thoughts. Mathematicians are really comfortable with it being undefined, and they really don't have any motivation to change that.
      (2 votes)
  • aqualine sapling style avatar for user phineartsacademy
    I'm just thinking about why x/x=1. Division is repeated subtraction. 10/5=2 cause 10-2-2-2-2-2=0. You subtract 2 five times from ten to get zero. You subtract ten one time from ten to get zero. You subtract zero zero times from zero to get zero. Therefore, 0/0=0. And then it works in reverse. 0*0=0. And the rule x/0=0 isn't broken. x/x=1 is ignored, but I already explained why. Also, it makes sense because zero isn't a number. Zero is a symbol that represents nothing. And nothing is undefined.
    (6 votes)
    Default Khan Academy avatar avatar for user
    • male robot donald style avatar for user Venkata
      Zero isn't a symbol. It is a number just how 1 or 45 is. Just how every number tells us the existence of something, zero tells us the existence of nothing.

      As for 0/0, you can't define it as 0 because a lot of problems arise, some of which are far beyond the current scope here, and will be dealt with in a really interesting branch of Mathematics called Calculus.

      And to add on, there's a reason why your logic doesn't work. So, okay, suppose we say 0/0 is 0. What is 0/1? That's also 0. What about 0/2? That's also zero. 0/1 = 0 says "I subtract 0 from 0 one time to get 0". 0/2 = 0 says "I subtract 0 from 0 twice to get 0". These are fair statements to make. But, subtracting 0 from 0, 0 times is a bit absurd, as that's similar to not doing anything at all. So, the idea of division being repeated subtraction breaks down.

      Finally, I'll leave you with this. 0/0 has a special name in the context of limits. It's called an indeterminate form, and it pops up a lot in advanced Math, along with other indeterminate forms like infinity/infinity. Just know that 0/0 can be any number you want it to be. Think of it like this. Let 0/0 = x. So, we have x * 0 = 0. Now, what value of x satisfies this equation? Well, any number does! That's why 0/0 can be any number, but in the future, you'll learn methods to narrow down what number it is, as you'll need it to solve problems!
      (10 votes)
  • stelly blue style avatar for user mk
    Imagine having 0 cookies to give among 0 friends. How many cookies would each person get? See, it doesn`t make sense. And the cookie monster is sad that you have 0 cookies, and you are sad that you have 0 friends.
    (10 votes)
    Default Khan Academy avatar avatar for user
  • winston default style avatar for user Redapple8787
    From what I understand, whenever mathematicians can't find a good answer for something, they leave it undefined. Do they leave things undefined quickly? How many undefined things are there in the world? Can undefined cases even exist?
    (6 votes)
    Default Khan Academy avatar avatar for user
    • mr pink green style avatar for user David Severin
      In real life, there is no reason to divide something by nothing, as it is undefined in math, it has no real applications in the real world. But in theory, it can be undefined. Similarly, infinity is also just a concept in math, it cannot have a real application either, even if you ask how many atoms are in the universe, it is a certain number even though it might be a very large number. There are not a lot of undefined things in math, but all of Geometry is based on three undefined terms, a point, line and plane none of which can exist in the real world, but without the theory of these, real world applications would be more difficult. You act like having something undefined is just an arbitrary thing mathematicians do, it is not at all.
      (8 votes)
  • blobby purple style avatar for user dooshbag
    i think 0/0 is 0 bcs if u have zero friends and zero cookies how many cookies will each friend get?
    ZERO!
    (5 votes)
    Default Khan Academy avatar avatar for user
    • stelly blue style avatar for user Kim Seidel
      The Multiplication Property of 0 tells us that any number times 0 = 0. If you turn that into an equation, you have:
      x*0 = 0, where "x" is any number.
      Solve the equation and your got x = 0/0. So, 0/0 can be made to equal any number. Since there is no single agreed upon solution, we say that 0/0 is indeterminate.
      (8 votes)
  • orange juice squid orange style avatar for user Brian Trzepacz
    If 0 x 5 = 0 then I divide both sides by 0, 0/0 = 5. So,
    0 x y = 0, y = 0/0, so nothing divided by nothing can be anything?
    (4 votes)
    Default Khan Academy avatar avatar for user
  • aqualine ultimate style avatar for user T.J.King2002
    Could 0/0 be equal to aleph-null?
    (6 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Harpreet Chandi
    What? At wouldn't 0/0 be left as indeterminate, not undefined?
    (3 votes)
    Default Khan Academy avatar avatar for user

Video transcript

In the last video we saw why when we take any non-zero number divided by zero why mathematicians have left that as being undefined. But it might have raised a question in your brain. What about zero divided by zero? Isn't there an argument why that could be defined? So we're gonna think about zero divided by zero. Well there's a couple of lines of reasoning here. One, you could start taking numbers closer and closer to zero and dividing them by themselves. So for example, you take 0.1 divided by 0.1. Well that's gonna be one. Let's get even closer to zero: 0.001 divided by 0.001. Well, that also equals one. Let's get super close to zero: 0.000001 divided by 0.000001. Well once again, that also equals one. And it didn't even matter whether these were positive or negative. I could make these negative and I'd still get the same result. Negative this thing divided by negative this thing still gets me to one. So based on this logic you might say, "Hey, well this seems like a pretty reasonable argument for zero divided by zero to be defined as being equal to one. " But someone could come along and say, "Well what happens if we divide zero by numbers closer and closer to zero; not a number by itself, but zero by smaller and smaller numbers, or numbers closer and closer to zero." And so they say, "For example, zero divided by 0.1, well that's just going to be zero. Zero divided by 0.001, well that's also going to be to zero. 0 divided by 0.000001 is also going to be equal to zero." And it didn't matter whether we were dividing by a positive or negative number. Make all of these negatives, you still get the same answer. So this line of reasoning tells you that it's completely legitimate, to think at least that maybe zero divided by zero could be equal to zero. And these are equally valid arguments. And because they're equally valid, and frankly neither of them is consistent with the rest of mathematics, once again mathematicians have left zero divided by zero as undefined.