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

Undefined & indeterminate expressions

Revisiting the problems of dividing any number by zero and dividing zero by zero. Using general mathematical considerations, we see why those are undefined and indeterminate problems. Created by Sal Khan.

Want to join the conversation?

  • purple pi purple style avatar for user Allyson P
    If you have 0/0, how come it isn't 1? If you cancel it out you will get 1, right? Also, lets say you have 0 apples, and 0 people. So for each (nonexistent) person you have a (nonexistent) apple. I'm so confused...
    (238 votes)
    Default Khan Academy avatar avatar for user
  • leafers seed style avatar for user sai krishna
    what is the difference between undefined and indeterminate?
    (91 votes)
    Default Khan Academy avatar avatar for user
  • piceratops ultimate style avatar for user Firedrake969
    So what would ∞*0 be?
    (25 votes)
    Default Khan Academy avatar avatar for user
  • leaf grey style avatar for user Conad
    Couldn't you just simplify 0/0=1, because they would just cancel each other out?
    (24 votes)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user tesseraktik
      One of the biggest issues with this reasoning is that you'd have to introduce exceptions to one of the most cherished properties of multiplication: It's associative.

      What this means is that for any values a, b and c, it holds true that (a*b)c = a(b*c)
      That is, the order of multiplication does not affect the result.

      If you introduce 0/0 = 1, you'll get situations such as this:

      4 * (0 * 1/0) = 4 * 0/0 = 4*1 = 4
      (4 * 0) * 1/0 = 0 * 1/0 = 0/0 = 1
      So, you either give up associativity, or accept that 4 = 1. Neither of these options is very tempting.

      It's also worth remembering that the very idea behind division is that it undoes multiplication; if you multiply X by 4, you can then divide by 4 and end up where you started.
      Division by zero does not accomplish this: Because X*0 = 0 for all values of X, 0/0 would have to take on different values each time in order to be an anti-multiplier, and that's not allowed.

      Lastly, one must ask oneself: What would the use of adding 0/0 = 1 be?
      If it is useful to your purposes, then, sure, use it. However, usually when you have a need to do anything similar to dividing zero by zero (such as when calculating limits in calculus), you usually require much more complex methods.

      We've also established that this would constitute an exception to the associative nature of multiplication, and to the rule that division undoes multiplication. Since both of these properties are used in a lot of mathematical proofs, these proofs would be invalid whenever they involved a factor 0/0; another reason why defining 0/0 just isn't very useful a lot of the time.
      (27 votes)
  • aqualine ultimate style avatar for user sarikap09609
    How does this relate to algebra?
    (11 votes)
    Default Khan Academy avatar avatar for user
    • leaf blue style avatar for user Stefen
      Great Question!
      As I am sure you know by now, algebra has lots of rules, just like any game. To play the game, you follow the rules. The rules of the game of math are so constructed that if you follow them, you will ALWAYS arrive a TRUE result.
      One of the rules you learned was that division by zero is not allowed. We say that division by zero is undefined. WHY? It is because if you do divide something by zero, you can arrive at an UNTRUE result. Most of the time this happens when it is not obvious that we are dividing by zero. How so? Follow this simple example.
      Suppose we have two variables called a and b.
      Let a = 1 and let b = 1
      Obviously then, a = b is true since a=1 and b = 1 thus a = b means 1 = 1, which is true.

      Now multiply both sides of the equation a = b by a and we get:
      a·a = a·b, and we can rewrite that as a² = a·b

      Now let us subtract from both sides of the equation so a²=a·b becomes:
      a² - b² = a·b - b²

      Now we can factor the expressions on each side of the equals sign.
      I hope you recognize that the expression a² - b² is a difference of squares and has a well defined factor form : a² - b² = (a + b)(a - b)
      We can factor the expression a·b - b² like so: a·b - b² = b(a - b)

      To summarize so far:
      1) a = b // given
      2) a² = a·b // we multiplied both sides of 1) by a
      3) a² - b² = a·b - b² // we subtracted from both sides of 2)
      4) (a + b)(a - b) = b(a - b) // we factored the expressions in 3)

      Now, since there is an (a - b) term on both sides of the equation, we can get rid of it by dividing both sides of the equation by (a - b) like so:
      5) (a + b)(a - b)/(a - b) = b(a - b)/(a - b), giving
      6) (a + b) = b

      Now, remember we said at the beginning that a = 1 and b = 1
      Well let's substitute those values into 6)
      (a + b) = b means (1 + 1) = 1 which means 2 = 1 But that is not true

      So what happened?
      Well, since we said a = 1 and b = 1, that meant that when we divided both sides of the equation in step 5) by (a - b) we were actually dividing by zero! since if a = 1 and b = 1, then a - b = 1 - 1 = 0. Therefore, Step 5 was an error, that is, it was an illegal step since it violated the do not divide by zero rule.

      To move forward in algebra we must be aware at all times if the forms we are working with are undefined or indeterminate. What we did in step 5) turned the absolutely fine equation we had in step 4) into an indeterminate form that gave us the wacky 2 = 1 result, which, I hope you agree, is not true.

      I hope this helped a bit - I know it was a bit long winded.
      This topic becomes more and more important as you get further into algebra.
      (50 votes)
  • aqualine seed style avatar for user Safayet Khan
    What is the Difference Between Undefined and Indeterminate ? It seems to me they are the same things ?
    (6 votes)
    Default Khan Academy avatar avatar for user
    • leaf blue style avatar for user Matthew Daly
      The difference is subtle. When I say that a/b = c, that is another way of saying that c is the unique number that satisfies the equation bc = a. For instance 6/2 = 3, because 2 * 3 = 6.

      Given that, you can see the difference between 1/0 and 0/0. We say that 1/0 is undefined because there is no number c that satisfies 0c = 1. On the other hand, any number c satisfies 0c = 0 and there's no reason to choose one over any of the others, so we say that 0/0 is indeterminate.
      (40 votes)
  • aqualine ultimate style avatar for user marciastewart22062
    To quote Siri from Apple "image you have 0 cookies and zero friend, and you want to divide those cookies evenly. See? It doesn't make sense. Cookie Monster is sad that there are no cookies, and you are sad because because you have no friends"
    (23 votes)
    Default Khan Academy avatar avatar for user
  • aqualine ultimate style avatar for user stahmaseb0001
    nobody ever comments about how good his drawings are :(
    (16 votes)
    Default Khan Academy avatar avatar for user
  • piceratops tree style avatar for user Adorah Allard
    If any non-zero number divided by zero equals zero, why doesn't 0/0 equal zero?
    (8 votes)
    Default Khan Academy avatar avatar for user
    • leafers ultimate style avatar for user Rohan
      Your first statement is incorrect. Anything divided by zero is impossible. Take this case:5=20/4 This means that 4*5=20 yes? If this is the same with zero, here is the equation: x/0=0. X is any number that is not 0. If we rearrange the equation like the one before: x=0*0 BUT this cannot be true. X is any number that is not zero so x could be 1, 2, 3, 4,... but it's not zero. but 0 times zero can't equal 1, 2, 3, 4,... It only equals 0. So mathematicians left it undefined. 0/0 is infinite. If we make another equation, 0/0=x, 0*x=0 so ANYTHING times zero is 0. So the answer is infinitely many answers.
      (15 votes)
  • blobby green style avatar for user Michael Somerby
    Is it possible that zero is not a number?
    Mathematics is a system of the relationships of numbers, real or imaginary.
    Zero is not a number. It is the absence of a number.
    (10 votes)
    Default Khan Academy avatar avatar for user
    • male robot hal style avatar for user Eric McCormick
      In truth, you are nudging up against one of the most powerful and changing ideas in mathematics. For a ling time we used Roman numerals: I, IV, LMXII, and so on. They added nothing to this, AKA: zero, and it changed math completely. Allowing for nothing opens up worlds of possibilities in math. Now, this new and special nothing has been given a special place. I is part of the number sets, but it does not behave like other numbers. So it is a number because we have defined it as a number, but as you point out it does not really play nice like other numbers so it sort of is not a number.

      Oh, and as for the relationships part, 0 does have relationships for both real and imaginary. It just has a few special cases, just like i, e, π, and many others do. The only place 0 gets funky is in division, which makes it special, but that does not kick it back out of being a number. It just makes it a special number.
      (10 votes)

Video transcript

So, once again let's think of yourself as some type of ancient philosopher/mathematician, who is trying to extend mathematics as much as possible and try to make sure that you're not being lazy and leaving things undefined, when you might be able to define them. Whenever you start extending mathematics, especially in the realm of multiplication and division, there are few things that you hold dear to. You feel that if you define some type of division operation, that needs to be undone by multiplication; this is close to your heart. So you assume... You want to assume... You would like to assume that any type of division operation, if you start with some number and if you divide with a number over which... - division by that number is defined - so when I divide by some number and then multiply by that same number that this should get me this original number right over here, this should give me x right over here. And this happens when we just multiply and divide with regular numbers. If I get 3 divided by 2 times 2, that's gonna get me 3. If I say 10 divided by 5 times 5, that's going to get me 10. The other things that I want to assume... - and this is very close to my heart - I feel that any type of definitions I make have to be constant with the idea x*0 has to be 0 or any x. So these are close to my heart. I wanna extend mathematics. These two things are things that cannot be contradicted, cannot be untrue. Now, that out of the way. You wanna start exploring the divide-by-0 question. So the first thing that you say: "Well, let me just try to define it." So let's start, let's assume that I have, so this is... So let's make a further assumption... that x is some non-zero number. Let's just say, well,maybe the best way of finding out what x divided by 0 should be, how I should divide it, let's just assume there is define, and then come up with any results that there might be, there might be a resolve for. So let's say that x divided by 0 is equal to k. Well, if this is true and if we are defining what it means to divide by zero, then we are assuming that if we multiply by zero, we'll get our original number right over here. This is something that we are not willing to contradict. So let's see what happens: x divided by 0 is equal to k. On the left hand side we have a divide by zero and than multiplied by zero. Well then if two things are equal, if I do something to one thing inorder for them to stay equal, I have to do it to the other thing. This has to be equal to that. I have to multiply the left hand AND the right hand side by zero. Well, then by this assumption that I am never willing to give up, this left hand side right over here, must be equal to x. And by this assumption right over here, that I am not willing to give up, This right hand side right over here must be equal to 0. But I just hit a contradiction! I assume that x does not equal to 0, and now I am being forced to say that x=0. And I am not willing to give up the idea, I am not willing to give up either of these ideas. I am defining what it means to divide by zero. Or if I am defining what it means to divide by anything... ...that if I then multiply by that something, that I should get my original number. And I am not willing to give up the idea that anything times 0 is 0. So all of these things... The only thing that I can give up is this right over here. And I'll say, well, I guess k will have to stay undefined. This whole contradiction happened because I attempted to define what x/0 is. Now that out of the way... OK... This was a situation when x does not equal zero. But what about when x DOES equal zero. So let's think about that a little bit. And once again, I will try to define it. So I will assume... that 0 divided by 0 is equal to some number. Well once again, so let's say it is equal to k again. And so, once again... we are trying to do the same logic, so we'll write 0/0 is equal to k. Actually, let me colourcode these zeros. This will be a magenta zero and this is a blue zero right over here. And once again, I am not willing to give up the idea that if I start with a number x, I divide it by something over which division is defined, and then I multiply by that something, I should get my original x again. I can't give this up. Otherwise it doesn't seem like a good definition for the division. So what I am gonna do - I am gonna multiply the left-hand side times 0 and by this property that I am not willing to give up, the left-hand side should simplify to this magenta zero. It should simplify to this over here. But once again, anything I do to once side of the equation, inorder for the equation to hold true, I need to the other side of the equation. And these two were equal beforehand. Any operation I do to this inorder for it to still be equal , I need to do to that. So let me multiply the right-hand side by zero. So the left I get 0, I just get this magenta 0, and on the right I could just write the zero here, but I won't simplify it. I get k times 0. Well, this I see right over here... This actually is not a contradiction. This actually is true for any k, This is one of the core assumptions that I've made in my mathematics that I am not willing to give up. So this is true True for any k. It's not a contradiction. But the problem here is I wanna come up with the k, I'd like a resolve for a k. It would not be nice if this turned out to be 0. if this turned out to be one, or if this turned out to be negative one But now I see, given the assumptions right here this could be ANY... this could be absolutely any k I cannot determine what k this should be This could be a hundred thousand, this could be 75, it could be anything true for any k I cannot determine what k this should be and that's why when you get a little bit more nuance in early math people will say, well 0 divided by 0, well we don't know what that's gonna be there's no consistent answer there so we're just going to call it undefined there's no good answer that seems better than any other answer but now we see a little bit nuance here one divided by zero... you just couldn't define it it led to direct contradictions zero divided by zero... it could be anything you can't determine it and so that's why, when you do higher level math and you'll often hear this when you take a calculus course we see that zero divided by zero is indeterminate