Wednesday, January 22, 2014

3: 0/0 = 1, or 0, or undefined?

Consider the following:

0 ÷ 4 is clearly 0.
4 ÷ 0 is undefined.
How should we deal with 0 ÷ 0?

Is 0 ÷ 0 = 1 because anything divided by itself is 1?
Is 0 ÷ 0 = 0 because 0 divided by anything is 0?
Is 0 ÷ 0 indeterminate because you can't determine it?
Is 0 ÷ 0 undefined because you're not Neo?


  1. For me it is undefined. Division is essentially sharing therefore if you are not sharing or dividing nothing into nothing it will not work.

  2. It is indeterminate. Unlike 4/0 where there is no answer, 0/0 sometimes leads to an answer by taking a different path to the solution. Think limits and L'Hôpital's rule. This also makes sense when considering division as the inverse of multiplication. If a/b = c, then cb = a. If a and b are both 0, then c could be anything since c•b = a (c•0=0).

    My two cents, worth considerably fewer.

  3. It is not about preference. You don't get to choose. 0/0 is undefined. Simple fact. But the more interesting facts arise when you take limits whose form is 0/0 and then must be determined. But 0/0 is undefined for everyone. period.

  4. This site is about the discussion leading to a greater understanding. Otherwise, Wikipedia or WolframAlpha is a better choice.

  5. Responding to the last two posts: Wolfram says indeterminate as does wikipedia.
    I did not look at these prior to my original post.
    In a collegial, not-trying-to-be-a-jerk-sort-of way, can anonymous please cite a source indicating it is undefined.

  6. Saying that 0/0 is indeterminate is just shorthand for something more complicated. It doesn't literally mean that dividing zero by itself gives different answers depending on the situation. It means that when calculating a limit at c, sometimes plugging in c itself gets you the formal expression 0/0, and then you have to work a little harder to figure out the limit, if it exists. This formal expression 0/0 doesn't actually represent the (undefined) arithmetic operation 0/0.

    0/0 is undefined because defining it in a consistent way isn't possible without making big changes to the rest of arithmetic, which just isn't worth it. However, I sometimes hear the term "wheel theory" as an example of where division by zero is defined. I haven't really looked into it and it might be pretty interesting, maybe even applicable in the "real world", but it's necessarily pretty different from the real number system.

  7. Not buying what your selling. Forget limits, simply put:

    0 * x = 3 has no solutions. Since 0*x=3 implies x=3/0, x=3/0 has no solutions and therefore is undefined.

    0 * x = 0 has infinite solutions. 0*x=0 implies x=0/0 and therefore x has infinitely many solutions since 0*x-0 has infinitely many solutions. I just cannot determine x and making it unable to be determined or indeterminable.

  8. Illustration 2, proof by contradiction. Assume 0/0 is undefined.

    Take the following system of equations:
    x * y = 0
    x = 10

    Therefore y = 0.

    Solving the top equation for x yields x = 0 / y. Substituting for y yields x = 0/0. By the assumption, x is undefined; however, x isn't undefined, x = 10 yielding a contradiction. Therefore, the assumption must be disregarded and therefore 0/0 is not undefined. QED.

    You may not want to call 0/0 indeterminable but you can't call it undefined.

  9. But you're the one who brought up limits in the first place, and 0/0 is just a formal expression that arises in that context. Saying it's indeterminate says something about whatever LIMIT is under consideration, not about the actual NUMBER 0/0 (if it exists).

    It's not that I don't want to call it indeterminate, and in fact I often do, but I'm always talking about a limit and not a number when I do. If we're sticking to more basic concepts of whole number division, then 0/0 does not correspond to any real number, i.e., it's undefined. Even the wikipedia page says it has no defined value. The fact that it's also called an indeterminate form in more advanced math doesn't change this, and it doesn't mean that somehow 0/0 equals 1 and 2 and 3 and 4 and...

    Your proofs don't work because they both assume the existence of a multiplicative inverse of 0 (every time you divide by zero, you assume this), which doesn't exist (in usual real number arithmetic).

  10. Also, late Jan.-early Feb. Anonymous (me) isn't the same person as Jan. 22 Anonymous. I agree with him/her, but I'm not the same person trying to "sell" anything. Just giving my own two cents, worth an indeterminate amount :)

  11. When I meant forget limits, which I did indeed bring up originally, I was not faulting you for bringing them up. I only meant that I was taking a different approach. Also, thanks for indicating that you are a different anonymous.

    Maybe we are just arguing semantics. I agree with you that "0/0 does not correspond to any real number;" however, in the example I gave in illustration 2, it does correspond to a number in this case the number is 10. Therefore, in this example, it is defined. One just cannot determine it simply by looking at the 0/0. This is different than with any other numerator. Put anything else above 0, and the expression has no answer and is therefore not defined. No manipulation and no further deduction will produce an answer. Based on the fact that 0/0 behaves differently than anything_else/0, it seems illogical to define it in the same fashion but despite our belief the mathematicians are logical, we are illogical on occasion.

  12. I saw a "proof" the other day mentioning the following: 0/0=x then 0=0x and therefore x can be any #.
    But if you multiplied each side my 0 to "cancel out"..aren't you assuming that 0/0 is 1?