# Why 0 ÷ 0 = 1 **URL:** \ **Category:** Great Debates\ **Created:** [February 28, 2003, 8:14pm UTC](https://boards.straightdope.com/t/why-0-0-1/158219 "2003-02-28T20:14:42Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![Ramanujan](https://avatars.discourse-cdn.com/v4/letter/r/51bf81/32.png) [@Ramanujan](https://boards.straightdope.com/u/Ramanujan)\ **Post date:** [March 1, 2003, 5:30am UTC](https://boards.straightdope.com/t/why-0-0-1/158219/21 "2003-03-01T05:30:00Z")
the division operation is just not defined when the denominator is 0. it’s just that simple. no way to justify how it is defined is valid. limits are not valid, because taking the limit is not the same operation as division. most operators we use these days, including multiplication, division, addition, and subtraction, have their definitions in group and ring theory (abstract algebra). an operator is just a function. like all functions, it needs a definition. the division operator we most often talk about is defined from an ordered pair of reals to a real. if the second of the pair it operates on is 0, there is no number it yields. it’s just defined that way. asking what a number divided by 0 yields is like asking what radiohead marrying a prime number yields. it’s just nonsensical. the words and operators just aren’t defined to be meaningful with those terms. ---
**Author:** ![I\_Brian](https://avatars.discourse-cdn.com/v4/letter/i/c89c15/32.png) [@I\_Brian](https://boards.straightdope.com/u/I_Brian)\ **Post date:** [March 1, 2003, 8:19am UTC](https://boards.straightdope.com/t/why-0-0-1/158219/22 "2003-03-01T08:19:23Z")
Ah yes - I can see I was quite wrong - I blame it having been late in the evening for myself! 😉 Yes - undefined - quite right. ---
**Author:** ![AV8R](https://avatars.discourse-cdn.com/v4/letter/a/898d66/32.png) [@AV8R](https://boards.straightdope.com/u/AV8R)\ **Post date:** [March 1, 2003, 5:00pm UTC](https://boards.straightdope.com/t/why-0-0-1/158219/23 "2003-03-01T17:00:10Z")
On one of my first calculators (I think it was a TI-55), if you entered **0÷0** the answered returned as **1**. (of course, it was blinking, which indicated error, but I still thought it was kinda cool to have a calculater that produced a wrong answer). ---
**Author:** ![nogginhead](https://avatars.discourse-cdn.com/v4/letter/n/5e9695/32.png) [@nogginhead](https://boards.straightdope.com/u/nogginhead)\ **Post date:** [March 1, 2003, 7:55pm UTC](https://boards.straightdope.com/t/why-0-0-1/158219/24 "2003-03-01T19:55:15Z")
* * * Originally posted by nogginhead Take the limit of x/x as x -\> 0. ## x/x = 1, so x/x = 1 as x -\> 0. > [@](#): > > \*Originally posted by ultrafilter \* > \*\*Now take the limit of 1/x as x -\> 0. Then take the limit of x[sup]2[/sup]/x as x -\> 0. Which one is right? > \*\* Why are 1/x and x[sup]2[/sup]/x relevant? We’re not talking about _anything_ divided by 0, only 0. > [@](#): > > 0/0 is indeterminate. In order that 0/0 = a, it must be the case that 0a = 0. That works for any real number a. So under this argument, any number is right. Including 1. > [@](#): > > a/b is defined to be the _unique_ number c such that bc = a. If c is not unique, a/b is indeterminate. If c doesn’t exist (as in the case of 1/0), a/b is undefined. \*\* Excluding toy math systems (i.e. those without applications outside mathematics) I believe this definition only applies when b=0. So it’s a specific rule about what happens when you divide by 0, and is thus an arbirtrary rule. I can just as easily make an arbitrary rule that x/0 = +infty if x\>0, = -infty if x \< 0, and =1 if x = 0. We also define 0! =1, for example, because it’s convenient. ---
**Author:** ![december](https://avatars.discourse-cdn.com/v4/letter/d/838e76/32.png) [@december](https://boards.straightdope.com/u/december)\ **Post date:** [March 1, 2003, 8:10pm UTC](https://boards.straightdope.com/t/why-0-0-1/158219/25 "2003-03-01T20:10:51Z")
> [@](#): > > \*Originally posted by nogginhead \* > I can just as easily make an arbitrary rule that x/0 = +infty if x\>0, = -infty if x \< 0, and =1 if x = 0. > > We also define 0! =1, for example, because it’s convenient. However, defining 0! = 1 really is convenient. The usual formulas work with this definition. E.g., (n+1)!/n! = n+1, for all positive integer values of n, including n=0. Defining _x/0 = +infty_ doesn’t actually mean anything, because _+infty_ isn’t a real number. If you want to extend the real number system to include +infty and -infty, you will lose a good many relationships. Or, you would have to have two classes of numbers, the finite ones and the infinite ones. Then you could try to derive different threorems for various subsets. The bottom line is that defining _0/1 = +infty_ is not convenient. _A little learning is a dangerous thing; drink deep, or taste not the Pierian spring: there shallow draughts intoxicate the brain, and drinking largely sobers us again._ Alexander Pope ---
**Author:** ![imthjckaz](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/imthjckaz/32/3223_2.png) [@imthjckaz](https://boards.straightdope.com/u/imthjckaz)\ **Post date:** [March 1, 2003, 8:44pm UTC](https://boards.straightdope.com/t/why-0-0-1/158219/26 "2003-03-01T20:44:53Z")
I was just wondering about the marbles everyone is talking about, because I lost mine and I think that maybe they might be mine. ---
**Author:** ![nogginhead](https://avatars.discourse-cdn.com/v4/letter/n/5e9695/32.png) [@nogginhead](https://boards.straightdope.com/u/nogginhead)\ **Post date:** [March 1, 2003, 9:10pm UTC](https://boards.straightdope.com/t/why-0-0-1/158219/27 "2003-03-01T21:10:33Z")
> [@](#): > > _december_ > The bottom line is that defining 0/1 = +infty is not convenient. > > A little learning is a dangerous thing; > drink deep, or taste not the Pierian spring: > there shallow draughts intoxicate the brain, > and drinking largely sobers us again. > > Alexander Pope OK, december, as long as we’re sniping with quotations, how about: > [@](#): > > _Oscar Wilde_ > There are two ways of disliking poetry; one way is to dislike > it, the other is to read Pope. and, more to the point > [@](#): > > _Shakespeare_ > There are more things in heaven and earth, > Horatio, Than are dreamt of in your philosophy My point was that _if_ one were to find it convenient to define 0/0 as 1, it would be as acceptable an arbitrary rule as to define it as undefined. Same for x/0=infty, though that wasn’t what I was driving at especially. ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [March 1, 2003, 9:10pm UTC](https://boards.straightdope.com/t/why-0-0-1/158219/28 "2003-03-01T21:10:53Z")
> [@](#): > > \*Originally posted by nogginhead \* > **Why are 1/x and x[sup]2[/sup]/x relevant? We’re not talking about _anything_ divided by 0, only 0.** Fine. So what’s the limit of 0/x as x approaches 0? > [@](#): > > \*\* > So under this argument, any number is right. Including 1. \*\* Yes, _any number_. That’s the point. 0/0 is undefined because there’s no _unique_ solution to the equation 0\*x=0. An operation which doesn’t yield a unique result is not well defined. > [@](#): > > \*\* > Excluding toy math systems (i.e. those without applications outside mathematics) I believe this definition only applies when b=0.\*\* It’s the mathematical definition of division, I’m afraid. You believe wrong. Well, unless you’re referring to the more common-sense, “how many A’s in a B” definition of division…which, as **Mangetout** and **astorian** have pointed out, _still_ doesn’t define 0/0. The definition using limits doesn’t let you define it either, 'cause x/x and 0/x don’t have the same limit as x approaches 0. And that’s it; we’re out of definitions of division. Except, of course, the arbitrary one: > [@](#): > > \*\* > So it’s a specific rule about what happens when you divide by 0, and is thus an arbirtrary rule. I can just as easily make an arbitrary rule that x/0 = +infty if x\>0, = -infty if x \< 0, and =1 if x = 0.\*\* Except that as **silverfish** demonstrated, if you assume 0/0=1 then you can “prove” that x=1 _for any x_, which makes your arbitrary definition about as useful as hammer made of sponge cake. I feel kind of silly just coming in here and pointing to everyone else’s posts, but it seems the points need to be repeated. ---
**Author:** ![RSC0318](https://avatars.discourse-cdn.com/v4/letter/r/a587f6/32.png) [@RSC0318](https://boards.straightdope.com/u/RSC0318)\ **Post date:** [March 1, 2003, 9:31pm UTC](https://boards.straightdope.com/t/why-0-0-1/158219/29 "2003-03-01T21:31:20Z")
I haven’t read many of the replys on this post, buthere are my thoughts. Not necessarily going by what I learned in school, but anyways. 0= a number, but it holds no value (i.e. it is neutral) 0/0=0. Yet it is still a number (and without zero, you cannot get to one, because it is at the beginning of the number scale (excluding negative integers in this discussion)). 0/0=1. Does this make sense to anyone else?😕 ---
**Author:** ![nogginhead](https://avatars.discourse-cdn.com/v4/letter/n/5e9695/32.png) [@nogginhead](https://boards.straightdope.com/u/nogginhead)\ **Post date:** [March 1, 2003, 9:34pm UTC](https://boards.straightdope.com/t/why-0-0-1/158219/30 "2003-03-01T21:34:42Z")
… officially abandoning line of argument … Just to say to orbifold that aspect of the definition I was referring to is the part about uniqueness/definedness, which only exists to deal with the 0 problem. my point is that just defining something away doesn’t make it not exist. ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [March 1, 2003, 9:49pm UTC](https://boards.straightdope.com/t/why-0-0-1/158219/31 "2003-03-01T21:49:39Z")
That’s a reasonable point **nogginhead** , except that we’re not “defining something away”. There’s no reasonable definition of 0/0 in the first place. Saying that 0/0 is undefined isn’t an arbitrary decision, it’s a recognition of the fact the definition of division we already have for other numbers just doesn’t work for 0/0, and there are good reasons why we can’t make it work. ---
**Author:** ![nogginhead](https://avatars.discourse-cdn.com/v4/letter/n/5e9695/32.png) [@nogginhead](https://boards.straightdope.com/u/nogginhead)\ **Post date:** [March 1, 2003, 9:57pm UTC](https://boards.straightdope.com/t/why-0-0-1/158219/32 "2003-03-01T21:57:37Z")
> [@](#): > > \*Originally posted by Orbifold \* > \*\*That’s a reasonable point **nogginhead** , except that we’re not “defining something away”. There’s no reasonable definition of 0/0 in the first place. Saying that 0/0 is undefined isn’t an arbitrary decision, it’s a recognition of the fact the definition of division we already have for other numbers just doesn’t work for 0/0, and there are good reasons why we can’t make it work. \*\* I give in already. It’s best to define it as undefined. ---
**Author:** ![Canadagirl](https://avatars.discourse-cdn.com/v4/letter/c/f0a364/32.png) [@Canadagirl](https://boards.straightdope.com/u/Canadagirl)\ **Post date:** [March 2, 2003, 2:01am UTC](https://boards.straightdope.com/t/why-0-0-1/158219/33 "2003-03-02T02:01:57Z")
Excuse me for asking, but how the hell do you have zero groups of anything, even if it’s zero groups of zero? I would definately go with the undefined that I was taught, because it makes **sense**. ---
**Author:** ![verybdog](https://avatars.discourse-cdn.com/v4/letter/v/49beb7/32.png) [@verybdog](https://boards.straightdope.com/u/verybdog)\ **Post date:** [March 2, 2003, 4:21am UTC](https://boards.straightdope.com/t/why-0-0-1/158219/34 "2003-03-02T04:21:40Z")
> [@](#): > > \*Originally posted by 0rbytal \* > \*\* > x÷0=infinity > And any number divided by itself is 1. > x÷x=1 > > Also look at it this way: 0÷0 = 0 × (1/0) = zeroes cancel = 1! > > If that’s not enough find : > lim sin(x) > [sup]x-›0 X[/sup] > > By direct substitution you get sin(0)÷0 = 0÷0 but it = 1! > > Enough proof? \*\* x÷x=1, yes, but not for 0÷0 which is = 0 therefore not = 1; 0÷0 = 0 × (1/0) = zeroes cancel = 1! ===\> No way. You can’t treat 0 as an egg that is physical. 0 means nothing, therefore the operation “0 × (1/0)” is basically is “0 x something” which is 0. When x=0.001, Sinx = 0.84147098, therefore sinx/x = 0.99999983 = 1. ---
**Author:** ![emarkp](https://avatars.discourse-cdn.com/v4/letter/e/3be4f8/32.png) [@emarkp](https://boards.straightdope.com/u/emarkp)\ **Post date:** [March 2, 2003, 8:04am UTC](https://boards.straightdope.com/t/why-0-0-1/158219/35 "2003-03-02T08:04:30Z")
The amazing thing to me is using a limit to attempt to show what happens when you divide by zero. Especially since limits were introduced to precisely what happens when **not** at a specific point. lim x-\>0 specifically refers to x =/= 0. Sheesh. ---
**Author:** ![John\_Kentzel-Griffin](https://avatars.discourse-cdn.com/v4/letter/j/ccd318/32.png) [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)\ **Post date:** [March 2, 2003, 10:45am UTC](https://boards.straightdope.com/t/why-0-0-1/158219/36 "2003-03-02T10:45:52Z")
**emarkp** , you bring up a good point. The only reason to bring up limits would be to _suggest_ a reasonable definition of 0/0. Since x/x and 2x/x reach different limits as x approaches 0, this does not get us anywhere. The bottom line is: a/b is the unique number c that satisfies a = b_c. For b = 0, there is either no value for c (for a non-zero) or no unique number (for a = 0) that satisfies a = b_c. There is no useful definiton if a/0. ---
**Author:** ![jshore](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@jshore](https://boards.straightdope.com/u/jshore)\ **Post date:** [March 2, 2003, 6:28pm UTC](https://boards.straightdope.com/t/why-0-0-1/158219/37 "2003-03-02T18:28:07Z")
> [@](#): > > \*Originally posted by emarkp \* > The amazing thing to me is using a limit to attempt to show what happens when you divide by zero. Especially since limits were introduced to precisely what happens when **not** at a specific point. Well, this is a truly absurb way of looking at things! I would say limits were introduced precisely as a way of finding out what is happening at a specific point when it would be difficult to do so directly; we do it by getting arbitrarily close to the point and see what happens! Thus limits are extremely useful in this case because while we can’t understand 0/0, we can understand it in the sense where the numerator and denominator are continuous functions of x [call them f(x) and g(x)] that approach 0 as x-\>0. That limit leads us to the conclusion that we should consider 0/0 to be indeterminate we can find functions f(x)/g(x) that give any value in the limit that x-\>0. ---
**Author:** ![jshore](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@jshore](https://boards.straightdope.com/u/jshore)\ **Post date:** [March 2, 2003, 6:29pm UTC](https://boards.straightdope.com/t/why-0-0-1/158219/38 "2003-03-02T18:29:35Z")
> [@](#): > > \*Originally posted by jshore \* > That limit leads us to the conclusion that we should consider 0/0 to be indeterminate we can find functions f(x)/g(x) that give any value in the limit that x-\>0. There should be a “because” before the “we can”. ---
**Author:** ![John\_Kentzel-Griffin](https://avatars.discourse-cdn.com/v4/letter/j/ccd318/32.png) [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)\ **Post date:** [March 2, 2003, 10:16pm UTC](https://boards.straightdope.com/t/why-0-0-1/158219/39 "2003-03-02T22:16:56Z")
> [@](#): > > \*Originally posted by \*\*\*jshore \*\* > Well, this is a truly absurb way of looking at things! I would say limits were introduced precisely as a way of finding out what is happening at a specific point when it would be difficult to do so directly; we do it by getting arbitrarily close to the point and see what happens! No. You are not looking at what happens _at_ a specific point. You are looking at what happens “near” a point. The fact that lim (x-\>0) sin x/x = 1 does not mean that sin 0/0 = 1. Look at the definition of limit. If the limit is taken as x approaches a, the value at a is specifically excluded. It is interesting that limits are brought up. The primary motivation for using limits in calculus is to evaluate lim (delta-x-\>0) delta-y/delta-x. Delta-y and delta-x both approach zero as delta-x approaches 0. The fact that this limit is (with qualifications) defined, but 0/0 is not tells me that limits are no help here. ---
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [March 2, 2003, 11:56pm UTC](https://boards.straightdope.com/t/why-0-0-1/158219/40 "2003-03-02T23:56:19Z")
Let’s consider lim x^3- 8 [sup]x-\>2[/sup] x-2 [Previous page](https://boards.straightdope.com/t/why-0-0-1/158219.md?page=1) [Next page](https://boards.straightdope.com/t/why-0-0-1/158219.md?page=3)