# Zero divided by zero **URL:** \ **Category:** Factual Questions\ **Created:** [October 4, 2004, 8:02pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516 "2004-10-04T20:02:51Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![bot0004](https://avatars.discourse-cdn.com/v4/letter/b/858c86/32.png) [@bot0004](https://boards.straightdope.com/u/bot0004)\ **Post date:** [October 4, 2004, 8:02pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/1 "2004-10-04T20:02:51Z")
What is zero divided by zero anyways? I tried researching it and got a slew of answers, I hope someone can solve this for me. ---
**Author:** ![zev\_steinhardt](https://avatars.discourse-cdn.com/v4/letter/z/97f17d/32.png) [@zev\_steinhardt](https://boards.straightdope.com/u/zev_steinhardt)\ **Post date:** [October 4, 2004, 8:07pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/2 "2004-10-04T20:07:06Z")
> [@bot0004](#): > > What is zero divided by zero anyways? I tried researching it and got a slew of answers, I hope someone can solve this for me. I believe the answer would be the same as any other number divided by 0 - undefined. Consider: 6/0 == you have six items and you put 0 in each box. How many boxes do you use? Likewise: 0/0 == you have zero items and put zero in each box. How many boxes do you use? Zev Steinhardt ---
**Author:** ![Gary\_T](https://avatars.discourse-cdn.com/v4/letter/g/3ec8ea/32.png) [@Gary\_T](https://boards.straightdope.com/u/Gary_T)\ **Post date:** [October 4, 2004, 8:12pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/3 "2004-10-04T20:12:52Z")
> [@bot0004](#): > > What is zero divided by zero anyways? It isn’t anything. As **z** noted, it is undefined. There is no answer, because no answer exists. For more info and explanation, Google on \*_“division by zero” undefined_ . ---
**Author:** ![Gary\_T](https://avatars.discourse-cdn.com/v4/letter/g/3ec8ea/32.png) [@Gary\_T](https://boards.straightdope.com/u/Gary_T)\ **Post date:** [October 4, 2004, 8:14pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/4 "2004-10-04T20:14:20Z")
Let’s try this again: It isn’t anything. As **zev** noted, it is undefined. There is no answer, because no answer exists. For more info and explanation, Google on _“division by zero” undefined_ . ---
**Author:** ![Crafter\_Man](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/crafter_man/32/458_2.png) [@Crafter\_Man](https://boards.straightdope.com/u/Crafter_Man)\ **Post date:** [October 4, 2004, 8:14pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/5 "2004-10-04T20:14:58Z")
While 0/0 would certainly be undefined, what about **lim [sub]a → 0[/sub] a/a** Wouldn’t the answer be 1? ---
**Author:** ![bot0004](https://avatars.discourse-cdn.com/v4/letter/b/858c86/32.png) [@bot0004](https://boards.straightdope.com/u/bot0004)\ **Post date:** [October 4, 2004, 8:15pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/6 "2004-10-04T20:15:02Z")
But doesn’t nothing go into nothing 1 time? 3 \* x/4 \* x Normally x can equal any number since the x’s cancel out right? So why is this any different from 0? ---
**Author:** ![Punoqllads](https://avatars.discourse-cdn.com/v4/letter/p/d2c977/32.png) [@Punoqllads](https://boards.straightdope.com/u/Punoqllads)\ **Post date:** [October 4, 2004, 8:18pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/7 "2004-10-04T20:18:18Z")
> [@Crafter\_Man](#): > > While 0/0 would certainly be undefined, what about > > **lim [sub]a → 0[/sub] a/a** > > Wouldn’t the answer be 1? yep, and **lim[sub]a-\>0[/sub] 0/a = 0** and **lim[sub]a-\>0[/sub] 2a/a = 2** ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [October 4, 2004, 8:18pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/8 "2004-10-04T20:18:20Z")
Generally, when you say “X divided by Y gives Z”, you mean that Z is the unique number such that Z times Y is X. So, if “0 divided by 0 gives Z”, then z must be the unique number such that Z times 0 is 0. However, Z times 0 = 0 is true for all numbers (integers, fractions, real numbers, complex – it doesn’t matter). Therefore there is no **unique** number Z, and 0 divided by 0 has to be meaningless. ---
**Author:** ![Complex\_Conjugate](https://avatars.discourse-cdn.com/v4/letter/c/3bc359/32.png) [@Complex\_Conjugate](https://boards.straightdope.com/u/Complex_Conjugate)\ **Post date:** [October 4, 2004, 8:19pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/9 "2004-10-04T20:19:54Z")
Dvsion is usually defined as: a/b = a\*b[sup]-1[/sup] where b[sup]-1[/sup] is the multiplicative inverse of b. An mathematical object containing an additive identity (0) and a muplicative inverse (for all a !+ 0) is called a field (the real numbers are a field); a field cannot contain a muplicative inverse, so divison by zero is undefined. ---
**Author:** ![Complex\_Conjugate](https://avatars.discourse-cdn.com/v4/letter/c/3bc359/32.png) [@Complex\_Conjugate](https://boards.straightdope.com/u/Complex_Conjugate)\ **Post date:** [October 4, 2004, 8:20pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/10 "2004-10-04T20:20:56Z")
That should be for all a != 0. ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [October 4, 2004, 8:28pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/11 "2004-10-04T20:28:45Z")
Wikipedia on [Division by zero.](http://en.wikipedia.org/wiki/Division_by_zero) ---
**Author:** ![Scarlett67](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/scarlett67/32/7387_2.png) [@Scarlett67](https://boards.straightdope.com/u/Scarlett67)\ **Post date:** [October 4, 2004, 8:30pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/12 "2004-10-04T20:30:48Z")
> [@Giles](#): > > Generally, when you say “X divided by Y gives Z”, you mean that Z is the unique number such that Z times Y is X. > > So, if “0 divided by 0 gives Z”, then z must be the unique number such that Z times 0 is 0. > > However, Z times 0 = 0 is true for all numbers (integers, fractions, real numbers, complex – it doesn’t matter). > > Therefore there is no **unique** number Z, and 0 divided by 0 has to be meaningless. This is the explanation I have seen, with the term for 0/0 being “indeterminate” as opposed to “undefined.” ---
**Author:** ![Complex\_Conjugate](https://avatars.discourse-cdn.com/v4/letter/c/3bc359/32.png) [@Complex\_Conjugate](https://boards.straightdope.com/u/Complex_Conjugate)\ **Post date:** [October 4, 2004, 8:35pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/13 "2004-10-04T20:35:26Z")
Argh! the last line should be: " a filed cannot contain a multiplicative inverse for zero". ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [October 4, 2004, 8:42pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/14 "2004-10-04T20:42:13Z")
> [@Scarlett67](#): > > This is the explanation I have seen, with the term for 0/0 being “indeterminate” as opposed to “undefined.” This implicitly is assuming a notion of limit. 0 exists in any field independant of a topological structure sufficient to consider the limits some posters are suggesting. 0/0 is undefined because 0 is by definition noninvertible in _any_ ring. ---
**Author:** ![nivlac](https://avatars.discourse-cdn.com/v4/letter/n/3bc359/32.png) [@nivlac](https://boards.straightdope.com/u/nivlac)\ **Post date:** [October 4, 2004, 8:47pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/15 "2004-10-04T20:47:16Z")
Yes 0/0 is undefined. But no one said why. 0/0 is undefined because if it were defined to be any number, then you can literally prove that any number is equal to any other number. And we can’t let that happen, can we? ---
**Author:** ![Complex\_Conjugate](https://avatars.discourse-cdn.com/v4/letter/c/3bc359/32.png) [@Complex\_Conjugate](https://boards.straightdope.com/u/Complex_Conjugate)\ **Post date:** [October 4, 2004, 8:47pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/16 "2004-10-04T20:47:24Z")
> [@Mathochist](#): > > This implicitly is assuming a notion of limit. 0 exists in any field independant of a topological structure sufficient to consider the limits some posters are suggesting. > > 0/0 is undefined because 0 is by definition noninvertible in _any_ ring. So it appies to rings in general? I’m guessing that’s because a\*0 = 0 in any ring. ---
**Author:** ![Complex\_Conjugate](https://avatars.discourse-cdn.com/v4/letter/c/3bc359/32.png) [@Complex\_Conjugate](https://boards.straightdope.com/u/Complex_Conjugate)\ **Post date:** [October 4, 2004, 8:49pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/17 "2004-10-04T20:49:32Z")
> [@nivlac](#): > > Yes 0/0 is undefined. But no one said why. 0/0 is undefined because if it were defined to be any number, then you can literally prove that any number is equal to any other number. And we can’t let that happen, can we? It’s undefined because divison by zero cannot be defined in a ring full-stop. ---
**Author:** ![ZipperJJ](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/zipperjj/32/211_2.png) [@ZipperJJ](https://boards.straightdope.com/u/ZipperJJ)\ **Post date:** [October 4, 2004, 8:52pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/18 "2004-10-04T20:52:24Z")
“My brain hurts” ZipperJJ (mathless journalism student and former Algebra II dropout) ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [October 4, 2004, 9:02pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/19 "2004-10-04T21:02:34Z")
> [@Complex Conjugate](#): > > It’s undefined because divison by zero cannot be defined in a ring full-stop. Well, actually I may have to back up from what I said before. Under some circumstances it is beneficial to allow the trivial ring with one element. Note that most axiom systems for rings explicitly disallow that one because it’s often annoying, but doesn’t violate the other axioms. That is, in the unique field with one element, 0/0 = 0, where 0 acts as both additive and multiplicative identity. ---
**Author:** ![Bytegeist](https://avatars.discourse-cdn.com/v4/letter/b/9dc877/32.png) [@Bytegeist](https://boards.straightdope.com/u/Bytegeist)\ **Post date:** [October 4, 2004, 9:26pm UTC](https://boards.straightdope.com/t/zero-divided-by-zero/267516/20 "2004-10-04T21:26:30Z")
For a concise but insightful explanation, I recommend this [proof sketch](http://www.webfitz.com/lyrics/Lyrics/1974/371974.html) by Preston et al. [Next page](https://boards.straightdope.com/t/zero-divided-by-zero/267516.md?page=2)