# Divide by zero - huh? **URL:** \ **Category:** Factual Questions\ **Created:** [January 22, 2003, 5:25am UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165 "2003-01-22T05:25:42Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [January 22, 2003, 4:56pm UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/21 "2003-01-22T16:56:28Z")
> [@](#): > > \*Originally posted by ftg \* > **…the arcsin of 2…** Minor detail, but the arcsin of 2 might be defined. It would be a complex number. I’ll check when I get home. ---
**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:** [January 22, 2003, 5:06pm UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/22 "2003-01-22T17:06:48Z")
A few years ago, I copyedited a math book in which the author finally gave definitions that I could both remember and understand: x/0 is _undefined_, because there is no y such that y\*0 = x. 0/0 is _indeterminate_, because for _every_ y, y\*0 = 0. I had learned about undefined vs. indeterminate previously, but could never keep them straigh because I had never heard such a clear explanation before. ---
**Author:** ![MSU\_1978](https://avatars.discourse-cdn.com/v4/letter/m/ac8455/32.png) [@MSU\_1978](https://boards.straightdope.com/u/MSU_1978)\ **Post date:** [January 22, 2003, 5:07pm UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/23 "2003-01-22T17:07:01Z")
The results of dividing by zero depend on which zero you’re using. Let’s say you wish to evaluate sin x / x where x=0. Since sin 0 = 0, this is 0/0. However, by l’Hopital’s rule, we can state that the limit as x approaches zero of f(x)/g(x) = f’(x) / g’(x). In this case, f’(x) = cos x, g’(x) = 1. Thus the limit of sin x / x as x approaches zero = cos 0 / 1 = 1/1 = 1. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [January 22, 2003, 5:24pm UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/24 "2003-01-22T17:24:20Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\*The reason why we don’t generally say that it’s 0 is because that violates the division theorem. For any numbers a and b, there are numbers q and r such that a = bq + r, where 0 \< r \< b. You’re saying that q should be 0 if b is; in that case, what would you choose for r? > > There are some mathematicians who do define it that way, for technical reasons, but it’s not a popular view.\*\* Let me explain this a bit. We would like to have functions q(a, b) and r(a, b) defined on pairs natural numbers such that a = q(a, b)b + r(a, b), with 0 \< r(a, b) \< b. As of right now, these functions are partial functions, because q(a, 0) and r(a, 0) are undefined. Some people define q(a, 0) = 0 and r(a, 0) = a. These will satisfy the equation, but zero becomes a special case for the second clause. And most of us don’t like special cases. ---
**Author:** ![kabbes](https://avatars.discourse-cdn.com/v4/letter/k/e99b99/32.png) [@kabbes](https://boards.straightdope.com/u/kabbes)\ **Post date:** [January 22, 2003, 6:14pm UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/25 "2003-01-22T18:14:33Z")
Mr **Filter** , you ARE a special case 😃 ---
**Author:** ![Saxman](https://avatars.discourse-cdn.com/v4/letter/s/f6c823/32.png) [@Saxman](https://boards.straightdope.com/u/Saxman)\ **Post date:** [January 22, 2003, 6:40pm UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/26 "2003-01-22T18:40:46Z")
Per my high school AP Physics teacher: Dividing 0 by 0 is like saying, “I got no money and no one to give it to. What do I do?” ---
**Author:** ![Dewey\_Cheatem\_Undhow](https://avatars.discourse-cdn.com/v4/letter/d/13edae/32.png) [@Dewey\_Cheatem\_Undhow](https://boards.straightdope.com/u/Dewey_Cheatem_Undhow)\ **Post date:** [January 22, 2003, 6:50pm UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/27 "2003-01-22T18:50:54Z")
I never thought a single number could be completely and totally fascinating. Then I read Charles Seife’s [ZERO: The Biography of a Dangerous Idea](http://www.amazon.com/exec/obidos/tg/detail/-/0140296476/qid=1043261343/sr=1-1/ref=sr_1_1/002-2489978-8204829?v=glance&s=books). Division by zero is only the start – the concept of “zero” has some fairly mindboggling implications. A highly recommended read. ---
**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:** [January 22, 2003, 7:12pm UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/28 "2003-01-22T19:12:16Z")
> [@](#): > > \*Originally posted by Dewey Cheatem Undhow \* > \*\*I never thought a single number could be completely and totally fascinating. > > Then I read Charles Seife’s [ZERO: The Biography of a Dangerous Idea](http://www.amazon.com/exec/obidos/tg/detail/-/0140296476/qid=1043261343/sr=1-1/ref=sr_1_1/002-2489978-8204829?v=glance&s=books). > > Division by zero is only the start – the concept of “zero” has some fairly mindboggling implications. A highly recommended read. \*\* [schoolhouse rock] My hero zero… [/schoolhouse rock/ Zev Steinhardt ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [January 22, 2003, 7:17pm UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/29 "2003-01-22T19:17:43Z")
> [@](#): > > \*Originally posted by kabbes \* > \*\*Mr **Filter** , you ARE a special case 😃 \*\* I prefer to think of myself as a pathological case. 😉 ---
**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:** [January 22, 2003, 8:34pm UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/30 "2003-01-22T20:34:15Z")
When I was in High School, we had an old electronic calculator that divided by counting successive subtractions. It kept subtracting the denominator from the numerator until it was less than the denominator. If you tried to divide by zero, the gears would just spin until you unplugged it. > [@](#): > > \*Originally posted by \*\*\*MSU 1978 \*\* > The results of dividing by zero depend on which zero you’re using. Let’s say you wish to evaluate sin x / x where x=0. Since sin 0 = 0, this is 0/0. However, by l’Hopital’s rule, we can state that the limit as x approaches zero of f(x)/g(x) = f’(x) / g’(x). In this case, f’(x) = cos x, g’(x) = 1. Thus the limit of sin x / x as x approaches zero = cos 0 / 1 = 1/1 = 1. That’s all well and good. However, the fact remains that f(x) = sin x/x is not defined for x = 0. Lim (x-\>a) f(x) = L NOT imply that f(a) = L or even that it is defined. The limit is defined at x = 0, but the function is not. Look at the definition of limit. It says nothing about the value of the function at x = a. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [January 22, 2003, 8:40pm UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/31 "2003-01-22T20:40:21Z")
In general, there are two sorts of problems you can get with algebraic operations. Some problems, such as arcsin(2) or sqrt(-4), make no sense in the real numbers, but do make sense in the complex numbers (note, however, that there are multiple complex solutions in each of these cases). All of the other “undefined” things in algebra are essentially division by zero problems. For instance, tan([symbol]p[/symbol]/2) is undefined, because tangent is sine over cosine, and cos([symbol]p[/symbol]/2) = 0. Now here’s the thing. We can consistently define square roots of negative numbers in a way which makes sense, for all (or very nearly all) applications where we would want to use them. The same method works in all cases. But we can’t do that in the case of division by zero: There is no way to define “1/0” which would make sense in all applications. ---
**Author:** ![Early\_Out](https://avatars.discourse-cdn.com/v4/letter/e/6f9a4e/32.png) [@Early\_Out](https://boards.straightdope.com/u/Early_Out)\ **Post date:** [January 22, 2003, 10:56pm UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/32 "2003-01-22T22:56:27Z")
> [@](#): > > \*Originally posted by DrMatrix \* > **When I was in High School, we had an old electronic calculator that divided by counting successive subtractions. It kept subtracting the denominator from the numerator until it was less than the denominator. If you tried to divide by zero, the gears would just spin until you unplugged it.** We had at least one model that, when you plugged it back in, would pick up precisely where it left off, and continue chugging. The only way to cure the sucker involved considerable dismantling to get at the offending gear. It takes very little to amuse a 15-year-old. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [January 22, 2003, 11:17pm UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/33 "2003-01-22T23:17:56Z")
> [@](#): > > \*Originally posted by Chronos \* > All of the other “undefined” things in algebra are essentially division by zero problems. For instance, tan([symbol]p[/symbol]/2) is undefined, because tangent is sine over cosine, and cos([symbol]p[/symbol]/2) = 0. How can you write 0[sup]0[/sup] in terms of division by zero? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [January 22, 2003, 11:29pm UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/34 "2003-01-22T23:29:12Z")
> [@](#): > > \*Originally posted by Achernar \* > \*\*How can you write 0[sup]0[/sup] in terms of division by zero? \*\* 0[sup]0[/sup] = 1, or the binomial theorem fails. Try expanding (a + 0)[sup]1[/sup]. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [January 22, 2003, 11:44pm UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/35 "2003-01-22T23:44:37Z")
Ah that’s right I remember. Well, I never liked the binomial theorem anyway, but that’s okay. ---
**Author:** ![William\_Ashbless](https://avatars.discourse-cdn.com/v4/letter/w/a87d85/32.png) [@William\_Ashbless](https://boards.straightdope.com/u/William_Ashbless)\ **Post date:** [January 23, 2003, 12:52am UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/36 "2003-01-23T00:52:08Z")
0[sup]0[/sup] is, as I thought, undefined. x[sup]0[/sup] is 1 for all x except when x is 0, and 0[sup]y[/sup] is zero for all y except when y is 0. Possible proof: Let f = 0[sup]0[/sup] Then: log(f) = log(0[sup]0[/sup] ) log(f) = 0(log(0)) So whatever the value of f is, it’s 0 times the log of 0. The log of 0 is itself undefined as log(y) grows asymptotically towards minus infinity as y approaches 0. Hm. if f is 1, then: log(1) = 0(log(0)) 0 = 0(log(0)) Interesting. I can _almost_ buy this given that 0d = 0 for all d, but d being log(0) which is undefined really really bugs me. Anybody else got anything on 0[sup]0[/sup] ? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [January 23, 2003, 12:56am UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/37 "2003-01-23T00:56:41Z")
See my last post in this thread. ---
**Author:** ![William\_Ashbless](https://avatars.discourse-cdn.com/v4/letter/w/a87d85/32.png) [@William\_Ashbless](https://boards.straightdope.com/u/William_Ashbless)\ **Post date:** [January 23, 2003, 1:01am UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/38 "2003-01-23T01:01:18Z")
I’m curious, though, whether that’s a convenience for the binomial theorem alone. I wonder if there’s a way to justify it outside of the binomial theorem or a way to show it leads to a contradiction like 0/0 can be made to show. ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [January 23, 2003, 1:04am UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/39 "2003-01-23T01:04:52Z")
Notation: log denotes natural log, frequently written ln. Let f(x) = x[sup]x[/sup]. Then log f = x log x = log x/x[sup]-1[/sup]. By L’Hôpital’s Rule, the limit of log f as x -\> 0 is the same as the limit of (1/x)/(-1/x[sup]2[/sup]) = -x ( provided the latter limit exists) and this second limit is plainly 0. Thus f = e[sup]log f[/sup] -\> e[sup]0[/sup] = 1 as x -\> 0, as log is comtinuous. So certainly x[sup]x[/sup] tends to 1 as x tends to 0. I prefer to regard 0[sup]0[/sup] as undefined, though. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [January 23, 2003, 1:07am UTC](https://boards.straightdope.com/t/divide-by-zero-huh/150165/40 "2003-01-23T01:07:18Z")
Truth be told, the only places you generally run into 0[sup]0[/sup] are in the multinomial theorem (a generalized version of the binomial theorem) and in the bivariate exponential function x[sup]y[/sup]. The multinomial theorem is very important, and elegant if you don’t need a special case. So that’s a compelling argument for 0[sup]0[/sup] = 1. Also, you can’t really define an arithmetic operation only in the context of a certain theory. The most compelling argument that 0[sup]0[/sup] != 1 is that it makes the bivariate exponential function discontinuous at the origin. So what? Have you seen the damn thing to the left of the origin? In fact, even for fixed values of x, the exponential function can’t be single-valued once you extend the domain to **C**. So trying to save its continuity at a given point is just a waste of time. [Previous page](https://boards.straightdope.com/t/divide-by-zero-huh/150165.md?page=1) [Next page](https://boards.straightdope.com/t/divide-by-zero-huh/150165.md?page=3)