# Divide by zero **URL:** \ **Category:** Factual Questions\ **Created:** [June 10, 1999, 3:46am UTC](https://boards.straightdope.com/t/divide-by-zero/1728 "1999-06-10T03:46:00Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 10, 1999, 3:46am UTC](https://boards.straightdope.com/t/divide-by-zero/1728/1 "1999-06-10T03:46:00Z")
I know there’s enough of you math munchkins out there to address this. While I do a lot of grunts in the trenches math every day, I don’t frequent the frontiers. So I must humbly ask the answer to this question. What if, instead of division by zero being undefined (“we won’t go there”) it was defined as equalling infinity? I’m not claiming some revelation or anything, I just want one of you who can speak to the subject to let me understand. Our regular math (i.e., quantifiable small numbers) wouldn’t change. Carpenters would still be able to get a set of stairs put up as before. What cosmological cataclysm would this change in assuumptions entail? ---
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 10, 1999, 4:01am UTC](https://boards.straightdope.com/t/divide-by-zero/1728/2 "1999-06-10T04:01:00Z")
Maybe not equal to infinity, but why not equal to zero? If X_0=0, then why does not X/0=0? Is it because X_Y can never equal X/Y, assuming the same value for X and Y in each equation? * * * The overwhelming majority of people have more than the average (mean) number of legs. – E. Grebenik ---
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 10, 1999, 4:39am UTC](https://boards.straightdope.com/t/divide-by-zero/1728/3 "1999-06-10T04:39:00Z")
First, you might want to know how division is defined in the first place. Roughly speaking, we define division by its relation to multiplication: division is the inverse (or opposite) of multiplication. For example, the reason we say that 10/5 = 2 is BECAUSE 2_5 = 10…multiplication is the “main event”, and division simply undoes multiplication. Now, apply this to division by zero. Suppose we say that 2/0 = infinity. Fine, then by defintion, we are saying that 0 \* infinity = 2. Now, suppose we say that 3/0 = infinity…uh oh, 0 \* infinity = 3! Which is it? Hence, the term “undefined”–it can’t be defined in a way which is consistent for all problems. (The same argument holds for divsion by zero not equaling 0 either: 2/0 = 0 means that 0_0 =2). What we do have to express the idea that dividing by something very small should result in something very large is the concept of a limit (remember that from freshman calc?). So, we can say that as the denominator of a fraction APPROACHES zero, the value approaches infinity. So, we do “go there” – we just never reach where we’re going! (oh, and hi! i’ve been lurking here for a couple weeks now waiting for a good math question to jump in on 🙂 ) ---
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 10, 1999, 4:42am UTC](https://boards.straightdope.com/t/divide-by-zero/1728/4 "1999-06-10T04:42:00Z")
The reason, I think, is pretty straight forward. If you’re going to divide, you have to divide by SOMETHING. Zero, by definition, is nothing. Therefore, division by zero is not actually division. * * * “The day after tomorrow is the third day of the rest of your life.” -George Carlin ---
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 10, 1999, 4:45am UTC](https://boards.straightdope.com/t/divide-by-zero/1728/5 "1999-06-10T04:45:00Z")
Welcome on board, lynne. Gee, you made my answer look kinda cheesy. Where’s a good music topic when you need one. * * * “The day after tomorrow is the third day of the rest of your life.” -George Carlin ---
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 10, 1999, 4:51am UTC](https://boards.straightdope.com/t/divide-by-zero/1728/6 "1999-06-10T04:51:00Z")
Lynne, your definition is wonderful: \>\>First, you might want to know how division is defined in the first place.\<\< However, good as it is, can I offer a dumbed down one, from my second grade teacher? If you have four apples and you divide them between two people, each gets two; if between three people, each gets one and one-third. If you divide four apples “among” one person, that person gets all the apples. If there are NO people to divide the apples among, you don’t divide them at all, you put them back in the bowl, or whatever. To summarize: when there are no people among whom to divide the apples, then you are NOT DOING DIVISION. If you say “What about the bowl?” That’s dividing by one-- one bowl. x/0 just isn’t possible. * * * * * * –Rowan Shopping is still cheaper than therapy. --my Aunt Franny ---
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 10, 1999, 4:04pm UTC](https://boards.straightdope.com/t/divide-by-zero/1728/7 "1999-06-10T16:04:00Z")
Nice job Lynne. I jumped in here with a ready answer, only to find that you had beat me to it. There’s getting to be too many bright people here! ---
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 10, 1999, 4:51pm UTC](https://boards.straightdope.com/t/divide-by-zero/1728/8 "1999-06-10T16:51:00Z")
The problem is even deeper than that. It can be proven that if x/0 is defined as being equal to anything at all – you can even call it “foo”, and try to treat it as an extension to the number system of the same kind as imaginaries – then math breaks down altogether, and 1=2. The only way for math to work is to declare that x/0 is always illegal. * * * John W. Kennedy “Compact is becoming contract; man only earns and pays.” – Charles Williams ---
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 10, 1999, 7:30pm UTC](https://boards.straightdope.com/t/divide-by-zero/1728/9 "1999-06-10T19:30:00Z")
Things are even more interesting when you get into calculus. Using L’Hopital’s rule, you can get equations that look like: 0/0 = 3. The result can really be any number; it depends on the value of 0. (Don’t even get me started on dividing infinity by infinity…) ---
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 10, 1999, 8:21pm UTC](https://boards.straightdope.com/t/divide-by-zero/1728/10 "1999-06-10T20:21:00Z")
Lynne’s answer was great. I’d just add one little thing. When you do problems where you are mathematically modeling the real world, and division by zero comes up, it actually means (almost always) that your model doesn’t work, i.e there is something about the behavior of reality that forbids the situation that gave you division by zero. Saying that division by zero “is forbidden” cannot be defined away without losing this important aspect of mathematics, even if you could put it on a logical footing. Which you can’t for the reason Lynne showed. This is an important conceptual distinction between 1/0 and lim(x-\>0) of 1/x. The former gives you problems, the latter need not. ---
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 10, 1999, 8:29pm UTC](https://boards.straightdope.com/t/divide-by-zero/1728/11 "1999-06-10T20:29:00Z")
\>\>(Don’t even get me started on dividing infinity by infinity…) That’s an easy one, isn’t it? Any number divided by itself is 1. Infinity/Infinity = 1. I’m a math god. ---
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 10, 1999, 9:30pm UTC](https://boards.straightdope.com/t/divide-by-zero/1728/12 "1999-06-10T21:30:00Z")
Actually, Athena, that isn’t necessarily the case. Based on L’Hopital’s rule, infinity/infinity could be 1, could be something else entirely. Actually, the more proper way to phrase that would be that as the numerator and denominator _approach_ infinity, the answer could _approach_ 1, or it could _approach_ some other value. L’Hopital’s rule concerns itself with 3 scenarios that I remember off the top of my head, 10 years after Calculus II (someone fill me in if I miss anything). They are: (N/D) --\> ? as N --\> 0, D --\> 0 (N/D) --\> ? as N --\> infinity, D --\> infinity 1^X --\> ? as X --\> infinity In each of the above cases, N, D, and X may be variables **or** expressions. Someone who’s a little more fresh on this, please correct me if necessary (as if I have to tell anyone here to do THAT!). Great, now we’re going to have to use a message board forum to teach people how to take derivatives. * * * “I wept because I had no shoes, then I met a man with no feet. So I took his shoes” - Dave Barry ---
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 10, 1999, 9:44pm UTC](https://boards.straightdope.com/t/divide-by-zero/1728/13 "1999-06-10T21:44:00Z")
Wouldn’t you be a math _godess_, Athena? Actually, L’hopital’s rules aside, there are different sizes of infinity. Counterintuitive, I know, but check out G. Cantor’s set theoretic notions of transfinite numbers. Kind of neat, and completely useless, as far as I know. ---
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 10, 1999, 10:19pm UTC](https://boards.straightdope.com/t/divide-by-zero/1728/14 "1999-06-10T22:19:00Z")
Strainger, you’re right on target with L’Hopital’s rule. The two basic forms are (N/D) --\> ? as N --\> 0, D --\> 0 (N/D) --\> ? as N --\> infinity, D --\> infinity and there are a few other forms (like that 1^x) that can be rearranged into the two basic ones (hey, anybody want a refresher in logarithmic differentiation? yechh.) That whole idea of _approaching_ (as in limits) is absolutely critical in both theory and application. Infinity is a way to symbolize a very useful mathematical concept (the idea of growing without bounds), but it isn’t a number, and you have to be very careful to not act like it is(trying to cancel infinities and so on)…or at least, I do, since I actually use/teach this stuff…the rest of you needn’t stay awake nights worrying that you’ve improperly used infinity! (More than likely, infinity will NOT come seeking revenge and/or monetary compensation). How’s this for a tag line? “The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that mathematicians have made a covenant with the devil to darken the spirit and confine man in the bonds of Hell.” – Saint Augustine ---
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 11, 1999, 1:16pm UTC](https://boards.straightdope.com/t/divide-by-zero/1728/15 "1999-06-11T13:16:00Z")
In defense of St. Augustine, the “mathematicians” he was talking about were bozos similar to the current “Bible code” bozos. * * * John W. Kennedy “Compact is becoming contract; man only earns and pays.” – Charles Williams ---
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 11, 1999, 1:42pm UTC](https://boards.straightdope.com/t/divide-by-zero/1728/16 "1999-06-11T13:42:00Z")
Hey, hey, hey! Y’all should just shut up about this L’Hopital’s rule and limits and stuff. Don’t you know that calculus is _useless_ in the real world??? 🙂 ---
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 11, 1999, 4:23pm UTC](https://boards.straightdope.com/t/divide-by-zero/1728/17 "1999-06-11T16:23:00Z")
Seems to me the confusion is that people equate ‘zero’ with a thing. It’s no-thing or nothing. You know, like nothing there. ---
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 12, 1999, 3:08am UTC](https://boards.straightdope.com/t/divide-by-zero/1728/18 "1999-06-12T03:08:00Z")
Thanks to lynne, _et al_; sometimes this board works the way it’s envisioned - I got an answer that works. I like handy’s comment. With a math that quantifies things between zero and infinity (check the monkeys thread), zero is when we’re fresh out of quantities. Regards ---
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 12, 1999, 3:19am UTC](https://boards.straightdope.com/t/divide-by-zero/1728/19 "1999-06-12T03:19:00Z")
Athena: I _said_ not to start on the infinity thing. Now we’ve got math geeks all over the place, and you _know_ how hard it is to get them out of the upholstery. =B^) ---
**Author:** ![system](https://global.discourse-cdn.com/straightdope/original/2X/e/e489c3b7d8fce19c4b355dd4fc3f88cc39c34b87.png) [@system](https://boards.straightdope.com/u/system)\ **Post date:** [June 12, 1999, 4:29am UTC](https://boards.straightdope.com/t/divide-by-zero/1728/20 "1999-06-12T04:29:00Z")
AuraSeer Like that. We’re embedded now. [Next page](https://boards.straightdope.com/t/divide-by-zero/1728.md?page=2)