 # Have we 'discovered' how to divide by zero?

Given this post discussing faster-than-light travel, a Doper mentions that something would involve division by zero.

I took a lot of math at school, and as I understood it, division by zero is mathematically impossible. The standard Wikipedia search didn’t really answer my question. . . but I’m curious, has there been any serious work on it?

## Tripler

x^2
lim X->0

I’m weary to say this (because usually when I do someone comes up with an exemption or a recent breakthrough) but no… The problem is it doesn’t realistically (in 99% of mathematical fields) make any sense. It’s akin to saying fish = quarter.

I am not a mathematical genius, IANAMG, but if it could be done, it would only be as a useless mathematical construct. Look at it this way, I have zero pies. I want to share these pies with five friends. Ok, all get none. I have 5 pies and I split them evenly among…
Ah never mind, Damn Liberal Arts degree.

SSG Schwartz

You can sometimes find common goldfish or feeder fish at a quarter each so perhaps it is possible after all.

First remember that the multiplicative inverse of b is the unique element c such that bc=cb=1. Denote the multiplicative inverse of by b[sup]-1[/sup].

The formal definition of “a divided by b” is ab[sup]-1[/sup], i.e., a times the multiplicative inverse of b.

But zero times any number is zero; it has no multiplicative inverse. And so “a divided by 0” does not make sense by definition.

No and as long as we are doing algebra we never will. Why? Because it doesn’t result in a number. Suppose we divide 1 by 1/x and let x get bigger and bigger. The denominater thus gets smaller and smaller and goes to zero as a limit and so we are dividing 1 by zero.

``````
**1
1
x

following the rules of dividing by a fraction the result
is just x

**

``````

So as x get’s bigger and bigger our result gets bigger and bigger and there is no number we can settle on as an answer. The same thing holds if we divide any number by 1/x and let x increase without limit.

Let me be this first to say "huh? "

Liberal Arts Major here. I think about every year or two I have to get this broke down for me. I understand the inverse properties of division, but why is this a property. We can have imaginary numbers, so why not imagine the square root of zero?

SSG Schwartz

The square root of zero is zero.

Your point is valid, though. We could define a new number to be the result of dividing 1 by 0. I’m not sure if the resulting system would be at all useful(you’d have to go though all kinds of contortions to make this system consistent), but you could do it.

David Simmons’ proof using limits doesn’t work in all algebraic structures which have a zero, because limits don’t always make sense. For example, take the integers modulo 3, i.e. {0, 1, 2}, with the usual definition of multiplication, i.e.:

0 x 0 = 1 x 0 = 2 x 0 = 0
1 x 1 = 2 x 2 = 1
1 x 2 = 2

So, you can’t divide 1 or 2 by 0, because there’s no number m such that m x 0 = 1, or number n such that n x 0 = 2.

Yeah but … the imaginary numbers resulted from the need for a solution to equations like x[sup]2[/sup] + 1 = 0. It’s a little hard to imagine a need to define a special number for division by zero.

x = x + 1?

-Dan Coffey and Merle Kessler, The Official Dr. Science Big Book of Science, Simplified. New York: McGraw-Hill (!), 1986.

One could construct an algebra which contained division by zero. But such algebras usually aren’t too useful, so what mathematicians have done instead is construct algebras where division by zero isn’t necessary. Basically, what they do is remove the expectation that one ought to be able to divide by zero in the first place, by removing zero from the space before defining multiplication. It’s all much more elegant than I’m making it sound here, though I’m not comfortable enough with the formalism to present it properly.

It’s not a problem of making it consistent so much as you end up with a system where arithmetic doesn’t work the way you’d like. In particular, some step in the following derivation fails:

0a = 0a
0a + 0 = (0 + 0)a
0a + 0 = 0a + 0a
0 = 0a

Well, your example does demonstrate the futility of attempting to divide by 0, but it’s not quite analogous: you’ve got your zero as the numerator instead of the denominator. Here: let this IANAMG take a crack at it.

Look at it this way: I have five pies. I need to divide these pies evenly amongst all zero of my friends (I’m a loser, what can I say), and not have any left over for myself. I can slice each pie into however many pieces I want, but there will still be pie. I can turn them sideways, try to get the cat to eat them, mail them to Peru, what have you. But without any friends to actually take their shares, I’ve still got pie.

On preview, I realize you did try to show this at the end of your post. Oh well: it’s a shame not to post something with so much pie in it. Mmmmmm…

Ok, finally a post that makes sense. It is a law like the 55 mph speed limit. It is a law that can be violated like, well never mind that, but unlike the speed of light. So what I am understanding, is that under the right circumstances, I could divide by zero, but I would have to break rules that define math, and possibly face criminal sanctions?

SSG Schwartz

:smack: That’s what I get for not spending more than 3 seconds coming up with a pseudo-random example. See, this is what I warned about! So instead of being left with the question “Hm, damn, how can we divide by zero? I’d really like to do that”, people (at least, the sort of people who are likely to say the former) are left with the question “Hm, damn, how can we multiply by zero? I’d really like to do that”? Heh. I’m sure it’s useful or interesting in some way, though. Out of curiosity, what are the structures you’re referring to?

Expanding a bit on what Chronos and ultrafilter said: If you’re talking about 0 you’re probably talking about some system with addition, since 0 means the identity for addition. If you’re talking about the possibility of division, then you’re probably talking about a system with multiplication. So, likely, you’re interested in the sort of system mathematicians call a ring, which has both addition and multiplication, behaving in nice ways (including obeying the distributive laws). In a ring you must have 0a = 0 for every element a.

Now you say that you want to be able to divide by 0. What does that mean? If you mean you want to be able to divide every element b by 0, then you want to always be able to solve 0x = b for x. But 0x is always 0, so b must always be 0. There’s one way to do this: Have 0 be your only element. It works. It’s not terribly interesting, but it works. (Extra credit: 0[sup]-1[/sup] = 0. What’s the multiplicative identity “1”?)

This isn’t the only way to make some sense out of dividing by 0, but the point is that you have to give up some familiar fact about the way numbers behave in order to do it.

Now that makes sense. But does it matter if the pies are apple or cherry? :smack:

SSG Schwartz