bigideas

On the TES forums and The Scotsman we’ve recently seen an outbreak of confusion about division by zero, which actually turns out to be an interesting topic. Here’s our cut-out-and-keep guide to this oft-debated topic.

Are You Telling Me You Can’t Divide By Zero?

Yeah, that’s right, you can’t, as Alex Salmond seems to almost know. That is, this:

24
4

is 6, but this:

24
0

is the same sort of thing as this:

24
apple

I think this bothers people because zero is a number, and we thought you could divide numbers by other numbers. But you don’t have a right to be able to divide by it just because it’s a number. People generally seem to get that you’re not supposed to do it but keep trying to suggest a workaround, as if this were a flaw in mathematics that we should try to patch up.

A Bad Reason Why You Can’t Divide By Zero

Lots of people have read somewhere that “you can’t divide by zero”, but don’t know why; this doesn’t stop them from wanting to shout people down on the internet when they make a brave attempt to grapple with some maths. That’s annoying: maths is pretty hard and I like to see people having a go at understanding something hard. I don’t think they should be mocked for it unless they’re pompous and arrogant about it (oh, wait…), and I certainly don’t think they should be mocked when the mocker doesn’t really know any better.

The worst reason I’ve heard is that it doesn’t make sense because you just can’t divide anything into zero pieces. I’m pretty comfortable with

24	= 48
0.5

That is, the idea that dividing by a half is the same as doubling. I’m also good with

24	= -24
-1

but cutting up a cake into half a piece, or into minus one pieces, doesn’t make any more sense than cutting it up into zero pieces. This sort of mental image doesn’t help us figure out whether a particular kind of division “makes sense” or not. Thinking this way leads people to say things like

If it is a scientific impossibility to create something from nothing how would it be possible to divide anything like an apple an infinite number of times with nothing so that it would disappear?

This kind of reasoning would also prevent us from doing a sum like 5-7 (how can I give you seven apples if I only have five?) or solving x²=-1. Such appeals to instinct have served mathematicians very poorly in the past, and made some of them look rather silly. Negative numbers, imaginary numbers and infinite cardinals have all been resisted for similar reasons by people who really ought to have known better.

Why You Can’t Divide By Zero

The reason why we don’t let you divide by zero is very simple. Zero is a special number. Look:

12x = 24

Quick — what’s x? Well, you can divide both sides by 12:

x = 2

See? Multiplying and dividing are useful for solving equations. Let’s do another one:

0x = 24

Quick, what’s the solution? There isn’t one, right? Because regardless of what number x represents,

0x = 0

It just can’t be that 0x=24. This is a very special and very important thing. In the case of addition there’s no special number like this — call it µ — so that, for every x, you get

µ + x = µ

Addition is fundamentally different from multiplication because multiplication has this special behaviour with zero and addition doesn’t. So if your mathematical training only got as far as adding up you need to leave your intuitions at the door.

OK, so zero is special. Let’s imagine dividing by zero worked. We won’t say what it is, let’s just imagine we can do it and write “x” when we do. Then we can do ace proofs like this:

2/0	=	x
so 2	=	0x
so 2	=	0

What went wrong? We divided by zero. Yeah, but what really went wrong?

In using division to solve equations we’re exploiting the fact that dividing is kind of like undoing multiplying. Think of a number. Multiply it by five. Now divide it by five. Would you be surprised if you ended up with the number you first thought of? If so come into my office, I have some mortgage-backed securities you might be interested in.

Of course you’re not surprised: dividing is un-multiplying. Let’s play a game: think of a number. Now multiply it by any number you choose. Tell me the answer and the number you multiplied by, and I bet you a whole bucket of debt instruments that I can tell you which number you first thought of.

You have a way to win, but not if you multiply by a normal number like 5, because I can divide your answer by five and get your original number. But what if you say: “the result of the multiplication was zero, and I multiplied by zero”? Well I can divide zero by zero and get — what? Surely not the number you first thought of, which was 17 (I peeked).

In other words, you can un-multiply as long as you don’t try to un-multiply by zero. If you try that, it won’t work because, unlike with any other number, multiplying by zero always gives the same answer. It’s like baking a cake with six eggs and then trying to get one of the eggs back out. OK it’s not really like that — but the point is you can’t undo an operation that collapses lots of things into the same thing, because you can’t tell how to undo it to get back where you started.

Bad Suggestions For How To Divide By Zero (Which You Can’t)

People do like to come up with suggestions for fixing this situation. I really think they think it’s a problem, a sort of pothole that for millennia mathematicians have had to be careful to drive around whenever they did a big long division (which is basically what mathematicians do all day). From the helpful folks on the TES forum, The Scotsman, the BBC and Alex Salmond himself, here are some suggestions for mending the road that Euler, Gauss, Fermat and Ramanujan may not have thought of.

The following suggestion is that 24 divided by 0 should be 24:

What do you get when you have 24 and you take zero away?
Is it not true you are left with 24?
If you have 24 and you divide it by no number at all what would you have?
Is it not true you would be left with 24?
Why should there be a difference between dividing a number by no number at all and dividing it by zero?

OK, so we have 24/0=24. Now multiply through by 0 — a perfectly reasonable thing to do — and obtain 24=0. If your instinct here is to start defining more rules to get around the problem then I suspect you’re the kind of person who, trapped in a hole, calls out for a shovel. Don’t feel bad: this character had the same reaction, and he works at a university, although if you read some of his stuff he does unfortunately come across as a fruitcake.

This one gets the same answer (24/0=24), but it kind of blew my mind because of its remarkable near-lucidity:

division is repeated subtraction, and you can subtract 0 from 24 as often as you like and still have something left to subtract from

What’s happened here is that someone has swallowed the “multiplication is just repeated addition” line and remembered that division undoes multiplication, and so concluded that it must be just repeated subtraction. This is a sort of para-maths that demonstrates some raw talent for the subject while being not just wrong but quite bizarre with it. How is dividing 24 by 4 tantamount to “repeated subtraction”? Because you subtract 4 from 24, erm, four and a half times to get the answer? Really?

The following is offered as a proof that 24 divided by 0 should be 0:

The result tends towards infinity for a positive number and -infinity for a negative number. So on a continuous scale (if that is the PARTICULAR context of the series of calculations you are making) the point half way between +infinity and -infinity must be zero … or, to be more precise, a zero of some sort.

Unlike the previous one — which was a swing and a miss at understanding inverse operations — this is just madness. It’s a completely spurious geometrical-intuition-misfire. If 24/0=0 then, once again, 24=0. If you ever get this answer it’s time to screw up the paper and start again.

The following argues that 24 divided by 0 should be infinity:

24 / 0.1 = 240
24 / 0.01 = 2400
24 / 0.001 = 24000
24 / 0.0001 = 240000
etc…
The answers are clearly getting larger as x gets smaller.
So as x “tends towards” 0, the answer “tends towards” infinity.

This is a more subtle argument that a lot of people find appealing. And they feel as if it doesn’t get them into trouble because “infinity” isn’t really a proper number, so they can still say you can’t “really” divide by zero. (This puts them in the same class as husbands who say it’s not really cheating if you keep your socks on). This is the answer Salmond goes with, by the way: you can’t really do it, but if you do, it’s infinity. If you can’t be good, be careful.

Unlike the others, this argument is wrong in a way that requires you to pay attention. Mathematicians often say a series of numbers “tends towards” infinity when it just grows and grows without ever settling down. They also sometimes use “limits” as a way to define functions at special points like zero. When they do this they say something like “well, the function gets closer and closer to some value p as we get closer to zero, so we’ll define the function at zero to actually be p”. Sounds promising, doesn’t it?

But “p is the limit of a sequence of numbers” and “a sequence of numbers tends towards infinity” are not the same sorts of statements. The first one means something like “the numbers in the sequence get closer and closer to p”. You can’t let p be infinity, because you can’t get “closer and closer” to infinity. That doesn’t make any sense.

["This is basic stuff by the way," sneers the poster of the above argument. Well, no, it's not, unless you've done a first-year course on analysis, in which case it is and you'll have to re-sit the exam.]

But what’s so wrong with defining division by zero to be infinity even if that funny “limits” thing didn’t work? Well, now 24/0=infinity, so 24 is zero times infinity. But 42/0=infinity too, so 42 is zero times infinity. Which means 24=42. Ding! Next!

What Have We Learned Today?

For what it’s worth, I applaud everyone who had a go at working out how to divide by zero, and who stuck their necks out and posted their stab at an answer on the internet. I think it’s good that people want to understand something like this and give it a try. I don’t think the arguments they presented (including those above) were stupid or devoid of merit, although they all go wrong for the simple algebraic reasons I’ve tried to point out.

I don’t, though, applaud the people who put forward their arguments in an exasperated “you’re all idiots” tone of voice. They’re reminiscent of grammar snobs whining about unrestrictive “which”, only worse because at least the latter have a Strunk & White or something similar that they’ve actually read and understood. Tip #1 for being an amateur in an academic subject: be humble.

There’s also a widespread sense that there’s a single right answer that can be worked out. In fact, as I’ve tried to suggest above, this isn’t a question about what the answer is when you divide by zero, but about what you’d like it to be. All the answers given have disastrous effects on ordinary high-school algebra. As such, we’d prefer not to define division by zero like that or, in fact, at all. If you’d like to work in a space where the answer is infinity, say, welcome to the real projective line. Have fun visiting, but please don’t write any credit derivative pricing software while you’re there.

The main reason I wrote this post is to see if I could put a “correct” answer into layman’s terms in a way that might convince you if you’re attracted to one of the other answers. You still might feel as if the undefinedness of division by zero is a flaw in the system. I admit it looks weird and unsatisfactory, but only from a certain perspective. From other perspectives the number system we have is beautifully symmetrical and really awfully nice.