Divide (by Zero) and Conquer

Question/Scream of Anguish : But I WANNA divide by zero, I WANNAAAAAAAA.

Anna

***

Well, all right.

If we’re dealing with ordinary arithmetic (the kind 99% of the population uses) the answer would be a resounding “Good god man, are you insane?!” And before we do anything weird, we should have a look at why dividing by zero would make your high school math teacher explode (I have anecdotal evidence for this).

First we have to define division. There are a few ways to do this but I’m gonna go with the one that clearly illuminates why division by zero is Not Usually a Thing. Here we think of division as the reverse of multiplication. If we have this for multiplication:

c = a x b

then we define division like this:

c / a = b

This basically asks, what number when multiplied by a would give us c?

Notice that for any two numbers a and c, we only have one number b that would make the equation true. This is really important — the solution to the equation is unique. Here’s a really simple example:

10 = 5 x 2

and

10 / 2 = 5

Only 5 can be multiplied by 2 to make 10. Easy. Now let’s try this out with zero.

10 / 0 = ???

The equation is saying, what number when multiplied by zero would give us 10? But the problem is we know that any number multiplied by zero would always give us zero. That’s part of what it means to be zero. And if we’re dealing with high school maths and ordinary arithmetic this is where we’d stop: division by zero is undefined, end of story.

But you seem to be adamant (obssessed?) about dividing by zero, so let’s take it a step further. See, mathematicians are a stubborn bunch. If they can’t do something in one branch of mathematics, they just invent new maths where their weird obssessions are allowed. And they do this all the time!

Take negative numbers, which we’re ok with because we’re so used to the idea. The number line was extended to include negative numbers to solve problems like this:

? + 3 = 0

What number when added to 3 gives 0? Almost anyone with a basic math education can say that the answer is -3. But think about it for a while. Adding always makes a number bigger, so how can we add to a number to make it smaller? It’s pretty strange. But mathematicians just extended the concept of numbers to include negative numbers.

Later on they had to extend it again to include imaginary numbers to solve equations like this:

?² + 1 = 0

What number when squared and added to one gives 0? We know it has to be a negative number, and that’s ok since we’ve extended our number line to include negative numbers now, but then we come to the problem of finding a number that gives a negative number when squared. Which is not a thing that regular numbers do. So mathematicians added a new kind of number, imaginary numbers.

So dividing by zero is the same deal. What number when muliplied by zero doesn’t give zero? There’s not just one but several branches of mathematics where people have made up ways of getting into this. I’m gonna show you one from the branch of real analysis, involving what’s called the projectively extended real line. Don’t let the name scare you; as with most things in mathematics it’s only called something difficult-sounding to frighten off the non-mathematicians.

What it means is that the number line is extended by adding a point for infinity, denoted by ∞ . If you were looking at the old, boring, can’t-divide-by-zero number line, you’d think you need two infinities, one on the positive side and one on the negative:

But real analysis doesn’t do that. Instead they add just one infinity, and turn the line into a circle so both sides reach it. So this is what our new, super-cool, can-divide-by-zero, projectively extended real line looks like:

The added point, the point at infinity, allows us to say that as you move towards larger and larger positive or negative numbers, the sequence approaches infinity. But this infinity is neither positive nor negative.

Why does this matter? Well, let’s divide a number, say 1, by zero, but we’re going to do it in a weird way.

First we divide this 1 by a number that’s very close to zero, say 0.000000000000000000001. We’re going to get a very large number which isn’t infinity, but which is a positive number. Then we try dividing our 1 by an even smaller number, like 0.00000000000000000000000001, and even smaller again, like 0.000000000000000000000000000000000000000000000001, and smaller and smaller until we gets to 0. The answers we get are positive numbers that keep getting larger and larger until they approach positive infinity.

Now let’s do the same thing again but this time we start with negative -0.000000000000000000001. If we divide 1 by this number we get a large negative number. Well, we do the same procedure, we make the divisor smaller and smaller until it gets to zero. This time our answer is a large negative number that gets larger and larger until it approaches negative infinity.

So we’ve got two very different answers to the same problem. If we divide by zero by starting from the positive side we get positive infinity and if we start from the negative side we get negative infinity. That can’t be right. But if we define our new number line in a loop as we’ve done above, the problem goes away!

(Other problems crop up, but we’re not gonna go there.)

And boom, we’ve got division by zero. We can now say that for any number k (positive or negative) that isn’t zero,

k / 0 = ∞

What we have done may seem like a bit of a cheat, but mathematicians do this all the time. There’s nothing wrong with defining new numbers or, hell, inventing a whole new branch of mathematics to help us solve problems we can’t solve with our current system.

But remember, a good mathematician would only go this far if the new system solves not just one new problem, but a whole bunch of new problems. It’s easy to make something up — it’s much harder to make it useful.

Math Nerd signing out.

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P.S For those who are interested in other places where division by zero is allowed, have a look at Riemann Spheres. They’re like the projectively extended real line but for planes instead of lines. And just like how our number line becomes a circle, the complex plane becomes a sphere.

And for a much, much more technical explanation of the projectively extended real line, here’s an explanation from Wolfram Mathworld.