## Ask Professor Puzzler

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Medhavi wants to know, "Why can't I divide by zero?"

That's an excellent question, Medhavi. When I'm teaching, I tell my students that dividing by zero is illegal, immoral, and socially unacceptable, and if they do it, I'll have them arrested and thrown into math prison.

I also tell them that dividing by zero makes math problems explode.

But that's just me being silly; it doesn't really answer the question of *why *it's illegal. I'll try to give you a couple different ways of looking at the question, in hopes that at least one of them will help you feel confident that there really are good reasons to not allow it.

## Division as the Inverse of Multiplication

First, you can think of division as being the inverse of multiplication. Let me show you what I mean by example:

^{14}/_{2} = 7, because 7 x 2 = 14

^{32}/_{4} = 8, because 8 x 4 = 32.

So now let's apply that to a "division by zero" problem:

^{15}/_{0} = ?, because 0 x ? = 15.

Ah...in order to solve the division problem, you have to solve the multiplication problem. If you want to know what 15 divided by 0 is, you have to figure out what number you need to multiply 0 by to 15. And the answer is?

I'm waiting...

You can't find one, can you? Nope! Because *anything* times zero is just zero, so you can never make it work out to 15.

It's just plain silly to think you could.

## Division as Repeated Subtraction

Here's another way of looking at division: ^{14}/_{2} = 7 because you can subtract 2 from 14 seven times before you reach zero.

If that's the case, what is ^{20}/_{0}? Well, how many times can you subtract zero from 20 before you reach zero? WHAT? What kind of a question is that? You can't ever reach zero by subtracting zero from 20, no matter how many times you do it!

It's just plain silly.

## Real World Division

What does division mean in the real world? 100 divided by 5 could be understood like this: If you have 100 marbles and you divide them into 5 equal groups, how big is each group? The answer to ^{100}/_{5} is 20, because each group of marbles has 20 marbles in it.

^{100}/_{2} = 50, because if you divide 100 marbles into two equal groups, there are 50 marbles in each group.

Great. So what is ^{100}/_{0}? Well, you find that by dividing 100 marbles into zero groups.

What? How do you do that? How in the world can you take 100 marbles and split them up into zero groups? That's just silly!

It's tempting to say, you do it by taking a stick of dynamite to your marbles, so there won't be any marbles.

But that's not dividing it into zero groups. That's dividing it into billions of groups (pieces), and each piece is really, really tiny.

So, you just flat out can't do it.

## Dividing Zero by Zero

But what about dividing zero by zero? Can't you divide nothing into no groups?

I would argue that you can't. You can divide it into any positive number of groups, and each group has zero marbles in it, but it's meaningless to suggest that you're going to "split" something into zero groups.

But in case you disagree with me, let's go back to our definition of division as the inverse of multiplication:

^{0}/_{0} = ?, because 0 x ? = 0.

Can you find a number to put in place of the question mark? Yes! The problem isn't that you *can't* find a number that works; the problem is that *any *number will work. 0 x 1 = 0; 0 x 2 = 0; 0 x 3 = 0, etc.

We can't give an answer to the division problem, because there isn't just one answer to the multiplication problem. We end up with the a similar issue if we treat it as a repeated subtraction problem.

Since there's no way to give one answer, we reject the notion of giving *any *answer. Division by zero is illegal, immoral, and socially unacceptable. And if you do it, we'll throw you in math prison.

We might even chain you to a dodecahedron and whip you with a Moebius strip.