Dividing by Zero

Don't divide by zero or this could happen!


Just kidding.

  Divide by Zero


In truth:

Dividing by Zero is undefined.


To see why, let us look at what is meant by "division":

Division is splitting into equal parts or groups.

It is the result of "fair sharing".

Example: there are 12 chocolates, and 3 friends want to share them, how do they divide the chocolates?

12 chocolates   12 chocolates divided into 3
12 Chocolates     12 Chocolates Divided by 3

So they get 4 each: 12/3 = 4

Dividing by Zero

Now, let us try dividing the 12 chocolates among zero people, how much does each person get?

Does that question even make sense? No, of course it doesn't.

We can't share among zero people, and we can't divide by 0.

Another Good Reason

After dividing, can we multiply to get back again?

multiply by zero

But multiplying by 0 gives 0, so that won't work.

Once again, dividing by zero gives us difficulties!

Imagine We Could Divide by Zero

Okay, let us imagine we can divide by zero, and see what happens.

That means that things like 10 and 00 would behave like normal numbers.

Try Multiplying By Zero

So let us try using our new "numbers".

For example, we know that zero times any number is zero:

Example: 0×1 = 0, 0×2 = 0, etc

So that should also be true for 10:

0 × 10 = 0

But we could also rearrange it a little like this:

0 × 10 = 00 × 1 = 1

(Careful! I am not saying this is correct! We are assuming that we can divide by zero, so 00 should work the same as 55, which is 1).

Arrggh! If we multiply 10 by zero we could get 0 or 1.

In fact we can't have both possibilites, so we cannot define 10 to be a number.

So it is undefined.

So what is 0/0 ?

0/0 is like asking "how many 0s in 0?"

Are there no zeros in zero at all? Or perhaps there is exactly one zero in zero? Or many zeros?

So 0/0 is indeterminate (it could be any value).

In conclusion:

When we try to divide by zero, things stop making sense

That is all.

But Wait ...

There is a special method where we get closer and closer to zero ... just read Limits (An Introduction) to find out more.