Nib asked
What happens when you divide by zero?
Dividing by zero has no answer because no number can undo multiplying by zero; infinity is what a limit does, not a value you can land on.
8 steps, drawn by Nib. Read it here, or watch him draw it. Updated .
1the poll
What is twelve divided by zero? Many people say infinity, some say zero. Neither is right. There is no answer at all, and mathematicians call it undefined.
2division undoes
Why? Division is multiplication run backwards. Twelve divided by three is four, because four times three is twelve. So twelve divided by zero would need a number times zero that makes twelve.
3try guesses
So try some guesses. Zero times zero is zero. Five times zero is zero. A thousand times zero is zero. Every guess lands on zero, so nothing can ever make twelve.
4zero over zero
Zero divided by zero is stranger. Now the question is what number times zero makes zero, and every number works. With a million answers and no way to choose, we call it indeterminate.
5the graph
So where does the infinity idea come from? Picture one divided by x as x shrinks toward zero from the right. One over a tenth is ten, over a hundredth is a hundred, over a thousandth is a thousand.
6two sides
But come from the left, with x slightly below zero, and the answers plunge: minus ten, minus a hundred, minus a thousand. Up on one side, down on the other, so no single value fits.
7the whole idea
The whole story: twelve divided by zero has no number that fits, zero divided by zero has too many, and infinity is only what a shrinking denominator heads toward, never a value.
8computers
Computers face the same wall. Many programs stop with an error, and others return a special flag value, called infinity or not-a-number, so you know something broke.
Dividing by zero is not a number too big to write down; it is a question with no answer.