Sometimes you can. Sometimes you cannot.

While exploring and developing our idea of number we have invented new numbers that allow us to give answers to questions that we previously could not answer with a number.

Maybe you are a farmer (thousands of years ago) and you want to divide your field into three equal parts, you measure distance by ‘chains’ in your part of the world. There is actually a chain kept at the surveyor’s house which is carefully laid out to mark off distances. If we measured that chain today wr would say it is a bit over `20`m long. Your field is `5` chains wide. You try dividing it into sections `2` chains wide. But they are too big, after you mark off two of them there is not enough left for a third one the same size. So you try sections `1` chain wide. But you have a big part of your field left over, they are too small. How do you describe the width a field which is exactly right? It must be more than `1` but less than `2`, but `2` is right after `1`, you do not have any numbers in between!

So you make up fractions. The length you are after is \(1{2\over3}\) chains. The surveyor folds his chain into three equal parts, and marks the ‘thirds’.

Then one winter you run out of wheat, but your neighbors still have more than enough stored in their big jars. You now invent negative numbers, to represent the idea of borrowed wheat.

But that comes along with the notion of exactly zero.

How does that fit with your idea of dividing? Can you make a fraction with that new number as the denominator? Can you divide by zero? The answer must be bigger than any other number.

Maybe you should give that answer a name, and use it like a number.

Let’s try `infty`.

But then try `+` and `×`. What about `infty - infty`. Or `infty div infty`

We could find ways to do these with fractions, with negative numbers, with real numbers, even with Complex Numbers (after we invented \(\ \rm i\ \) as the solution to `\ x^2+1=0` ). But we cannot achieve a good working definition of `infty` that lets us use it in our number algebra.

Infinitessimals are a related problem.

Consider the wolf and the sprinter, a challenge has kept us wondering for millenia.