Question

How do you prove that zero exists?

Answer 1

Taking your question at face value (though I think you may have intended something else):

You don't have to. You can construct it.

Explanation:

Suppose the numbers you know about are the counting numbers:

`1, 2, 3,...`

You can quite happily add and multiply these numbers.

Then one day, you can introduce the concept of a number called zero, which you give the symbol `0`.

You give it certain properties to fit in with the schemes of addition and multiplication you already have, such as:

`0 + n = n + 0 = n` for all numbers `n`

`0 * n = n * 0 = 0` for all numbers `n`

You find that `0` fits in well with the other numbers and has this interesting additive identity property.

Next you might start to introduce negative numbers,...