Question

What is `5\div 0`?

Answer 1

Undefined

Explanation:

Division by `0` is not defined. When we divide a number `x` by a number `y`, it is the same as multiplying `x` by the multiplicative inverse of `y`, that is, the number `y^(-1)` which has the property `yxxy^(-1)=1`.

If `y` is not `0`, then that number is `1/y`, but if `y=0`, then there is no number `y^(-1)` fulfilling `0xxy^(-1) = 1`, because the product of `0` and any number is `0`. Thus we cannot divide by `0`.

Answer 2

`5 -: 0` is undefined

Explanation:

Another way of putting it:

Division by `0` is (almost) always undefined.

To see why in our current example, suppose there was some number `x` such that:

`x = 5 -: 0`

Then we could multiply both sides of the equation to find:

`x xx 0 = 5`

Note that any number multiplied by `0` is `0`, so this simplifies to:

`0 = 5`

which is false.

So our supposition that there was such a number `x` must have been false.