Question

What is x/y/0 ?

Answer 1

`x/(y/0)` and `(x/y)/0` are both undefined, except under special circumstances.

Explanation:

When dealing with arithmetic of ordinary numbers, division by `0` is always undefined, so any resulting expression is undefined.

So both `x/(y/0)` and `(x/y)/0` are undefined.

There are at least two non-ordinary contexts in which it may be defined.

They are called the "real projective line" `RR_oo` and the Riemann sphere `CC_oo`.

Both of these extensions of ordinary numbers add a single point "at infinity" `oo` with some arithmetic rules, but not all arithmetic results in determinate values.

For example:

For any `x in RR_oo`, we have:

`x+oo = oo+x = oo`

For any non-zero `x in RR`, we have:

`x/oo = 0`

`x/0 = oo`

The following expressions are all indeterminate:

`0/0`

`oo/oo`

`oo-oo`

`0*oo`

However, we do find that if `x, y in RR`, with `y != 0` then in the "arithmetic" of `RR_oo` we have:

`x/(y/0) = x/oo = 0`