# What is x/y/0 ?

Apr 8, 2017

$\frac{x}{\frac{y}{0}}$ and $\frac{\frac{x}{y}}{0}$ 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 $\frac{x}{\frac{y}{0}}$ and $\frac{\frac{x}{y}}{0}$ are undefined.

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

They are called the "real projective line" ${\mathbb{R}}_{\infty}$ and the Riemann sphere ${\mathbb{C}}_{\infty}$.

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

For example:

For any $x \in {\mathbb{R}}_{\infty}$, we have:

$x + \infty = \infty + x = \infty$

For any non-zero $x \in \mathbb{R}$, we have:

$\frac{x}{\infty} = 0$

$\frac{x}{0} = \infty$

The following expressions are all indeterminate:

$\frac{0}{0}$

$\frac{\infty}{\infty}$

$\infty - \infty$

$0 \cdot \infty$

However, we do find that if $x , y \in \mathbb{R}$, with $y \ne 0$ then in the "arithmetic" of ${\mathbb{R}}_{\infty}$ we have:

$\frac{x}{\frac{y}{0}} = \frac{x}{\infty} = 0$