# Addition on Numbers has no Zero Element

## Theorem

On all the number systems:

there exists no zero element for addition.

## Proof

Suppose $z$ is a zero element for addition in a standard number system $\F$.

Then:

 $\displaystyle \forall n \in \F \ \$ $\displaystyle n + z$ $=$ $\displaystyle z$ $\displaystyle \leadsto \ \$ $\displaystyle n$ $=$ $\displaystyle 0$ subtracting $z$ from both sides

As $n$ is arbitrary, and therefore not always $0$, it follows there can be no such $z$.

$\blacksquare$