Addition on Numbers has no Zero Element

Theorem
On all the number systems: there exists no zero element for addition.
 * natural numbers $\N$
 * integers $\Z$
 * rational numbers $\Q$
 * real numbers $\R$
 * complex numbers $\C$

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

Then:

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

Also see

 * Identity Element of Addition on Numbers
 * Zero Element of Multiplication on Numbers