Addition on Numbers has no Zero Element

From ProofWiki
Jump to navigation Jump to search

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:

\(\ds \forall n \in \F: \, \) \(\ds n + z\) \(=\) \(\ds z\)
\(\ds \leadsto \ \ \) \(\ds n\) \(=\) \(\ds 0\) subtracting $z$ from both sides

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

$\blacksquare$


Also see


Sources