Addition on Numbers has no Zero Element

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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 $z$.

$\blacksquare$


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