Addition on Numbers has no Zero Element

Theorem

On all the number systems:

  • natural numbers $\N$
  • integers $\Z$
  • rational numbers $\Q$
  • real numbers $\R$
  • complex numbers $\C$

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

  • 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.3$. Units and zeros: Example $74$
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