Addition on Numbers has no Zero Element
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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 $74$