Gaussian Integers are Closed under Negation
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Theorem
The set of Gaussian integers $\Z \sqbrk i$ is closed under negation:
- $\forall x \in \Z \sqbrk i: -x \in \Z \sqbrk i$
Proof
Let $x$ be a Gaussian integer.
Then:
\(\ds \exists a, b \in \Z: \, \) | \(\ds x\) | \(=\) | \(\ds a + b i\) | Definition of Gaussian Integer | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds -x\) | \(=\) | \(\ds -a - b i\) | Definition of Complex Negation Function | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds -x\) | \(\in\) | \(\ds \Z \sqbrk i\) | Definition of Negative Integer: $-a \in \Z$, and Integer Subtraction is Closed |
$\blacksquare$