Gaussian Integers are Closed under Negation

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

$\blacksquare$