Gaussian Integers are Closed under Addition

Theorem

The set of Gaussian integers $\Z \sqbrk i$ is closed under addition:

$\forall x, y \in \Z \sqbrk i: x + y \in \Z \sqbrk i$

Proof

Let $x$ and $y$ be Gaussian integers.

Then:

\(\ds \exists a, b \in \Z: \, \)

\(\ds x\)

\(=\)

\(\ds a + b i\)

Definition of Gaussian Integer

\(\ds \exists c, d \in \Z: \, \)

\(\ds y\)

\(=\)

\(\ds c + d i\)

Definition of Gaussian Integer

\(\ds \leadsto \ \ \)

\(\ds x + y\)

\(=\)

\(\ds \paren {a + c} + \paren {b + d} i\)

Definition of Complex Addition

\(\ds \leadsto \ \ \)

\(\ds x + y\)

\(\in\)

\(\ds \Z \sqbrk i\)

Integer Addition is Closed: $a + b \in \Z$ and $c + d \in \Z$

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Gaussian integer
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Gaussian integer