Gaussian Integers are Closed under Subtraction

Theorem

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

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

Proof

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

Then:

\(\ds x - y\)

\(=\)

\(\ds x + \paren {-y}\)

Definition of Complex Subtraction

\(\ds \leadsto \ \ \)

\(\ds x + \paren {-y}\)

\(\in\)

\(\ds \Z \sqbrk i\)

Gaussian Integers are Closed under Addition and Gaussian Integers are Closed under Negation

\(\ds \leadsto \ \ \)

\(\ds x - y\)

\(\in\)

\(\ds \Z \sqbrk i\)

Definition of Complex Subtraction

$\blacksquare$

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