Gaussian Integers are Closed under Addition
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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