Gaussian Integers are Closed under Multiplication
Jump to navigation
Jump to search
Theorem
The set of Gaussian integers $\Z \sqbrk i$ is closed under multiplication:
- $\forall x, y \in \Z \sqbrk i: x \times y \in \Z \sqbrk i$
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 | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x \times y\) | \(=\) | \(\ds \paren {a c - b d} + i \paren {a d + b c}\) | Definition of Complex Multiplication | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x \times y\) | \(\in\) | \(\ds \Z \sqbrk i\) | Integer Addition is Closed, Integer Subtraction is Closed, Integer Multiplication is Closed |
$\blacksquare$
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