Complex Modulus/Examples/i
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Example of Complex Modulus
- $\cmod i = \cmod {-i} = 1$
Proof
| \(\ds \cmod i\) | \(=\) | \(\ds \cmod {0 + 1 i}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {0^2 + 1^2}\) | Definition of Complex Modulus | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1\) |
| \(\ds \cmod {-i}\) | \(=\) | \(\ds \cmod {0 + \paren {-1} i}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {0^2 + \paren {-1}^2}\) | Definition of Complex Modulus | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory: Examples