Complex Modulus/Examples/1+2it-t^2 over 1+t^2
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Example of Complex Modulus
- $\cmod {\dfrac {1 + 2 i t - t^2} {1 + t^2} } = 1$
where:
- $t \in \R$ is a real number.
Proof
\(\ds \cmod {\dfrac {1 + 2 i t - t^2} {1 + t^2} }\) | \(=\) | \(\ds \cmod {\dfrac {\left({1 + i t}\right)^2} {\left({1 + i t}\right) \left({1 - i t}\right)} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {\dfrac {1 + i t} {1 - i t} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\cmod {1 + i t} } {\cmod {1 - i t} }\) | Complex Modulus of Quotient of Complex Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {1^2 + t^2} {1^2 + t^2}\) | Definition of Complex Modulus | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1$. Algebraic Theory of Complex Numbers: Exercise $4 \ \text{(iv)}$