Complex Modulus/Examples/1+2it-t^2 over 1+t^2

Example of Complex Modulus

$\cmod {\dfrac {1 + 2 i t - t^2} {1 + t^2} } = 1$

where:

$t \in \R$ is a real number.

$\ds \cmod {\dfrac {1 + 2 i t - t^2} {1 + t^2} }$

$=$

$\ds \cmod {\dfrac {\left({1 + i t}\right)^2} {\left({1 + i t}\right) \left({1 - i t}\right)} }$

$\ds $

$=$

$\ds \cmod {\dfrac {1 + i t} {1 - i t} }$

$\ds $

$=$

$\ds \dfrac {\cmod {1 + i t} } {\cmod {1 - i t} }$

$\ds $

$=$

$\ds \dfrac {1^2 + t^2} {1^2 + t^2}$

Definition of Complex Modulus

$\ds $

$=$

$\ds 1$

$\blacksquare$

1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1$. Algebraic Theory of Complex Numbers: Exercise $4 \ \text{(iv)}$