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

\(\displaystyle \cmod {\dfrac {1 + 2 i t - t^2} {1 + t^2} }\) \(=\) \(\displaystyle \cmod {\dfrac {\left({1 + i t}\right)^2} {\left({1 + i t}\right) \left({1 - i t}\right)} }\)
\(\displaystyle \) \(=\) \(\displaystyle \cmod {\dfrac {1 + i t} {1 - i t} }\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\cmod {1 + i t} } {\cmod {1 - i t} }\) Complex Modulus of Quotient of Complex Numbers
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {1^2 + t^2} {1^2 + t^2}\) Definition of Complex Modulus
\(\displaystyle \) \(=\) \(\displaystyle 1\)

$\blacksquare$


Sources