# 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.

## 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 $\displaystyle$ $=$ $\displaystyle 1$

$\blacksquare$