Topic: Review of complex numbers

## Question

After class today, a student asked me the following question:

$\left| e^{j \omega} \right| = ?$

$e^{j \omega} = cos( \omega) + i*sin( \omega)$

hence,

$\left| e^{j \omega} \right| = \left|cos( \omega) + i*sin( \omega) \right| = \sqrt{cos^2( \omega) + sin^2( \omega)} = 1$

TA's comments: Is this true for all $\omega \in R$? The answer is yes.
Instructor's comment: I would like to propose a more straightforward way to compute this norm using the fact that $|z|^2=z \bar{z}$. Can you try it out? -pm

becasue: $e^{jx} =cos(x)+ jsin(x)$

$| e^{j \omega}|=|cos(\omega) + i*sin(\omega)|=\sqrt{cos(\omega)^2 +sin(\omega)^2}=1$

TA's comments: The point here is to use Euler's formula to write a complex exponential as a complex number. Then the norm(magnitude) and angle(phase) of this complex number can be easily computed.
Instructor's comment: Again, I would argue that using the fact that $|z|^2=z \bar{z}$ is more straightforward. Can you try it out? -pm

$e^{j \omega} = cos( \omega) + i*sin( \omega)$
$\left| e^{j \omega} \right| = \left|cos( \omega) + i*sin( \omega) \right| = \sqrt{cos^2( \omega) + sin^2( \omega)} = 1$