Consider the complex numbers $a = \frac{(1+i)^5}{(1-i)^3}$ and $b = e^{3-\pi i}$.
How do I calculate the real and imaginary part of these numbers? What is the general approach to calculate these parts?
I thought about reforming them to the form $x + i\cdot y$ which might be possible for a, but what about b?
I just started occupying with complex numbers and don't yet understand the whole context.
Best Answer
Try to understand and prove each step:
$$\begin{align*}\bullet&\;\;\frac{1+i}{1-i}=i\implies \frac{(1+i)^5}{(1-i)^3}=\left(\frac{1+i}{1-i}\right)^3(1+i)^2=i^3\cdot2i=(-i)(2i)=2\\{}\\\bullet&\;\;e^{b-\pi i}=e^be^{-\pi i}=e^b\left(\cos\pi-i\sin\pi\right)=-e^b\end{align*}$$