I've learned that
$$\bbox[8px,border:1px solid black]{\operatorname{Re}(z)= \frac{z+\overline{z}}{2} \qquad \qquad \operatorname{Im}(z)=\frac{z-\overline{z}}{2i}} $$
And that in the number $z=a+bi$, $a$ is the real part and $b$ is the imaginary part. The formulas I mentioned above are used to get $a$ and $b$ alone. But by looking at $z$, I could get the real part just by taking $a$ and ignoring the rest. The same is valid for $b$ and in both cases, without using the formulas. So why are these formulas important? I just learned the basics of complex numbers and still don't know why one needs those formulas.
Best Answer
This is a nice question. We often learn formulae without asking why they are useful.
I've re-typed this post half a dozen times. Every time, I thought I had a nice use, but then found out that I didn't need the formulae after all. Having said that, I think that I have found one.
Using the exponential form $z=\mathrm{e}^{\mathrm{i}\theta} = \cos\theta + \mathrm{i}\sin\theta$, we see that
$$\begin{eqnarray*} \cos\theta &=& \frac{1}{2}\!\left(\mathrm{e}^{\mathrm{i}\theta} + \mathrm{e}^{-\mathrm{i}\theta}\right) \\ \\ \sin\theta &=& \frac{1}{2\mathrm{i}}\!\left(\mathrm{e}^{\mathrm{i}\theta} - \mathrm{e}^{-\mathrm{i}\theta}\right) \end{eqnarray*}$$ We can use these formulae to evaluate sine and cosine over the complex plane: $$\begin{eqnarray*} \cos(\mathrm{i}) &=& \frac{1}{2}\!\left(\mathrm{e}^{\mathrm{i}\mathrm{i}} + \mathrm{e}^{-\mathrm{i}\mathrm{i}}\right) \\ \\ &=& \frac{1}{2}\!\left(\mathrm{e}^{-1} + \mathrm{e}^{1}\right) \\ \\ &=& \frac{1+\mathrm{e}^2}{2\mathrm{e}}\approx 1.543 \end{eqnarray*}$$ I have ignored the multi-valued problem, i.e. $\mathrm{e}^{\mathrm{i}\theta} = \mathrm{e}^{\mathrm{i}(\theta+2\pi k)}$ for all $k \in \mathbb{Z}$.