Why is $ i^2 \neq (1 + i)^4$?

Question

Today I read that you can see the number $i$ as the rotation of 90° and therefore i^2 is the rotation of 180° or -1.

I also learned that $1+i$ is 45° but if this would be true I should be able to rotate 4 times with 45° and I should also get to 180° or -1 but $(1+i)^4 = -4$

It seems that $(1+i) * (1+i)$ also scales along the $i$ axes by 2.

I am still new to the imaginary number and I am trying to get an intuition.

Answer 1

You can write $i+1$ as $\sqrt{2} (\cos 45^{\circ}+i\sin 45^{\circ})$, so by De Moivre's formula $(i+1)^4=(\sqrt{2})^4(\cos 180^{\circ}+i\sin 180^{\circ})$.

Generally, multipling by $\cos \alpha+i \sin \alpha $ is rotation of $\alpha$, multipling by $\beta \in \mathbb{R}$ is scaling by $\beta$.

Answer 2

Another way of seeing it is that for $\theta = \frac{\pi}{4}, {\frak{Re}} (z) = 1, {\frak{Im}} z = 1$ (i.e. $r = \sqrt{2}$) Euler's expression will be $\sqrt{2} e^{i \frac{\pi}{4}}$, which, after putting it to the power 4 is certainly not equal to $-1$.