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If $\displaystyle |a|=1$ then $\displaystyle a=e^{i\theta}$ and so if $\displaystyle z = |z|e^{i\arg (z)}$ then $\displaystyle az = |z|e^{i\theta}e^{i\arg(z)} = |z|e^{i(\theta + \arg (z))}$ which is the same complex number by rotated $\displaystyle \theta$ .
This is Mine 85th Post!!!
Let us see what happens. Pick a number, say $\displaystyle 1+0i$ and graph it on the complex plane. If you multiply this by $\displaystyle i$ we get $\displaystyle 0+i$. Pick a point as $\displaystyle -1+0i$ multiply it by $\displaystyle i$ and it becomes $\displaystyle -i$. Try it for $\displaystyle \pm i$. Can you guess what type of rotation this is?