Modulus and Argument (I)

Try this

1. Drag $w$ onto the unit circle ($|w|=1$). Does multiplying by $w$ change the length of $z$? What is the only thing it does?
2. Drag $w$ onto the positive real axis. Does multiplying by $w$ change the direction? What is the only thing it does?
3. For a general $w$: do the light-red triangle $O\text{-}1\text{-}z$ and the green triangle $O\text{-}w\text{-}zw$ have the same shape?
4. Look at the arcs: the red arc is $\arg z$, the blue arc that follows it is $\arg w$, and the green dashed arc is $\arg (zw)$. When does the “argument” reading show a warning?

The point $Z$ corresponds to the complex number $z=a+bi$. The length $r=|z|=\sqrt{a^2+b^2}$ of the vector $\overrightarrow{OZ}$ is the modulus. The angle $\theta$ whose initial side is the positive $x$-axis and whose terminal side is $\overrightarrow{OZ}$ is an argument, written $\mathrm{Arg}z$. There are infinitely many arguments, differing by integer multiples of $360^{\circ }$; the one in $[0,2\pi )$ is the principal argument, written $\arg z$. So

This is the trigonometric form of a complex number. Its whole point is multiplication:

Why

Taking roots: regular polygons

Now solve $w^n=b$ the other way round. Let $b=\rho (\cos \phi +i\sin \phi )$ and $w=s(\cos \alpha +i\sin \alpha )$. De Moivre’s theorem gives $s^n=\rho$ and $n\alpha =\phi +360^{\circ }k$.

Remove one condition

Practice