De Moivre's formula and nth roots

Parametric equations

Write $z$ in polar form as $z=r{e}^{i\theta }$. A power and a root then change only two things: the power of the modulus, and the multiple of the argument.

This is De Moivre’s formula. In the other direction, $w=r{e}^{i\theta }$ has $n$ nth roots:

They lie on one circle, and neighboring arguments differ by $2\pi /n$. If $\theta$ is the principal argument of $w$, the root with $k=0$ is the principal root. When $w=0$ the only root is the origin, and the argument is undefined.

Interaction

• Power zⁿ / Roots: switch between the chain of arrows $z,z^2,\dots ,z^n$ and the $n$ roots of $w$
• Order n: increase $n$, and watch the argument get multiplied and the roots divide the circle into finer equal parts
• Drag $z$, or the $w$ under the root, in the complex plane, and read how the modulus and the argument rewrite the result

What to notice

• A power multiplies the argument by $n$ and raises the modulus to that power. When $|z|>1$ the arrows run outward. When $|z|<1$ they pull in toward the origin.
• When $w\ne 0$, the $n$ roots are equally spaced on the circle of radius $|w{|}^{1/n}$. Raising any one of them to the $n$th power returns $w$.
• This is the same fact as multiplying two complex numbers by adding their arguments. Raising $z$ to the $n$th power adds that same argument $n$ times.

Related

• Polar form of a complex number
• Geometric significance of complex arithmetic
• Euler’s formula, rotating
• Complex numbers and Euler’s formula

Further reading

• De Moivre’s formula (Wikipedia)
• nth root (Wikipedia)