Theory

Theory 1

The exponential notation leads to a formula for a complex $n^{\text{th}}$ root of any complex number:

$$\sqrt[n]{r{e}^{i\theta }}=\sqrt[n]{r}{e}^{i\frac{\theta }{n}}$$

$n$ distinct roots

Every complex number actually has $n$ distinct $n^{\text{th}}$ roots!

That’s two square roots, three cube roots, four $4^{\text{th}}$ roots, etc.

All complex roots

The complex roots of $z=r{e}^{i\theta }$ are given by this formula:

$$w_k=\sqrt[n]{r}\cdot e^{i(\frac{\theta }{n}+k\frac{2\pi }{n})}\text{for each }k=0,1,2,\dots ,n-1$$

In Cartesian notation:

$$w_k=\sqrt[n]{r}\cos (\frac{\theta }{n}+k\frac{2\pi }{n})+\sqrt[n]{r}\sin (\frac{\theta }{n}+k\frac{2\pi }{n})i$$

In words:

• Start with the basic root: $\sqrt[n]{r}\cdot e^{i\frac{\theta }{n}}$
• Rotate by increments of $\frac{2\pi }{n}$ to get all other roots
  • After $n$ distinct roots, this process repeats itself

Extra - Complex roots proof

We must verify that $w_k^n=r{e}^{i\theta }$:

$$\begin{array}{c}{(\sqrt[n]{r}\cdot e^{i(\frac{\theta }{n}+k\frac{2\pi }{n})})}^n\gg \gg r^{\frac{n}{n}}\cdot e^{i(\frac{\theta }{n}+k\frac{2\pi }{n})n}\\ \\ \gg \gg r\cdot e^{i(\theta +2\pi k)}\gg \gg r{e}^{i\theta }{e}^{i2\pi k}\gg \gg r{e}^{i\theta }\end{array}$$