Training Complex Numbers Complex Roots & Applications
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Complex Roots & Applications

26 min Complex Numbers
The nth roots of a complex number z = r·e^(iθ) are the n values w_k = r^(1/n)·e^(i(θ + 2πk)/n) for k = 0, 1, …, n − 1, equally spaced around a circle of radius r^(1/n). The roots of unity (roots of 1) form the vertices of a regular n-gon on the unit circle. Finding complex roots of polynomials combines the quadratic formula, synthetic division, and the rational root theorem. Applications include solving differential equations with complex characteristic roots (yielding oscillatory solutions), analyzing resonance in mechanical and electrical systems, and computing discrete Fourier transforms.

Complex Roots & Applications

$n$th Roots of a Complex Number

The $n$ distinct $n$th roots of $z = r\operatorname{cis}\theta$ are:

$$w_k = r^{1/n} \operatorname{cis}\left(\frac{\theta + 2\pi k}{n}\right), \quad k = 0, 1, \ldots, n-1$$

Fundamental Theorem of Algebra

Every polynomial of degree $n \geq 1$ with complex coefficients has exactly $n$ roots (counted with multiplicity) in $\mathbb{C}$.

Roots of Unity

The $n$th roots of $1$ are $w_k = \operatorname{cis}\left(\frac{2\pi k}{n}\right)$ for $k = 0,1,\ldots,n-1$. They are equally spaced on the unit circle.

Example 1

Find the square roots of $i$.

  1. $i = \operatorname{cis}(\pi/2)$, $r = 1$, $n = 2$.
  2. $w_0 = \operatorname{cis}(\pi/4) = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i$
  3. $w_1 = \operatorname{cis}(5\pi/4) = -\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i$
Example 2

Find all cube roots of $8$.

  1. $8 = 8\operatorname{cis}(0)$.
  2. $r^{1/3} = 2$.
  3. $w_0 = 2\operatorname{cis}(0) = 2$
  4. $w_1 = 2\operatorname{cis}(2\pi/3) = -1 + \sqrt{3}i$
  5. $w_2 = 2\operatorname{cis}(4\pi/3) = -1 - \sqrt{3}i$
Example 3

Solve $x^2 + 2x + 5 = 0$.

  1. Set up the problem.
  2. $$x = \frac{-2 \pm \sqrt{4-20}}{2} = \frac{-2 \pm \sqrt{-16}}{2} = \frac{-2 \pm 4i}{2} = -1 \pm 2i$$

Practice Problems

1. Find the square roots of $-1$.
2. Find the cube roots of $-8$.
3. List the 4th roots of unity.
4. Solve $x^2 + 4 = 0$.
5. Solve $x^2 - 6x + 13 = 0$.
6. Find $\sqrt{-25}$.
7. How many 5th roots does any nonzero complex number have?
8. The 3rd roots of unity: list them.
9. If $z = 2+i$ is a root of a polynomial with real coefficients, what other root must exist?
10. Solve $x^2 + x + 1 = 0$.
11. Find the square roots of $-4i$.
12. A polynomial of degree 5 has how many roots in $\mathbb{C}$?
Show Answer Key

1. $\pm i$

2. $-2$, $1+\sqrt{3}i$, $1-\sqrt{3}i$

3. $1, i, -1, -i$

4. $x = \pm 2i$

5. $x = 3 \pm 2i$

6. $\pm 5i$

7. $5$

8. $1$, $-\frac{1}{2}+\frac{\sqrt{3}}{2}i$, $-\frac{1}{2}-\frac{\sqrt{3}}{2}i$

9. $2-i$ (complex conjugate root theorem)

10. $x = \frac{-1 \pm \sqrt{-3}}{2} = -\frac{1}{2} \pm \frac{\sqrt{3}}{2}i$

11. $-4i = 4\operatorname{cis}(3\pi/2)$; roots: $2\operatorname{cis}(3\pi/4)$ and $2\operatorname{cis}(7\pi/4)$, i.e. $-\sqrt{2}+\sqrt{2}i$ and $\sqrt{2}-\sqrt{2}i$

12. $5$ (counting multiplicity)

√ nth Roots of a Complex Number
r^(1/n)
Root 0
Root 1
All roots (angular spacing)
Polar Form & De Moivre's Theorem Poles, Zeros & the Complex Plane