Training Complex Numbers Complex Roots & Applications
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Complex Roots & Applications

26 min Complex Numbers

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 $i$.

$i = \operatorname{cis}(\pi/2)$, $r = 1$, $n = 2$.

$w_0 = \operatorname{cis}(\pi/4) = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i$

$w_1 = \operatorname{cis}(5\pi/4) = -\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i$

Example 2

Find all cube roots of $8$.

$8 = 8\operatorname{cis}(0)$. $r^{1/3} = 2$.

$w_0 = 2\operatorname{cis}(0) = 2$

$w_1 = 2\operatorname{cis}(2\pi/3) = -1 + \sqrt{3}i$

$w_2 = 2\operatorname{cis}(4\pi/3) = -1 - \sqrt{3}i$

Example 3

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

$$x = \frac{-2 \pm \sqrt{4-20}}{2} = \frac{-2 \pm \sqrt{-16}}{2} = \frac{-2 \pm 4i}{2} = -1 \pm 2i$$

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)

Polar Form & De Moivre's Theorem Poles, Zeros & the Complex Plane