The roots of unity

Complex numbers, grade 12 (5 units) Lesson 13 of 18Learning

The roots of unity

The solutions of

\displaystyle {z^n}={1}

The solutions of $\displaystyle {z^n}={1}$ are called the roots of unity of order $\displaystyle n$ . The polar form finds all of them: we substitute $\displaystyle {z}={r\operatorname{cis}\theta}$ and use De Moivre’s theorem.

$\displaystyle {r^n\operatorname{cis}n\theta}={1\operatorname{cis}0}$ .
The moduli are equal: $\displaystyle {r^n}={1}$ , and $\displaystyle r>0$ , so $\displaystyle {r}={1}$ .
The angles are equal up to a multiple of $\displaystyle 2\pi$ : $\displaystyle {n\theta}={2k\pi}$ , that is $\displaystyle {\theta}={\frac{2k\pi}{n}}$ .
$\displaystyle {k}={0,1,\ldots,n-1}$ give $\displaystyle n$ different numbers; $\displaystyle {k}={n}$ gives $\displaystyle {\operatorname{cis}2\pi}={1}$ again.

\displaystyle z_{k}=\operatorname{cis}\frac{2k\pi}{n}

\displaystyle k=0,1,\ldots,n-1

A regular polygon

All the roots of unity lie on the unit circle, $\displaystyle {|z|}={1}$ , and the angle between two neighbouring roots is $\displaystyle \frac{2\pi}{n}$ . So they are the vertices of a regular polygon with $\displaystyle n$ sides, one of whose vertices is at 1.

The coordinate system below shows the six roots of unity of order 6 – the vertices of a regular hexagon: $\displaystyle \operatorname{cis}\frac{k\pi}{3}$ , that is $\displaystyle \pm1$ and $\displaystyle \pm\frac{1}{2}\pm\frac{\sqrt{3}}{2}i$ .

\displaystyle x^{2}+y^{2}=1

\displaystyle z_0=(1,0)

\displaystyle z_1=(0.5,0.866)

\displaystyle z_2=(-0.5,0.866)

\displaystyle z_3=(-1,0)

\displaystyle z_4=(-0.5,-0.866)

\displaystyle z_5=(0.5,-0.866)

Properties of the roots of unity

Write $\displaystyle {\omega}={\operatorname{cis}\frac{2\pi}{n}}$ . By De Moivre’s theorem $\displaystyle {z_k}={\omega^k}$ , so:

The roots of unity are a geometric sequence: $\displaystyle 1,\omega,\omega^2,\ldots,\omega^{n-1}$ , with ratio $\displaystyle \omega$ .
Their sum is 0 (when $\displaystyle n\geq2$ ): by the sum formula of a geometric sequence, $\displaystyle {\frac{\omega^n-1}{\omega-1}}={\frac{1-1}{\omega-1}}={0}$ .
They come in conjugate pairs: $\displaystyle {\bar{\omega}^k}={\omega^{n-k}}$ – the polygon is symmetric about the real axis.
Their product is $\displaystyle {\operatorname{cis}\frac{2\pi(0+1+\ldots+(n-1))}{n}}={\operatorname{cis}(n-1)\pi}$ : $\displaystyle 1$ when $\displaystyle n$ is odd, $\displaystyle -1$ when $\displaystyle n$ is even.

Exercise 1

Solve. Write each solution in algebraic or polar form.

Exercise 2

Solve $\displaystyle {z^3-1}={0}$ by factoring: $\displaystyle {z^3-1}={(z-1)(z^2+z+1)}$ .
Check that you got the solutions of Exercise 1, part a.

Exercise 3

$\displaystyle {\omega}={\operatorname{cis}\frac{2\pi}{3}}$ . Calculate.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

The roots of unity of order 4 are the vertices of a square.
Calculate the length of the square’s side and its area.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 5

Prove that the sum of the squares of the roots of unity of order $\displaystyle n$ is 0, when $\displaystyle n\geq3$ .
Hint: the squares are a geometric sequence too.

Formulas: complex numbers, grade 12 (5 units)

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The algebraic form and the four operations

The number $\displaystyle i$
$\displaystyle i^{2}=-1$
The algebraic form
$\displaystyle z=a+bi$ $\displaystyle a=\operatorname{Re}(z),\ b=\operatorname{Im}(z)$
Equality of complex numbers
$\displaystyle a+bi=c+di\iff a=c,\ b=d$
Multiplying
$\displaystyle (a+bi)(c+di)$ $\displaystyle =$ $\displaystyle (ac-bd)+(ad+bc)i$
The conjugate
$\displaystyle \bar{z}=a-bi$
A number times its conjugate
$\displaystyle z\cdot\bar{z}$ $\displaystyle =$ $\displaystyle a^{2}+b^{2}$
Dividing – times the denominator’s conjugate
$\displaystyle \frac{z_{1}}{z_{2}}$ $\displaystyle =$ $\displaystyle \frac{z_{1}\cdot\bar{z}_{2}}{z_{2}\cdot\bar{z}_{2}}$

The modulus and the conjugate

The modulus – the distance from the origin
$\displaystyle |z|$ $\displaystyle =$ $\displaystyle \sqrt{a^{2}+b^{2}}$
The distance between two points
$\displaystyle |z_{1}-z_{2}|$ $\displaystyle =$ $\displaystyle \sqrt{(a_{1}-a_{2})^{2}+(b_{1}-b_{2})^{2}}$
The product with the conjugate and the modulus
$\displaystyle z\cdot\bar{z}$ $\displaystyle =$ $\displaystyle |z|^{2}$
The modulus of a product
$\displaystyle |z_{1}\cdot z_{2}|$ $\displaystyle =$ $\displaystyle |z_{1}|\cdot|z_{2}|$
The conjugate of a sum
$\displaystyle \overline{z_{1}+z_{2}}$ $\displaystyle =$ $\displaystyle \bar{z}_{1}+\bar{z}_{2}$
The conjugate of a product
$\displaystyle \overline{z_{1}\cdot z_{2}}$ $\displaystyle =$ $\displaystyle \bar{z}_{1}\cdot\bar{z}_{2}$
A number plus its conjugate
$\displaystyle z+\bar{z}$ $\displaystyle =$ $\displaystyle 2\operatorname{Re}(z)$
The triangle inequality
$\displaystyle |z_{1}+z_{2}|\le|z_{1}|+|z_{2}|$

The quadratic equation

The quadratic formula
$\displaystyle z_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
The sum of the roots
$\displaystyle z_{1}+z_{2}$ $\displaystyle =$ $\displaystyle -\frac{b}{a}$
The product of the roots
$\displaystyle z_{1}\cdot z_{2}$ $\displaystyle =$ $\displaystyle \frac{c}{a}$

The polar form (angles in radians)

The polar form
$\displaystyle z=r(\cos\theta+i\sin\theta)$ $\displaystyle r=|z|$
The notation cis
$\displaystyle r\operatorname{cis}\theta$ $\displaystyle =$ $\displaystyle r(\cos\theta+i\sin\theta)$
The real part
$\displaystyle a=r\cos\theta$
The imaginary part
$\displaystyle b=r\sin\theta$
The angle – by the quadrant of the point
$\displaystyle \tan\theta=\frac{b}{a}$ $\displaystyle a\neq0$
A product in polar form
$\displaystyle z_{1}\cdot z_{2}=r_{1}\cdot r_{2}\left[\cos(\theta_{1}+\theta_{2})+i\sin(\theta_{1}+\theta_{2})\right]$
A quotient in polar form
$\displaystyle \frac{z_{1}}{z_{2}}=\frac{r_{1}}{r_{2}}\operatorname{cis}(\theta_{1}-\theta_{2})$
De Moivre’s theorem
$\displaystyle \left[R(\cos\varphi+i\sin\varphi)\right]^{n}$ $\displaystyle =$ $\displaystyle R^{n}(\cos n\varphi+i\sin n\varphi)$
The solutions of $\displaystyle {z^n}={R(\cos\varphi+i\sin\varphi)}$
$\displaystyle z_{k}=\sqrt[n]{R}\left[\cos\left(\frac{\varphi}{n}+\frac{2k\pi}{n}\right)+i\sin\left(\frac{\varphi}{n}+\frac{2k\pi}{n}\right)\right]$ $\displaystyle z^{n}=R(\cos\varphi+i\sin\varphi),\ k=0,1,\ldots,n-1$

Loci and rotation

The circle with centre $\displaystyle z_0$ and radius $\displaystyle R$
$\displaystyle |z-z_{0}|=R$
The perpendicular bisector of the segment $\displaystyle z_1z_2$
$\displaystyle |z-z_{1}|=|z-z_{2}|$
Apollonius’ circle
$\displaystyle |z-z_{1}|=m|z-z_{2}|$ $\displaystyle m>0,\ m\neq1$
Rotating by $\displaystyle \theta$ about $\displaystyle z_0$
$\displaystyle w=z_{0}+(z-z_{0})\operatorname{cis}\theta$