Chapter 2: The Complex Plane and Polar Form
The Gaussian plane, polar form, and De Moivre's theorem
Introduction (high-school level)
The Complex Plane
A complex number $z = a + bi$ can be regarded as a pair of two real numbers $(a, b)$. This lets us depict complex numbers as points in a plane.
Definition: the complex plane
The complex plane (also called the Gaussian plane) is the plane in which complex numbers are represented as points.
- Horizontal axis (x-axis): the real axis — represents the real part
- Vertical axis (y-axis): the imaginary axis — represents the imaginary part
The complex number $z = a + bi$ corresponds to the point $(a, b)$.
Basic quantities and their geometric meaning
For $z = a + bi$, the definitions of the basic quantities and their geometric meaning on the complex plane are as follows.
- Real number: when $b = 0$ (so $z = a$). Geometrically, a point on the real axis.
- Pure imaginary number: when $a = 0$ and $b \neq 0$ (so $z = bi$). Geometrically, a point on the imaginary axis (excluding the origin).
- Modulus $|z| = \sqrt{a^2 + b^2}$. Geometrically, the distance from the origin to $z$.
- Complex conjugate $\bar{z} = a - bi$. Geometrically, the reflection of $z$ across the real axis.
Polar Form
A complex number can be expressed not only in Cartesian coordinates $(a, b)$ but also in polar coordinates $(r, \theta)$.
Definition: polar form
Let $z \neq 0$ be a complex number. The polar form of $z$ is:
$$z = r(\cos\theta + i\sin\theta)$$where:
- $r = |z|$: the distance from the origin (the modulus, or radius)
- $\theta$: the angle from the positive real axis to the point $z$ (the argument)
The argument is written $\mathrm{arg}(z)$; restricting it to $-\pi < \theta \leq \pi$ gives the principal value, sometimes written $\mathrm{Arg}(z)$.
Relation between Cartesian and polar coordinates
From $z = a + bi = r(\cos\theta + i\sin\theta)$:
- $a = r\cos\theta$
- $b = r\sin\theta$
- $r = \sqrt{a^2 + b^2}$
- $\tan\theta = \displaystyle\dfrac{b}{a}$ (when $a \neq 0$)
Example: converting to polar form
Express $z = 1 + i$ in polar form.
- $r = |z| = \sqrt{1^2 + 1^2} = \sqrt{2}$
- $\tan\theta = \dfrac{1}{1} = 1$, and since $z$ lies in the first quadrant, $\theta = \dfrac{\pi}{4}$
Hence $z = \sqrt{2}\left(\cos\dfrac{\pi}{4} + i\sin\dfrac{\pi}{4}\right)$.
Example: complex numbers at special angles
| $\theta$ | Polar form | Cartesian form |
|---|---|---|
| $0$ | $\cos 0 + i\sin 0$ | $1$ |
| $\dfrac{\pi}{2}$ | $\cos\dfrac{\pi}{2} + i\sin\dfrac{\pi}{2}$ | $i$ |
| $\pi$ | $\cos\pi + i\sin\pi$ | $-1$ |
| $\dfrac{3\pi}{2}$ | $\cos\dfrac{3\pi}{2} + i\sin\dfrac{3\pi}{2}$ | $-i$ |
This table lists the angles for $0 \le \theta < 2\pi$. For the principal value ($-\pi < \theta \le \pi$), $\dfrac{3\pi}{2}$ corresponds to $-\dfrac{\pi}{2}$ (the same point $-i$).
Multiplication and Division in Polar Form
Theorem: multiplication in polar form
For $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$:
$$z_1 z_2 = r_1 r_2 \left(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)\right)$$That is, the moduli multiply and the arguments add.
Proof
\begin{align*} z_1 z_2 &= r_1 r_2 (\cos\theta_1 + i\sin\theta_1)(\cos\theta_2 + i\sin\theta_2) \\ &= r_1 r_2 \left[(\cos\theta_1\cos\theta_2 - \sin\theta_1\sin\theta_2) + i(\sin\theta_1\cos\theta_2 + \cos\theta_1\sin\theta_2)\right] \\ &= r_1 r_2 \left(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)\right) \end{align*}The last step uses the angle-addition formulas for sine and cosine.
Theorem: division in polar form
For $z_2 \neq 0$:
$$\dfrac{z_1}{z_2} = \dfrac{r_1}{r_2} \left(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)\right)$$That is, the moduli divide and the arguments subtract.
Proof
Multiply the numerator and denominator by the conjugate $\cos\theta_2 - i\sin\theta_2$ of $z_2$.
\begin{align*} \dfrac{z_1}{z_2} &= \dfrac{r_1(\cos\theta_1 + i\sin\theta_1)}{r_2(\cos\theta_2 + i\sin\theta_2)} \\ &= \dfrac{r_1}{r_2} \cdot \dfrac{(\cos\theta_1 + i\sin\theta_1)(\cos\theta_2 - i\sin\theta_2)}{(\cos\theta_2 + i\sin\theta_2)(\cos\theta_2 - i\sin\theta_2)} \\ &= \dfrac{r_1}{r_2} \cdot \dfrac{(\cos\theta_1\cos\theta_2 + \sin\theta_1\sin\theta_2) + i(\sin\theta_1\cos\theta_2 - \cos\theta_1\sin\theta_2)}{\cos^2\theta_2 + \sin^2\theta_2} \\ &= \dfrac{r_1}{r_2} \left(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)\right) \end{align*}The denominator becomes $\cos^2\theta_2 + \sin^2\theta_2 = 1$, and the numerator simplifies through the angle-subtraction formulas for sine and cosine.
Geometric meaning
Multiplying a complex number $z$ by $w$:
- rotates it about the origin by $\mathrm{arg}(w)$
- scales its distance from the origin by a factor of $|w|$
In particular, when $|w| = 1$, multiplying by $w$ is a pure rotation.
De Moivre's Theorem
Theorem: De Moivre's theorem
For any integer $n$:
$$(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$$Proof (for $n \geq 1$, by induction)
For $n = 1$: $(\cos\theta + i\sin\theta)^1 = \cos\theta + i\sin\theta$. Clearly true.
Assume it holds for $n = k$: $(\cos\theta + i\sin\theta)^k = \cos(k\theta) + i\sin(k\theta)$.
For $n = k+1$:
\begin{align*} (\cos\theta + i\sin\theta)^{k+1} &= (\cos\theta + i\sin\theta)^k \cdot (\cos\theta + i\sin\theta) \\ &= (\cos(k\theta) + i\sin(k\theta))(\cos\theta + i\sin\theta) \\ &= \cos((k+1)\theta) + i\sin((k+1)\theta) \end{align*}The last equality uses multiplication in polar form.
For $n = 0$: $(\cos\theta + i\sin\theta)^0 = 1 = \cos 0 + i\sin 0$. True.
For $n < 0$: writing $n = -m$ ($m > 0$),
$$(\cos\theta + i\sin\theta)^{-m} = \dfrac{1}{(\cos\theta + i\sin\theta)^m} = \dfrac{1}{\cos(m\theta) + i\sin(m\theta)}$$$= \cos(-m\theta) + i\sin(-m\theta) = \cos(n\theta) + i\sin(n\theta)$.
Example: applying De Moivre's theorem
Compute $(1 + i)^8$.
From $1 + i = \sqrt{2}\left(\cos\dfrac{\pi}{4} + i\sin\dfrac{\pi}{4}\right)$:
\begin{align*} (1 + i)^8 &= \left(\sqrt{2}\right)^8 \left(\cos\dfrac{\pi}{4} + i\sin\dfrac{\pi}{4}\right)^8 \\ &= 16 \left(\cos\dfrac{8\pi}{4} + i\sin\dfrac{8\pi}{4}\right) \\ &= 16(\cos 2\pi + i\sin 2\pi) \\ &= 16 \cdot 1 = 16 \end{align*}nth Roots
In the complex numbers, every nonzero number has $n$ distinct $n$th roots.
Theorem: nth roots of a complex number
Let $w \neq 0$ be a complex number with $w = r(\cos\phi + i\sin\phi)$. The solutions of $z^n = w$ are:
$$z_k = \sqrt[n]{r}\left(\cos\dfrac{\phi + 2k\pi}{n} + i\sin\dfrac{\phi + 2k\pi}{n}\right) \quad (k = 0, 1, 2, \ldots, n-1)$$That is, there are exactly $n$ solutions.
Geometric interpretation
The $n$th roots lie on the circle of radius $\sqrt[n]{r}$ centred at the origin, equally spaced (by an angle $\dfrac{2\pi}{n}$). The first root $z_0$ has argument $\dfrac{\phi}{n}$, and the remaining roots follow at steps of $\dfrac{2\pi}{n}$. When $\phi \neq 0$, $z_0$ does not lie on the real axis, and the whole regular polygon is rotated about the origin by $\dfrac{\phi}{n}$.
Example: the cube roots of $w = 8(\cos 120^\circ + i\sin 120^\circ)$
Find the cube roots of $w = 8(\cos 120^\circ + i\sin 120^\circ) = -4 + 4\sqrt{3}\,i$. From the polar form $|w| = 8$, so by the $n = 3$ formula:
\begin{align*} z_k &= \sqrt[3]{8}\left(\cos\dfrac{120^\circ + 360^\circ k}{3} + i\sin\dfrac{120^\circ + 360^\circ k}{3}\right) \\ &= 2\big(\cos(40^\circ + 120^\circ k) + i\sin(40^\circ + 120^\circ k)\big) \quad (k = 0, 1, 2) \end{align*}- $z_0 = 2(\cos 40^\circ + i\sin 40^\circ) \approx 1.53 + 1.29\,i$
- $z_1 = 2(\cos 160^\circ + i\sin 160^\circ) \approx -1.88 + 0.68\,i$
- $z_2 = 2(\cos 280^\circ + i\sin 280^\circ) \approx 0.35 - 1.97\,i$
Each root lies on the circle of radius $\sqrt[3]{8} = 2$, and the argument $40^\circ = \dfrac{120^\circ}{3}$ of the first root $z_0$ shows that taking the cube root trisects the argument of $w$.
Summary
- In the complex plane, a complex number $z = a + bi$ is drawn as the point $(a, b)$
- In polar form $z = r(\cos\theta + i\sin\theta)$, $r$ is the modulus and $\theta$ the argument
- Under multiplication the moduli multiply and the arguments add
- De Moivre's theorem: $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$
- A complex number has exactly $n$ $n$th roots, equally spaced around a circle
Frequently Asked Questions
Q1: What is the complex plane?
A: The plane in which a complex number $z = x + iy$ is drawn as the point $(x, y)$ is called the Gaussian plane (complex plane, or Argand diagram). The horizontal axis is the real part Re$(z) = x$ and the vertical axis the imaginary part Im$(z) = y$. $|z| = \sqrt{x^2 + y^2}$ is the modulus (the magnitude of the complex number) and $\arg(z) = \arctan(y/x)$ is the argument.
Q2: What is polar form?
A: Writing a complex number $z = x+iy$ as $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$ (with $r = |z|$, $\theta = \arg z$) is called its polar form. Multiplication becomes simple: $z_1 z_2 = r_1 r_2 e^{i(\theta_1+\theta_2)}$ (the moduli multiply, the arguments add).
Q3: What is De Moivre's theorem?
A: It is the identity $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$ for every integer $n$. It follows naturally because the $n$th power of $e^{i\theta}$ is $e^{in\theta}$, and it is used to find the roots of $z^n = 1$ (the $n$th roots of unity).