The Complex Plane

In this lesson you’ll treat complex numbers as points in a plane, learn modulus and argument, convert to polar and exponential form, and use them to take powers and roots almost without calculation.

Write $z = x + iy$ and plot it at $(x, y)$. That is the entire idea behind the complex plane, sometimes called the Argand diagram, and once you have it nothing about complex numbers is mysterious.

• The real axis is the horizontal one, where $y = 0$.
• The imaginary axis is the vertical one, where $x = 0$.
• Addition is vector addition, exactly as in $\mathbb{R}^2$.
• Conjugation $z \mapsto \bar z$ is reflection across the real axis.

Multiplication is where $\mathbb{C}$ stops behaving like $\mathbb{R}^2$. There is no useful multiplication of plane vectors, but there is for complex numbers, and it has a clean geometric meaning we get to shortly.

The modulus is the distance to the origin:

$$|z| = \sqrt{x^2+y^2} = \sqrt{z\bar z}$$

The argument is the angle from the positive real axis, measured counterclockwise. It is only defined up to multiples of $2\pi$, since rotating a full turn returns you to the same point. The principal argument $\operatorname{Arg} z$ is the unique choice in $(-\pi, \pi]$.

Useful properties, all worth knowing cold:

• $|z_1z_2| = |z_1||z_2|$ and $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$
• $\arg(z_1z_2) = \arg z_1 + \arg z_2$, and arguments subtract on division
• $|z|^2 = z\bar z$, which is how you divide complex numbers
• $\overline{z_1+z_2} = \bar z_1 + \bar z_2$ and $\overline{z_1z_2} = \bar z_1 \bar z_2$
• $|z_1+z_2| \le |z_1|+|z_2|$, the triangle inequality, and it is genuinely the triangle inequality for the picture

Note that $|z_1 + z_2|$ has no simple formula. Moduli behave well under multiplication and badly under addition, which is the reverse of what you might expect.

Reading the right triangle off the diagram,

$$x = r\cos\theta, \qquad y = r\sin\theta, \qquad z = r(\cos\theta + i\sin\theta)$$

Euler’s formula compresses this to

$$e^{i\theta} = \cos\theta + i\sin\theta, \qquad\text{so}\qquad z = re^{i\theta}$$

This is the single most useful identity in the section. Setting $\theta = \pi$ gives $e^{i\pi} = -1$, usually written $e^{i\pi}+1=0$ and often called the most beautiful equation in mathematics. It is also just a statement that rotating 1 by half a turn lands on $-1$.

Multiplication becomes rotation and scaling. If $z_1 = r_1e^{i\theta_1}$ and $z_2 = r_2e^{i\theta_2}$ then

$$z_1z_2 = r_1r_2\,e^{i(\theta_1+\theta_2)}$$

Multiply the lengths, add the angles. In particular multiplying by $i = e^{i\pi/2}$ is exactly a quarter-turn counterclockwise, which is the honest answer to “what does $i$ mean.”

De Moivre’s theorem is just the multiplication rule applied $n$ times:

$$\big(re^{i\theta}\big)^n = r^ne^{in\theta}$$

Roots run the same machinery backwards, and this is where polar form earns its place. The equation $z^n = w$ with $w = \rho e^{i\varphi}$ has exactly $n$ solutions:

$$z_k = \rho^{1/n}\,e^{i(\varphi+2\pi k)/n}, \qquad k = 0, 1, \ldots, n-1$$

All have the same modulus, so they sit on a circle, and their arguments differ by $\frac{2\pi}{n}$, so they are the vertices of a regular $n$-gon. The $n$th roots of unity are the case $w=1$.

Two facts that follow immediately and are used constantly:

• The $n$th roots of unity sum to zero for $n \ge 2$, by symmetry of the polygon.
• They form a cyclic group of order $n$ under multiplication, which is the connection back to abstract algebra.

Complex analysis is done on subsets of $\mathbb{C}$, so it helps to read set descriptions geometrically:

• $|z - a| = r$ - the circle of radius $r$ centred at $a$
• $|z - a| < r$ - the open disc
• $|z - a_1| = |z - a_2|$ - the perpendicular bisector of the segment joining them
• $\operatorname{Re} z > 0$ - the right half plane
• $\alpha < \operatorname{Arg} z < \beta$ - a sector
• $r_1 < |z-a| < r_2$ - an annulus, which is where Laurent series live

Example 1: Put $z = -1 + i\sqrt{3}$ into exponential form.

Solution. Modulus:

$$|z| = \sqrt{(-1)^2+(\sqrt3)^2} = \sqrt{4} = 2$$

The point is in the second quadrant, so the reference angle $\arctan(\sqrt3/1) = 60°$ gives

$$\operatorname{Arg} z = 180° - 60° = 120° = \frac{2\pi}{3}$$ $$z = 2e^{2\pi i/3}$$

Always check the quadrant. A calculator’s $\arctan(y/x)$ returns $-60°$ here, which is the wrong point; the picture is what tells you to add $180°$.

Example 2: Compute $(1+i)^{10}$.

Solution. In Cartesian form this is a nightmare. In polar form, $|1+i| = \sqrt2$ and $\operatorname{Arg}(1+i) = \pi/4$, so

$$(1+i)^{10} = \left(\sqrt2\right)^{10} e^{10i\pi/4} = 32\,e^{5i\pi/2}$$

Reduce the angle mod $2\pi$: $\frac{5\pi}{2} - 2\pi = \frac{\pi}{2}$, so

$$(1+i)^{10} = 32e^{i\pi/2} = 32i$$

∎

This is the example that sells polar form. Ten binomial expansions replaced by one multiplication and one angle reduction.

Example 3: Find all solutions of $z^3 = 8i$.

Solution. Write the right side in polar form: $|8i| = 8$ and $\operatorname{Arg}(8i) = \pi/2$, so $8i = 8e^{i\pi/2}$.

The three cube roots have modulus $8^{1/3} = 2$ and arguments

$$\frac{\pi/2 + 2\pi k}{3} = \frac{\pi}{6},\ \frac{5\pi}{6},\ \frac{3\pi}{2}$$

for $k = 0,1,2$. So

$$z = 2e^{i\pi/6},\quad 2e^{5i\pi/6},\quad 2e^{3i\pi/2}$$

which in Cartesian form is $\sqrt3 + i$, $-\sqrt3+i$, and $-2i$.

Check: the three are $120°$ apart on the circle of radius 2, and $(\sqrt3+i)^3 = 8i$. ✓

Example 4: Show the $n$th roots of unity sum to zero for $n\ge2$.

Solution. They are $1, \omega, \omega^2,\ldots,\omega^{n-1}$ where $\omega = e^{2\pi i/n}$. This is a geometric series:

$$\sum_{k=0}^{n-1}\omega^k = \frac{\omega^n-1}{\omega-1} = \frac{1-1}{\omega-1} = 0$$

valid because $\omega \ne 1$ when $n \ge 2$. ∎

Geometrically these are the vertices of a regular polygon centred at the origin, and the centroid of a regular polygon is its centre. Algebra and geometry give the same answer, which is the normal situation in this subject.

Example 5: Divide $\frac{3+2i}{1-4i}$.

Solution. Multiply above and below by the conjugate of the denominator:

$$\frac{3+2i}{1-4i}\cdot\frac{1+4i}{1+4i} = \frac{3+12i+2i+8i^2}{1+16} = \frac{-5+14i}{17}$$ $$= -\frac{5}{17}+\frac{14}{17}i$$

The trick is that $z\bar z = |z|^2$ is real, so conjugating the denominator clears the $i$ from it. This is the only division algorithm you need.

Example 6: Describe the set $|z-1| = 2|z+1|$.

Solution. Square both sides using $|w|^2 = w\bar w$, with $z = x+iy$:

$$(x-1)^2+y^2 = 4\big[(x+1)^2+y^2\big]$$

Expanding and collecting,

$$3x^2+10x+3+3y^2 = 0 \quad\Longrightarrow\quad x^2+\tfrac{10}{3}x+y^2 = -1$$

Completing the square:

$$\left(x+\tfrac53\right)^2+y^2 = \tfrac{25}{9}-1 = \tfrac{16}{9}$$

A circle centred at $-\frac53$ with radius $\frac43$. ∎

Sets defined by a ratio of distances are circles, called Apollonius circles, and they reappear when we study Möbius transformations.

Example 7: Use polar form to derive a trig identity.

Solution. By De Moivre with $n=3$ and $r=1$,

$$(\cos\theta+i\sin\theta)^3 = \cos3\theta+i\sin3\theta$$

Expand the left side:

$$\cos^3\theta + 3i\cos^2\theta\sin\theta - 3\cos\theta\sin^2\theta - i\sin^3\theta$$

Match real parts:

$$\cos3\theta = \cos^3\theta-3\cos\theta\sin^2\theta = 4\cos^3\theta-3\cos\theta$$

∎

Multiple-angle formulas are a corollary of complex multiplication. This is the cleanest way to generate them, and it is why trigonometry looks so much easier once you have $e^{i\theta}$.

Phasors in electrical engineering. A sinusoidal voltage $V\cos(\omega t+\phi)$ is stored as the single complex number $Ve^{i\phi}$. Adding two sinusoids of the same frequency then becomes adding two complex numbers, which converts a trigonometry problem into arithmetic. This is the standard method in every power systems course.

Rotations in graphics. Multiplying by $e^{i\theta}$ rotates a 2D point about the origin, which is cheaper and more numerically stable than a matrix multiply. The 3D generalization is the quaternion, used in essentially every game engine for smooth camera and character rotation.

Roots of unity and the FFT. The fast Fourier transform is built on the algebraic relations among the $n$th roots of unity. It is arguably the most-executed nontrivial algorithm in the world, and it exists because those roots form a group.

Signal representation. Amplitude and phase are precisely modulus and argument. Any tool that displays a spectrum is showing you $|z|$ and $\arg z$ for a family of complex numbers.

Impedance matching. The Smith chart, standard in radio engineering, is a plot in the complex plane using a Möbius transformation, and the circles on it are Apollonius circles like the one in Example 6.

Crystallography and symmetry. Regular polygons of roots of unity describe the rotational symmetries available to a crystal lattice, which is why fivefold symmetry was long thought impossible in crystals.