The Complex Plane: Polar Form and Geometric Operations

Complex numbers as points

A complex number $a + bi$ carries two real numbers, so it fits naturally on a plane. The real part goes across and the imaginary part goes up.

$$3 + 4i \;\longleftrightarrow\; (3, 4)$$

Real numbers sit on the horizontal axis, and numbers like $2i$ sit on the vertical axis. Every complex number has exactly one point, and every point is exactly one complex number.

Polar form: a distance and an angle

The same point can be located a second way: how far it is from the origin, and in which direction.

The distance is the modulus, written $|z|$:

$$|a + bi| = \sqrt{a^2 + b^2} \qquad |3 + 4i| = \sqrt{9 + 16} = 5$$

The direction is the argument $\theta$, the angle from the positive real axis. For $3 + 4i$ it is about $53.13°$.

A point at distance $r$ in direction $\theta$ has coordinates $(r\cos\theta, r\sin\theta)$ — the unit circle scaled up by $r$. So

$$a + bi = r(\cos\theta + i\sin\theta)$$

Both forms name the same point, because $a = r\cos\theta$ and $b = r\sin\theta$ are two descriptions of one location: one by how far across and up, the other by how far out and at what angle.

Adding, subtracting and conjugating

Adding adds the real parts and the imaginary parts separately, which is exactly how arrows add: place one after the other.

$$(3 + i) + (1 + 2i) = 4 + 3i$$

The sum is the far corner of the parallelogram built on the two arrows. Subtraction reverses one arrow: $z - w$ is the arrow that goes from $w$ to $z$.

The conjugate of $a + bi$ is $a - bi$, the reflection of the point across the real axis, as the first figure shows.

Why multiplying multiplies lengths and adds angles

Multiply two numbers in polar form:

$$r_1(\cos\alpha + i\sin\alpha) \cdot r_2(\cos\beta + i\sin\beta)$$

Expand, using $i^2 = -1$:

$$r_1 r_2\bigl[(\cos\alpha\cos\beta - \sin\alpha\sin\beta) + i(\sin\alpha\cos\beta + \cos\alpha\sin\beta)\bigr]$$

The two brackets are exactly the sum formulas:

$$= r_1 r_2\bigl(\cos(\alpha + \beta) + i\sin(\alpha + \beta)\bigr)$$

Multiplying complex numbers multiplies their moduli and adds their arguments. Every multiplication is a stretch combined with a rotation.

Multiplying by $i$, which has modulus $1$ and argument $90°$, is a pure quarter turn: $i(3 + 4i) = -4 + 3i$, the point $(3, 4)$ rotated $90°$ counterclockwise.

Powers by angle and length

Powers repeat the rule. Take $w = -1 + \sqrt{3}\,i$: its modulus is $\sqrt{1 + 3} = 2$ and its argument is $120°$. Cubing it cubes the modulus and triples the angle:

$$w^3 = 2^3\bigl(\cos 360° + i\sin 360°\bigr) = 8(1 + 0i) = 8$$

Expanding $(-1 + \sqrt{3}\,i)^3$ by hand gives the same $8$, after considerably more work.

Distance and midpoint

The distance between two complex numbers is the length of the arrow from one to the other, which is the modulus of their difference:

$$\text{distance} = |z - w|$$

The midpoint is their average:

$$\text{midpoint} = \frac{z + w}{2}$$

These are the distance and midpoint formulas of the coordinate plane, written in complex numbers.

Worked examples

Common mistakes

Practice problems