The Complex Plane — Mathematics Knowledge System

Overview

The complex plane (also called the Argand plane or z-plane) is the plane formed by identifying each complex number z = x + iy with the point (x, y) in a Cartesian coordinate system. The horizontal axis, called the real axis, carries the real numbers; the vertical axis, called the imaginary axis, carries the purely imaginary numbers. This identification turns complex-number arithmetic into plane geometry: addition becomes vector addition, and multiplication becomes a combined scaling and rotation. The complex plane is the natural setting for complex analysis, where functions of a complex variable are studied via their geometric action on this plane.

Intuition

A real number lives on a single line, but a complex number a + bi needs two coordinates to specify — its real part a and its imaginary part b — so it naturally lives in a plane rather than on a line. Once you plot z = a + bi as the point (a, b), everyday operations get a geometric meaning: adding two complex numbers is exactly vector addition (tip-to-tail), and multiplying by a complex number of modulus r and argument θ scales every point by r and rotates it by θ about the origin. This is why multiplying by i (modulus 1, argument 90°) rotates a point a quarter turn counterclockwise.

Interactive Graph

Formal Definition

Definition

Every complex number z can be written in rectangular (Cartesian) form or in polar form, both realized as coordinates in the complex plane:

z = x + iy,\quad x,y \in \mathbb{R}

x is the real part (horizontal coordinate), y is the imaginary part (vertical coordinate)

Rectangular form

z = r e^{i\theta} = r(\cos\theta + i\sin\theta)

r = |z| = \sqrt{x^2+y^2} is the modulus (distance from the origin); θ = \arg(z) is the argument (angle from the positive real axis)

Polar form

Notation

Notation	Meaning
$\operatorname{Re}(z),\ \operatorname{Im}(z)$	The real and imaginary parts of z, i.e. the horizontal and vertical coordinates
$\|z\| = \sqrt{x^2+y^2}$	Modulus — Euclidean distance of the point z from the origin
$\arg(z) = \theta$	Argument — the angle the ray from the origin to z makes with the positive real axis, usually taken in (-π, π]
$\mathbb{C}$	The set of complex numbers, identified with the plane ℝ²

Properties

Addition as vector addition

(x_1+iy_1)+(x_2+iy_2) = (x_1+x_2) + i(y_1+y_2)

Example: Geometrically, tip-to-tail vector addition of the two points

Multiplication as scale + rotate

r_1e^{i\theta_1} \cdot r_2e^{i\theta_2} = r_1r_2\, e^{i(\theta_1+\theta_2)}

Example: Moduli multiply, arguments add

Modulus of a difference is distance

|z_1 - z_2| = \text{Euclidean distance between the points } z_1 \text{ and } z_2

Conjugation as reflection

\bar{z} = x - iy \text{ is the reflection of } z \text{ across the real axis}

Applications

The complex plane underlies phasor diagrams in AC circuit analysis, where impedances and voltages are plotted as points/vectors whose angles represent phase.

Worked Examples

z has real part -2 and imaginary part 3, so it is the point (-2, 3), in the second quadrant.
$(x,y) = (-2, 3)$
Compute the modulus.
$|z| = \sqrt{(-2)^2 + 3^2} = \sqrt{13} \approx 3.606$
Compute the argument, adjusting for the second quadrant since x < 0, y > 0.
$\arg(z) = \pi - \arctan\!\left(\frac{3}{2}\right) \approx \pi - 0.9828 \approx 2.159 \text{ rad} \approx 123.69^\circ$

Answer: |z| = √13 ≈ 3.606, arg(z) ≈ 123.69°

Practice Problems

Difficulty 4/10

Convert z = 1 + i√3 to polar form.

Difficulty 5/10

If z1 = 3 + 4i and z2 = 1 - 2i, find the distance between the points z1 and z2 on the complex plane.

Common Mistakes

Common Mistake

Computing arg(z) as plain arctan(y/x) without adjusting for the quadrant.

arctan(y/x) only gives the correct argument directly in the first and fourth quadrants (x > 0). For x < 0, add or subtract π (180°) as appropriate; for x = 0, the argument is ±π/2 depending on the sign of y.

Common Mistake

Confusing the imaginary axis with the imaginary part itself, e.g. plotting bi at height 'bi' instead of height b.

The vertical coordinate of the point representing x + iy is the real number y (the imaginary part), not the imaginary quantity iy.

Summary

The complex plane identifies z = x + iy with the Cartesian point (x, y): real part → horizontal axis, imaginary part → vertical axis.
Polar form z = re^{iθ} encodes the same point via modulus r = |z| (distance from origin) and argument θ = arg(z) (angle from positive real axis).
Addition of complex numbers is vector addition; multiplication multiplies moduli and adds arguments (scale + rotate).
Distance between two complex numbers z1, z2 in the plane is |z1 − z2|.
The complex plane is the geometric foundation for complex analysis, conformal mapping, and phasor/z-plane techniques in engineering.

References

WebsiteWikipedia — Complex plane