Cartesian Representation of Complex Numbers¶

Definition¶

The complex plane or $z$ -plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal $x$ -axis, called the real axis, is formed by the real numbers, and the vertical $y$ -axis, called the imaginary axis, is formed by the purely imaginary numbers. The form $a+\mathrm{i} b$ is called the Cartesian form of a complex number.

Interpretation¶

The complex plane draws each complex number $a+\mathrm{i}b$ as the point $(a,b)$ of the Cartesian plane. It can therefore be seen as the plane $\mathbb{R}^2$ equipped with additional structure, such as complex multiplication.

A grid in the complex plane with the real axis horizontal and the imaginary axis vertical, and the points 2 + i, -1 + 2i, -2 - i and 1.5 - 1.5i plotted.

Each plotted point is read off the grid: $2+\mathrm{i}$ lies $2$ to the right and $1$ up, $-1+2\mathrm{i}$ lies $1$ to the left and $2$ up, and $-2-\mathrm{i}$ and $1.5-1.5\mathrm{i}$ lie below the real axis because their imaginary parts are negative. The label $\mathrm{i}$ on the vertical axis marks the number $0+1\mathrm{i}$ $1 i$ , one unit up; its coordinate is the real number $1$ .

Each complex number $a+\mathrm{i}b$ is the point $(a,b)$ . Drag $z$ along the axes to see the real and the purely imaginary numbers.

The two axes separate the parts of a number: the real axis holds the numbers with $b=0$ $0$ , the imaginary axis those with $a=0$ $0$ , and they meet at $0$ . Operations that work part by part have simple pictures here. Complex addition shifts a point by an arrow, the conjugate reflects it in the real axis, and the modulus $|a+\mathrm{i}b|=\sqrt{a^2+b^2}$ is its distance from $0$ by Pythagoras, for example $|3+4\mathrm{i}|=5$ $5$ .

Multiplication has no such simple description in coordinates. It becomes clear in the polar form, which describes the same point by its distance from $0$ and its angle. The Cartesian form is the natural one for adding, the polar form for multiplying.

Key Results¶

Each complex number $a+\mathrm{i} b$ corresponds to unique coordinates $(a,b)$ .
In the Cartesian coordinate system, the addition of complex numbers can be visualized as vector addition.

Examples¶

$1+\mathrm{i}$ and $2-\mathrm{i}$ are elements in the complex plane $\mathbb{C}$ .
For $z=3+4\mathrm{i}$ $4 i$ , the real part is $3$ , the imaginary part is $4$ , and $|z|=5$ $5$ .

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 1

Aliases: Complex Plane (Cartesian Coordinate System), Argand plane, Argand diagram, complex plane, z-plane, Gaussian plane Keywords: Cartesian form, rectangular form, algebraic form Symbols: complex numbers, real numbers, square root

Last updated: 20 October 2026 03:02