What Is the Complex Plane (Argand Diagram)?

The complex plane, also called an Argand diagram, is a coordinate plane used to represent complex numbers. A complex number $z=a+bi$ is plotted as the point ((a,b)), where $a$ is its real part and $b$ is its imaginary part.

This is an IB Mathematics: Analysis and Approaches HL concept from AHL 1.12: Complex numbers: Cartesian form and Argand diagram.

How the Argand Diagram Works

Unlike an ordinary Cartesian plane, the horizontal and vertical axes represent parts of a complex number rather than two independent variables.

For example, consider

$$z=-3+4i.$$

It is represented by the point $(-3,4)$: move three units left along the real axis and four units upward parallel to the imaginary axis. Its modulus is its distance from the origin:

$$\lvert z\rvert=\sqrt{(-3)^2+4^2}=5.$$

Its argument is measured anticlockwise from the positive real axis. Since $(-3,4)$ lies in quadrant II, its principal argument is

$$\arg(z)=\pi-\arctan\left(\frac{4}{3}\right).$$

The same point can also be viewed as the position vector from the origin to ((a,b)). This makes addition geometric: adding complex numbers adds their horizontal and vertical components. The conjugate $\overline{z}=a-bi$ is the reflection of $z$ in the real axis, while (-z) is a rotation of $180^\circ$ about the origin on the diagram.

A common misconception is that the vertical coordinate is (bi). It is not: the coordinate is the real number $b$, while the corresponding complex number contains the term (bi).

IB Exam Technique

On AA HL Paper 1 or Paper 2, you may be asked to plot complex numbers, identify their real and imaginary parts, or calculate modulus and argument. Label both axes, place points accurately, use the correct quadrant for the argument, and show working rather than giving only a calculator value.