A precise definition
Complex numbers extend the real number line to a 2D plane. A complex number has the form a + bi, where a is the 'real part', b is the 'imaginary part', and i = √(-1). The word 'imaginary' is unfortunate — it suggests these numbers are made up. They are not. Complex numbers are the natural number system for anything that oscillates (waves, currents, signals) or rotates (in 2D or 3D space).
The problem it was invented to solve
16th-century Italian mathematicians (Cardano, Tartaglia, Bombelli) encountered √(-1) when solving cubic equations. They found that even when the final answer was a real number, the intermediate steps sometimes required using imaginary numbers. Bombelli in 1572 established rules for calculating with them, noting pragmatically that they 'work.' It took another two centuries before Argand (1806) and Gauss gave them a geometric interpretation, establishing that complex numbers are points in a 2D plane.
Where you find it in the world — including South Africa
These are not contrived textbook examples. Each application below is currently in use, driven by real institutions, and producing real outcomes.
Eskom's AC power grid — electrical engineering
South Africa's electrical grid runs on alternating current (AC). The analysis of AC circuits — voltage, current, impedance, power factor — is most naturally done using complex numbers. Every electrical engineer at Eskom, every student in an EEE degree at Wits or UCT, uses complex number phasors daily.
Signal processing: filtering and Fourier transforms
Digital audio, digital television (OpenView HD, DStv), and mobile data (5G in SA cities) all rely on signal processing algorithms expressed in complex numbers. The Fourier transform — which decomposes signals into frequencies — is a function from real numbers to complex numbers.
Quantum mechanics — the physics of the very small
Quantum mechanics is formulated in complex Hilbert spaces. The probability amplitudes that describe the behaviour of electrons, photons, and quantum computers are complex numbers. SA physicists at iThemba LABS (Somerset West) work with complex-number formalisms daily.
Control systems: stability analysis
The stability of any automated system — from a Transnet locomotive control system to a drone's flight controller — is determined by the location of complex roots (eigenvalues) in the complex plane. A system is stable if and only if all roots have negative real parts.
You've already encountered this
The number i is not imaginary in the sense of fictional. It is a rotation operator — multiplying a vector by i rotates it 90° in the plane. This is why complex numbers appear wherever rotation or oscillation occurs: i is the mathematics of turning.
What you study — and when
- ›Definition: i = √(-1), i² = -1
- ›Standard form: a + bi
- ›Addition, subtraction, and multiplication of complex numbers
- ›Conjugate pairs and division of complex numbers
- ›Argand diagram: complex numbers as points in a plane
- ›Modulus of a complex number
Related topics and institutions
Complex numbers are the bridge between Grade 12 mathematics and university physics and engineering.
The Continuum introduces complex numbers with the geometric intuition (rotation in the plane) that makes them feel natural — not arbitrary — so the imaginary becomes obvious.
No card required. South African curriculum. Grade 8–12.