Complex Numbers

Quantum state amplitudes are complex numbers. Quantum interference relies on phases. Without comfort with the complex plane, quantum mechanics stays opaque no matter how good your intuition for other parts of the subject is.

This short course builds exactly the fluency needed as a prerequisite for quantum computing study - not through dry calculation, but through Brilliant’s visual, interactive approach that makes complex numbers genuinely intuitive.

What you’ll learn

• What complex numbers are: why the equation x² = -1 has no real solution and how introducing i solves it
• The complex plane: representing complex numbers as points in two dimensions, with the real part on one axis and imaginary part on the other
• Arithmetic with complex numbers: addition, subtraction, multiplication, and division treated geometrically
• Polar form: expressing complex numbers as a magnitude (modulus) and angle (argument) instead of real and imaginary parts
• Why polar form is useful: multiplication becomes rotation and scaling, which maps directly to quantum gate operations
• Euler’s formula: e^(iθ) = cos(θ) + i·sin(θ), one of the most important relationships in physics and the foundation of quantum state representation
• Complex conjugates: what they are and why squaring the modulus of a complex amplitude gives the measurement probability (Born’s rule)
• Phase in quantum mechanics: how two states can have equal probabilities but interfere differently depending on their relative phases

Course structure

The course follows a build-up approach. You begin with the motivation: real numbers cannot describe rotations cleanly, which is a problem for physics. Complex numbers are introduced as the solution, with immediate geometric interpretation.

Arithmetic is covered first in rectangular form (a + bi), then revisited in polar form to show how multiplication becomes rotation - a much cleaner picture. Euler’s formula is approached through the series expansions of e^x, cos(x), and sin(x), connecting calculus and complex numbers without requiring heavy calculation.

The later sections shift explicitly toward physics applications. You see how a qubit state |ψ⟩ = α|0⟩ + β|1⟩ uses complex amplitudes α and β, and why the phase difference between them determines interference patterns. Each concept is explored through interactive diagrams before symbolic manipulation is introduced.

Who is this for?

• Anyone preparing to study quantum computing who wants mathematical confidence
• Students who covered complex numbers in school but never felt they truly understood them
• Self-learners who have tried reading quantum computing textbooks and got stuck on amplitudes and state vectors
• Anyone taking the Brilliant Quantum Computing or Quantum Objects courses who wants thorough preparation before starting
• Physicists returning to study who want a refresher with modern interactive tools

Prerequisites

Basic algebra is required: solving simple equations, working with fractions and exponents. Familiarity with the unit circle and basic trigonometry (sine and cosine) is helpful for the polar form sections - but the course provides reminders. No calculus is needed. No prior exposure to complex numbers is assumed.

Hands-on practice

All practice is browser-based and interactive. You will:

• Drag vectors around the complex plane to see how multiplication changes them
• Adjust the phase angle θ in e^(iθ) and watch the unit circle point move
• Enter complex number calculations and receive immediate feedback
• Manipulate amplitude phases in a two-path quantum interference setup and see the constructive and destructive interference patterns change

The emphasis is on building geometric intuition before algebraic fluency - you understand what an operation does before you practise calculating it.

Why take this course?

Complex numbers sit at the heart of every quantum computing formalism. A qubit’s state lives in a two-dimensional complex vector space. Quantum gates are unitary matrices with complex entries. Quantum interference, the phenomenon that gives quantum algorithms their power, is entirely a consequence of complex amplitude arithmetic - amplitudes for unwanted outcomes cancel while amplitudes for correct outcomes reinforce.

Without fluency here, every quantum computing textbook and course requires constant back-referencing to definitions. Brilliant’s interactive approach makes this fluency enjoyable to build. The course is short enough to complete in a weekend and will pay dividends across everything you study afterwards.