Calculus×Computing
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Complex numbers

Level UniversityDifficulty ★★★★★Concept⌖ Open in the map

What is it?

Numbers z=a+biz = a + bi with i2=−1i^2 = -1. Geometrically they are points of the plane, and multiplying by eiθe^{i\theta} is a rotation — which is why they are the natural language of oscillations, waves and signals.

Why does it exist?

Historically, to solve cubic equations: the formula for real roots passed through square roots of negative numbers. Today the reason is different: every polynomial of degree nn has exactly nn complex roots (fundamental theorem of algebra), and rotations and oscillations become simple multiplications.

Intuition

Write z=reiθz = r e^{i\theta}: a length r=∣z∣r = |z| and an angle θ=arg⁡z\theta = \arg z. Multiplying two complex numbers multiplies the lengths and adds the angles. So eiωte^{i\omega t} is a point spinning around the unit circle at angular speed ω\omega; its shadow on the real axis is cos⁡ωt\cos \omega t. A signal is a sum of such spinning arrows, and that is the whole idea of Fourier analysis.

Formulas

eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i \sin\theta
Euler's formula
∣z∣=a2+b2,zˉ=a−bi,zzˉ=∣z∣2|z| = \sqrt{a^2 + b^2}, \qquad \bar z = a - bi, \qquad z \bar z = |z|^2
modulus and conjugate
ωN=e−2πi/N,ωNN=1\omega_N = e^{-2\pi i / N}, \qquad \omega_N^N = 1
roots of unity (the heart of the FFT)

Where it shows up in computing

  • Fast Fourier transform (FFT)★★★★★fundamentalSignals, media and vision

    The FFT is built on the symmetries of the NN-th roots of unity e−2πik/Ne^{-2\pi i k/N}.

  • Quantum computing★★★★★fundamentalQuantum computing and physics

    Qubit amplitudes are complex numbers; interference is addition of complex phases.

  • Signal processing★★★★★fundamentalSignals, media and vision

    Sinusoids are handled as phasors Aei(ωt+φ)Ae^{i(\omega t + \varphi)}: amplitude and phase in one number.

  • Wave function★★★★★fundamentalQuantum computing and physics

    A wave function takes complex values; only ∣ψ∣2|\psi|^2 is a probability.

  • Control theory★★★★★frequentRobotics and control

    A linear system is stable when the poles of its transfer function lie in the left half of the complex plane.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

⚛ Physics and simulation

∿ Signals, media and vision

⊕ Cryptography and security

ψ Quantum computing and physics

3D Computer graphics

⚙ Robotics and control

ℒ AI and machine learning

What depends on it

Exercises

1Computation

Compute (1+i)8(1+i)^8 using the polar form.

Solution

1+i=2 eiπ/41 + i = \sqrt 2\, e^{i\pi/4}, so (1+i)8=24e2πi=16(1+i)^8 = 2^4 e^{2\pi i} = 16.

2Graphical

Where in the plane are the solutions of z6=1z^6 = 1? What shape do they form?

Solution

At e2πik/6e^{2\pi i k/6}, k=0,…,5k = 0,\dots,5: the vertices of a regular hexagon inscribed in the unit circle.

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