Complex numbers: an extension of the reals

A complex number is an ordered pair of reals written z = a + bi, where a = Re(z) is the real part, b = Im(z) the imaginary part, and i the imaginary unit defined by i² = −1. The set ℂ extends the reals ℝ so that every polynomial equation can be solved (fundamental theorem of algebra, Gauss 1799). Every z ∈ ℂ can be represented as a point (a, b) in the Argand-Gauss plane; reals lie on the horizontal axis, pure imaginaries on the vertical axis.

Complex arithmetic

Sum: (a+bi) + (c+di) = (a+c) + (b+d)i. Product: (a+bi)·(c+di) = (ac−bd) + (ad+bc)i. Division (by multiplying by the conjugate): (a+bi)/(c+di) = [(ac+bd) + (bc−ad)i]/(c²+d²). Conjugate: z̄ = a − bi; property: z·z̄ = a² + b² = |z|². Modulus: |z| = √(a² + b²). Argument: arg(z) = atan2(b, a) ∈ (−π, π].

Polar and exponential form

Every non-zero complex number can be written z = r·(cos θ + i·sin θ) = r·e^(iθ), where r = |z| and θ = arg(z). Euler's famous formula e^(iπ) + 1 = 0 links five fundamental constants. Polar form simplifies multiplications and powers: z₁·z₂ = r₁·r₂·e^(i(θ₁+θ₂)) (multiply moduli, add arguments); z₁/z₂ = (r₁/r₂)·e^(i(θ₁−θ₂)).

De Moivre's theorem

For any integer n: z^n = r^n·e^(inθ) = r^n·(cos nθ + i·sin nθ). Powerful tool for computing high powers without expanding. Example: (1 + i)^10 = ? r = √2, θ = π/4. z^10 = (√2)^10·e^(i·10π/4) = 32·e^(i·5π/2) = 32·e^(i·π/2) = 32·i. Verify: ((1+i)²)^5 = (2i)^5 = 32·i^5 = 32i. ✓

n-th roots

A non-zero complex number z = r·e^(iθ) has exactly n distinct n-th roots: w_k = r^(1/n)·e^(i(θ + 2πk)/n), for k = 0, 1, ..., n−1. They lie on a circle of radius r^(1/n) at the vertices of a regular n-gon. Example: the 3 cube roots of 1: w_0 = 1, w_1 = −1/2 + (√3/2)i, w_2 = −1/2 − (√3/2)i. The n-th roots of unity w_k = e^(2πik/n) form a cyclic group fundamental in group theory and discrete Fourier transforms.

Applications

Electrical engineering (impedances Z = R + iωL − i/(ωC) in AC, RLC circuit analysis); quantum physics (wave function ψ ∈ ℂ, Hermitian operators, Pauli matrices); signal processing (Fourier transform: complex spectrum); control engineering (root locus, Nyquist, Bode diagrams); aerodynamics (conformal potential functions, Joukowski); geometry (Möbius transformations, hyperbolic geometry); fractals (Mandelbrot z_{n+1} = z_n² + c, Julia sets); cryptography (elliptic-curve algorithms over complex finite fields).

How to use

Select mode: Arithmetic (z₁, z₂ → sum, difference, product, quotient), Polar (a+bi → r, θ in degrees and radians, and vice versa), De Moivre (z, n → z^n), Roots (z, n → n n-th roots). Real and imaginary parts as decimal numbers.

A brief history: from Cardano to Euler and Gauss

Complex numbers were born in sixteenth-century Italy: Gerolamo Cardano met them in 1545 while solving cubic equations in his Ars Magna, and Rafael Bombelli gave the first coherent calculation rules in 1572, treating the square root of −1 as a strange but operationally valid quantity. Only with Euler, who introduced the symbol i in 1777, and with Gauss, who gave them a geometric interpretation on the plane that now bears his name, did complex numbers become full citizens of mathematics. Euler's relation e^(iθ) = cosθ + i·sinθ links the exponential to trigonometry and yields, for θ = π, the identity e^(iπ) + 1 = 0, connecting the five fundamental constants of mathematics in a single line.

Applications: from AC circuits to quantum mechanics

In electrical engineering the impedance of an AC circuit is a complex number Z = R + iX unifying resistance and reactance; the phasor method reduces the differential equations of circuits to complex algebra. In signal processing the Fourier transform decomposes any signal into complex exponentials and underlies MP3, JPEG and telecommunications. In quantum mechanics probability amplitudes are intrinsically complex — the squared modulus |ψ|² gives the observable probability — and fractals such as the Mandelbrot set arise from the simple complex iteration z → z² + c.

Updated July 2026