Complex numbers

In the lesson on quadratic equations a negative discriminant meant “no roots”: x² + 1 = 0 has no real solution, because no real square is negative. In the 16th century Italian mathematicians, Cardano and Bombelli among them, noticed that the formulas for cubic equations sometimes pass through square roots of negative numbers even when the final answer is an ordinary real number. They decided to calculate with such numbers anyway, and today complex numbers are an everyday tool in electrical engineering, signal processing, control theory and quantum physics.

The imaginary unit and the algebraic form

Real numbers are the complex numbers with b = 0; numbers with a = 0, such as 3i, are called purely imaginary. Two complex numbers are equal only when both their real parts and their imaginary parts are equal. Unlike real numbers, complex numbers cannot be ordered: an inequality such as 2 + i < 3 has no meaning.

Operations and the conjugate

Complex numbers are added and subtracted part by part, and multiplied like ordinary brackets, replacing i² with −1:

Geometrically, adding complex numbers means adding vectors (the parallelogram rule), and |z₁ − z₂| is the distance between the points z₁ and z₂. For example, |z − 2| = 1 describes the circle of radius 1 centred at the point 2. In this way complex numbers turn plane geometry into algebra.

Modulus, argument, trigonometric and exponential form

Draw z = a + bi as the point (a, b) of the complex plane: the horizontal axis is real and the vertical axis is imaginary. The distance from the origin is the modulus r = |z|, and the angle measured from the positive real axis is the argument φ = arg z. Then a = r cos φ and b = r sin φ.

De Moivre's formula and roots

The solutions of zⁿ = 1 are the n-th roots of unity e^(2πik/n). For n = 3 they are 1 and −1/2 ± (√3/2)i; for n = 4 they are 1, i, −1 and −i. For n ≥ 2 they add up to 0, because the regular polygon is balanced around the origin. Roots of unity are the heart of the fast Fourier transform used in audio and image processing.

Quadratics with a negative discriminant

A polynomial with real coefficients has its non-real roots in conjugate pairs a ± bi. And by the fundamental theorem of algebra, every polynomial of degree n ≥ 1 has exactly n complex roots counted with multiplicity: in ℂ every polynomial equation can be solved.

import cmath

z1, z2 = 3 + 2j, 1 - 4j
print(z1 * z2)
print(z1 / z2)
print((1 + 1j) ** 10)
print(cmath.polar(1 + 1j))
print(cmath.exp(1j * cmath.pi) + 1)

(11-10j)
(-0.29411764705882354+0.8235294117647058j)
32j
(1.4142135623730951, 0.7853981633974483)
1.2246467991473532e-16j