Imaginary Numbers, Real Answers
The shortest path between two truths in the real domain passes through the complex domain.
— Jacques Hadamard
In 1545 Gerolamo Cardano published a formula for solving cubic equations, and found something disturbing. For some cubics with three perfectly good real roots, the formula required taking square roots of negative numbers along the way. The square roots cancelled out at the end, leaving real answers, but in the middle of the calculation there were quantities that, by every standard of the time, did not exist. Cardano called the manipulation “as subtle as it is useless”. For two centuries, mathematicians used these “imaginary” numbers uneasily, as a trick that happened to work.
It took until the early nineteenth century, when Caspar Wessel, Jean-Robert Argand and Gauss pictured complex numbers as points in a plane, for them to become respectable. Once they did, they turned out to be not a trick but a revelation. Complex numbers explain why real phenomena behave as they do, and they keep providing the shortest route to results about real numbers, exactly as Hadamard said.
Why complex numbers matter
Every polynomial equation has a solution in the complex numbers: the fundamental theorem of algebra. So the complex numbers are, in a precise sense, where algebra is complete. They are also the natural language of rotation and oscillation. Multiplying by a complex number of modulus \(1\) rotates the plane, so anything involving angles, waves or periodic behaviour is simpler with complex numbers. Electrical engineers use them to analyse alternating currents; physicists cannot write down quantum mechanics without them; signal processing, from mobile phones to MRI scanners, runs on the complex Fourier transform. And in pure mathematics, functions of a complex variable turn out to be astonishingly rigid and beautiful: knowing a complex function on a tiny region determines it everywhere.
What you will meet at university
You will use complex numbers from the first week, but the big step comes with complex analysis, usually in the second year. Its central objects are holomorphic functions, complex functions that can be differentiated, and the theory is full of surprises. Every holomorphic function is automatically infinitely differentiable and equal to its Taylor series. Integrals round closed curves can be evaluated by looking only at the points where the function blows up (Cauchy’s residue theorem), which lets you compute real integrals that resist every other method. The theory leads on to conformal maps (Section 5.7), Riemann surfaces, and the Riemann zeta function that controls the primes (Section 1.2). In parallel, the algebra of roots of unity (Section 5.2, Section 5.3) leads to Galois theory, and complex geometry leads to the non-Euclidean geometry of Section 5.8.
The questions that run through this chapter
How do complex numbers help with real problems? Why should a detour through the complex plane produce facts about \(\pi\), binomial coefficients or cosines?
What is special about the roots of unity? Why do the solutions of \(z^n = 1\) keep appearing, in sums, products, polygons and angles?
What does a complex function look like? How can we picture functions whose inputs and outputs are both points of a plane, and what geometry do they reveal?
The chapter moves from arguments and multiplication (Machin’s formula), through the algebra of roots of unity (filters, factorisations, polygon distances and irrational angles), to geometry with complex numbers (the Fermat point) and finally to complex functions as maps (the exponential, and a hidden distance in the unit disc).
Ideas in this chapter
| § | Topic | Idea |
|---|---|---|
| 5.1 | Machin’s formula | Arguments add when complex numbers multiply |
| 5.2 | Every third binomial coefficient | Averaging over roots of unity filters by residue |
| 5.3 | Products of cosines | Factorise over the roots, then substitute |
| 5.4 | Chords of a regular polygon | Distances are moduli; \(z^n - 1 = \prod(z - \omega^k)\) |
| 5.5 | \(\cos^{-1}\frac13\) is irrational | De Moivre, parity and a minimal counterexample |
| 5.6 | The Fermat point | Rotation is multiplication; good coordinates |
| 5.7 | \(e^w\) as a map | Picturing complex functions; many-to-one maps |
| 5.8 | A hidden distance | An inequality as a triangle inequality in hyperbolic geometry |
How to read this chapter
Draw pictures. Almost every result in this chapter has a geometric meaning, and a quick sketch of the complex plane (where the roots of unity are, which way a multiplication rotates, where the grid lines go) often shows you what to do before any algebra. Then do the algebra carefully, because the pictures can mislead about signs and multiples of \(2\pi\).