We began with Cardano’s unsettling discovery that real answers can require imaginary steps, and with three questions: how complex numbers help with real problems, what is special about roots of unity, and what complex functions look like. Here is what the chapter’s eight questions showed.
1. How do complex numbers help with real problems?
Again and again, a fact about real numbers had a short proof through the complex plane.
Angles become products. Machin’s formula for \(\pi\) is the statement that \((5 - \mathrm{i})^4(1 + \mathrm{i})\) is a real multiple of \(239 - \mathrm{i}\), checked with whole-number arithmetic (Section 5.1).
Counting becomes averaging. Sums of every third binomial coefficient came from evaluating \((1 + x)^n\) at the cube roots of unity (Section 5.2).
Trigonometric products become polynomial values. Products of cosines at equally spaced angles came from factorising \(z^{2n} + 1\) and putting \(z = \mathrm{i}\) (Section 5.3).
Geometry becomes algebra. Distances became moduli, so the product of chords from a vertex of a regular \(n\)-gon is \(|1 + 1 + \cdots + 1| = n\) (Section 5.4); rotations became multiplications, so the Fermat point construction collapsed to the identity \(1 + \alpha^2 = \alpha\) (Section 5.6).
Irrationality from De Moivre. Expanding \((\cos\theta + \mathrm{i}\sin\theta)^k\) and arguing about parity showed that \(\cos^{-1}\frac13\) is not a rational number of degrees, the key fact in Dehn’s solution of Hilbert’s third problem (Section 5.5).
2. What is special about the roots of unity?
The \(n\)th roots of unity are the vertices of a regular \(n\)-gon, they are the roots of the simplest possible polynomial \(z^n - 1\), and they form a group under multiplication. Those three facts together explain their ubiquity. Because they are evenly spread round a circle, they sum to zero, which gives the filter of Section 5.2 and, in general, the discrete Fourier transform. Because they are the roots of \(z^n - 1\), products over them are values of simple polynomials (Section 5.3, Section 5.4). And because they are closed under multiplication, powers of a root of unity cycle round, which is exactly what fails for an angle like \(\cos^{-1}\frac13\) (Section 5.5). In Chapter 6 roots of unity become the heart of algebra, leading towards Galois theory.
3. What does a complex function look like?
In Section 5.7 we pictured \(e^w\) as a map that rolls a strip into a plane, sending vertical lines to circles and horizontal lines to rays, and wrapping round infinitely often, so that the logarithm is many-valued. We saw that it preserves angles, the first example of a conformal map, and that it underlies Mercator’s chart. In Section 5.8 a strange inequality turned out to be the triangle inequality of hyperbolic geometry, where the unit disc is an infinite plane, straight lines are circular arcs, and distances combine by the relativistic velocity-addition formula.
4. A toolkit for complex numbers
| Method | Where you used it |
|---|---|
| Arguments add under multiplication | Machin’s formula (5.1) |
| Checking multiples of \(2\pi\) | Machin’s formula (5.1) |
| Averaging over roots of unity | Every third coefficient (5.2) |
| Factorising over roots, then substituting | Cosine products (5.3); chords (5.4) |
| Dividing out a vanishing factor | 5.3; 5.4 |
| De Moivre’s theorem | Multiple-angle formulas (5.5) |
| Parity and minimal counterexamples | Irrational angles (5.5) |
| Rotation as multiplication | Equilateral triangles (5.6) |
| Mapping grids | The exponential map (5.7) |
| \(|a|^2 = aa^*\) | The Poincaré disc inequality (5.8) |
Where next
Roots of unity drive Chapter 6, including the cyclotomic polynomials of Section 6.1 and Gauss’s construction of the regular \(17\)-gon in Section 6.2.
The connection between hyperbolic geometry, \(\tanh\) and relativity is developed in Section 7.6.
Gaussian integers, which appeared in Machin’s formula, were the subject of Section 1.6.
Tristan Needham, Visual Complex Analysis: the most beautiful book on complex numbers ever written, full of pictures. Start with Chapter 1.
Paul Nahin, An Imaginary Tale: the history of \(\sqrt{-1}\), from Cardano to modern physics.
Liang-shin Hahn, Complex Numbers and Geometry: Euclidean geometry with complex numbers, at exactly the level of this chapter.
Hilary Priestley, Introduction to Complex Analysis: for a first look at the university course.
Cardano thought his imaginary quantities were as useless as they were subtle. Five centuries later, it is hard to find a corner of mathematics or physics where they do not appear. The lesson of this chapter is Hadamard’s: when a problem about real numbers is hard, try stepping off the real line. The complex plane is bigger, and in it, many things that look complicated turn out to be simple.