Machin's Formula

The hook

In 1706 the London mathematician John Machin computed \(\pi\) to \(100\) decimal places, by hand. Nobody had come close before. His secret was the formula in part (ii), which expresses \(\frac\pi4\) in terms of angles whose tangents are tiny fractions. Tiny fractions make the series \(\tan^{-1}x = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots\) converge very fast, and Machin’s formula, or variants of it, remained the standard way to compute \(\pi\) for two and a half centuries, until computers and the methods of Section 2.6 took over.

The formula looks like a miracle of trigonometric identities. The question shows it is really a fact about multiplying complex numbers: when you multiply complex numbers, their arguments add. One multiplication of whole-number complex numbers proves the whole thing.

Before you start

\(\arg(5 - \mathrm{i}) = -\tan^{-1}\frac15\), a small negative angle, and \(\arg(1 + \mathrm{i}) = \frac\pi4\). When you multiply complex numbers, the moduli multiply and the arguments add (up to multiples of \(2\pi\)). So \[\arg\bigl((5 - \mathrm{i})^4(1 + \mathrm{i})\bigr) = \frac\pi4 - 4\tan^{-1}\frac15 \quad\text{(possibly plus a multiple of } 2\pi\text{)}.\] If we can compute the product and find its argument another way, we get an identity.

Working it through

Stage 1 Bounds on the modulus

Upper bound: \((|x| + |y|)^2 = x^2 + y^2 + 2|x||y| \geq x^2 + y^2 = |z|^2\), and both sides are non-negative, so \(|z| \leq |x| + |y|\).

Lower bound: \(|x|\cos\alpha + |y|\sin\alpha\) is the dot product of the vector \((|x|, |y|)\), of length \(|z|\), with the unit vector \((\cos\alpha, \sin\alpha)\). A dot product is at most the product of the lengths, so it is at most \(|z|\). (Algebraically: \(|z|^2 - (|x|\cos\alpha + |y|\sin\alpha)^2 = (|x|\sin\alpha - |y|\cos\alpha)^2 \geq 0\).)■

Stage 2 The key multiplication

\((5 - \mathrm{i})^2 = 25 - 10\mathrm{i} - 1 = 24 - 10\mathrm{i}\). Then \((24 - 10\mathrm{i})^2 = 576 - 480\mathrm{i} - 100 = 476 - 480\mathrm{i}\). Finally \[(476 - 480\mathrm{i})(1 + \mathrm{i}) = 476 + 476\mathrm{i} - 480\mathrm{i} + 480 = 956 - 4\mathrm{i}.\]

\(956 - 4\mathrm{i} = 4(239 - \mathrm{i})\), which has argument \(-\tan^{-1}\frac{1}{239}\). So \[\frac\pi4 - 4\tan^{-1}\frac15 = -\tan^{-1}\frac{1}{239} + 2k\pi\] for some integer \(k\). But \(0 < \tan^{-1}\frac15 < \frac15\), so the left-hand side lies between \(\frac\pi4 - \frac45\) and \(\frac\pi4\), that is, between about \(-0.02\) and \(0.79\); and the right-hand side without the \(2k\pi\) lies between \(-0.005\) and \(0\). Since \(2\pi\) is much bigger than the gap, \(k = 0\). Rearranging, \[\frac\pi4 = 4\tan^{-1}\frac15 - \tan^{-1}\frac{1}{239}. \qed\] Figure 5.1.1 shows what happened: starting at \(45^\circ\), each multiplication by \(5 - \mathrm{i}\) rotates clockwise by about \(11.3^\circ\), and after four of them we have overshot the real axis by just \(0.24^\circ\).

Stage 3 The second formula

Consider \((4 + \mathrm{i})^3(20 + \mathrm{i})(1985 + \mathrm{i})\), whose argument is \(3\tan^{-1}\frac14 + \tan^{-1}\frac{1}{20} + \tan^{-1}\frac{1}{1985}\). Now \((4 + \mathrm{i})^2 = 15 + 8\mathrm{i}\), so \((4 + \mathrm{i})^3 = (15 + 8\mathrm{i})(4 + \mathrm{i}) = 52 + 47\mathrm{i}\). Then \((52 + 47\mathrm{i})(20 + \mathrm{i}) = 1040 + 52\mathrm{i} + 940\mathrm{i} - 47 = 993 + 992\mathrm{i}\), and \[(993 + 992\mathrm{i})(1985 + \mathrm{i}) = (1971105 - 992) + (993 + 1969120)\mathrm{i} = 1970113 + 1970113\mathrm{i}.\] Real part equals imaginary part, so the argument is \(\frac\pi4\). All three angles are small and positive (their sum is less than \(3 \cdot \frac14 + \frac1{20} + \frac{1}{1985} < 1 < 2\pi\)), so there is no multiple of \(2\pi\) to worry about, and the formula follows.■

Looking back

• Arguments add under multiplication. That single fact turned a product of Gaussian integers into an identity between angles.

• Mind the multiples of \(2\pi\). Arguments are only defined up to \(2\pi\), so each identity needed a size estimate to pin down the multiple.

• Whole numbers do the work. All the arithmetic was with integers; the identity for \(\pi\) came out exactly, with no approximation.

What’s really going on

Gaussian integers again

The numbers \(a + b\mathrm{i}\) with \(a\), \(b\) whole numbers are the Gaussian integers of Section 1.6. The reason \(5 - \mathrm{i}\) works so well is a factorisation: \(239^2 + 1 = 57122 = 2 \times 13^4\), which in the Gaussian integers reads \((239 + \mathrm{i})(239 - \mathrm{i}) = (1 + \mathrm{i})(1 - \mathrm{i})\,(3 + 2\mathrm{i})^4(3 - 2\mathrm{i})^4\), and \(5 - \mathrm{i} = (1 - \mathrm{i})(3 + 2\mathrm{i})\) is built from the same Gaussian primes. Finding Machin-type formulas is really a problem about factorising numbers of the form \(n^2 + 1\). In 1899 Carl Størmer used exactly this idea to prove that there are only four formulas of the form \(\frac\pi4 = a\tan^{-1}\frac1m + b\tan^{-1}\frac1n\) with whole numbers \(a\), \(b\), \(m\), \(n\): Machin’s, and three found by Euler, Hermann and Hutton.

Rotations

Multiplying by a complex number of modulus \(1\) is a rotation of the plane. That is why complex numbers are the natural language for anything involving rotation and angles, from the geometry later in this chapter to alternating currents in electrical engineering and the wave functions of quantum mechanics. In Section 4.2 the addition formula for \(\tan^{-1}\) came from a clever substitution; here it comes from the multiplication of complex numbers, and the two are the same fact seen from different sides.

The thinking skill

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Further reading

• Jonathan Borwein & Peter Borwein, Pi and the AGM (Wiley). Includes the history of Machin-type formulas.

• Jörg Arndt & Christoph Haenel, Pi Unleashed (Springer). A lively account of methods for computing \(\pi\), old and new.

• Tristan Needham, Visual Complex Analysis (Oxford University Press). The best book for seeing complex numbers geometrically. Chapter 1 is about multiplication as rotation and scaling.