1667 – 1754 · Probability and the Complex Plane
The Huguenot refugee who became Newton's closest mathematical confidant, founded probability theory, and revealed the deep connection between complex numbers and trigonometry.
Born on 26 May 1667 in Vitry-le-François, Champagne, France, into a Huguenot (French Protestant) family. His father was a surgeon.
De Moivre studied at the Protestant academy at Sedan until its forced closure in 1681, then at Saumur, and finally in Paris under the private tutelage of the mathematician Jacques Ozanam.
In 1685, Louis XIV revoked the Edict of Nantes, stripping French Protestants of their rights. De Moivre was imprisoned in the Priory of Saint-Martin for over a year before escaping to London around 1688.
As a refugee, he never obtained a university position despite his brilliance. He spent his entire career as a private mathematics tutor, meeting students in Slaughter's Coffee House in St. Martin's Lane.
The revocation of the Edict of Nantes caused a mass exodus of French Protestants. De Moivre was one of roughly 200,000 refugees who fled, many bringing skills that enriched their host countries.
Legend has it that de Moivre learned higher mathematics by tearing pages from Newton's Principia and studying them between tutoring sessions as he walked across London.
Despite being one of the finest mathematicians in Europe, de Moivre never received a salaried academic position. His refugee status and French accent were permanent barriers.
De Moivre's intellectual brilliance was recognized early by the English mathematical establishment. He was elected a Fellow of the Royal Society in 1697 and became a close personal friend of Isaac Newton.
Newton regarded de Moivre so highly that when visitors asked him about mathematics, Newton would reportedly say, "Go to Mr. de Moivre; he knows these things better than I do."
His major work, The Doctrine of Chances (1st ed. 1718, expanded 1738, 1756), was the first textbook of probability theory and remained the standard reference for over a century.
De Moivre also served on the Royal Society commission that adjudicated the Newton-Leibniz priority dispute, firmly siding with his friend Newton.
He lived to the remarkable age of 87, dying on 27 November 1754 in London. A famous legend claims he predicted the day of his own death by noting that he was sleeping 15 minutes longer each day.
Their friendship lasted from the 1690s until Newton's death in 1727 — one of the most important intellectual friendships of the era.
Doctrine of Chances (1718), Miscellanea Analytica (1730), Annuities upon Lives (1725)
Slaughter's Coffee House was de Moivre's "office." Gamblers, insurance brokers, and students sought him out there for calculations involving probability and risk.
De Moivre worked at the intersection of several revolutions: in mathematics, in commerce, and in the understanding of uncertainty.
Pascal and Fermat had exchanged letters on probability in 1654. Huygens published the first treatise in 1657. De Moivre transformed these beginnings into a systematic theory.
The late 17th century saw the birth of modern insurance (Lloyd's of London, 1688) and government annuities. There was urgent demand for mathematical tools to price risk.
The aristocratic gambling culture of Georgian England created a market for probability calculations. The Doctrine of Chances was partly a manual for sophisticated gamblers.
The Huguenot refugees brought immense talent to England, the Netherlands, and Prussia. De Moivre was the most distinguished mathematician among them.
Complex numbers were still controversial — called "imaginary" and viewed with suspicion. De Moivre's formula helped legitimize them by connecting them to geometry.
De Moivre was part of Newton's inner circle alongside Halley, Cotes, and Taylor. This group dominated British mathematics in the early 18th century.
De Moivre's formula, first stated around 1707, connects complex exponentiation to trigonometry:
(cos θ + i sin θ)n = cos nθ + i sin nθ
In modern notation using Euler's formula eiθ = cos θ + i sin θ, this becomes simply:
(eiθ)n = einθ
The formula means that raising a point on the unit circle to the n-th power corresponds to multiplying its angle by n. This elegant geometric interpretation reveals that complex multiplication is really rotation.
De Moivre used this to find n-th roots of complex numbers, derive trigonometric identities, and solve equations that were intractable by other means.
De Moivre's formula is the key that unlocks a vast territory of mathematics:
Trigonometric identities: Expanding (cos θ + i sin θ)n by the binomial theorem and separating real and imaginary parts gives expressions for cos(nθ) and sin(nθ) in terms of cos θ and sin θ.
Roots of unity: The n-th roots of 1 are zk = cos(2πk/n) + i sin(2πk/n) for k = 0, 1, ..., n−1. These form a regular n-gon inscribed in the unit circle.
Chebyshev polynomials: The formula leads directly to the Chebyshev polynomials Tn(cos θ) = cos(nθ), fundamental in approximation theory.
De Moivre's formula predates Euler's formula eiθ = cos θ + i sin θ by several decades. Euler generalized de Moivre's insight, but de Moivre discovered the fundamental connection first.
The formula is the seed from which complex analysis grew. Without the geometric interpretation of complex multiplication as rotation, the theory of analytic functions might have developed much later.
The connection between complex exponentials and trigonometry is the mathematical foundation of Fourier analysis, the DFT, and all of modern signal processing.
In 1733, de Moivre published a Latin pamphlet (later included in the Doctrine of Chances) proving that the binomial distribution approaches a continuous bell-shaped curve as the number of trials grows large.
He showed that for a fair coin flipped n times, the probability of getting k heads approaches:
P(k) ≈ (1/√(2πn · pq)) · e−(k−np)²/(2npq)
This is the normal approximation to the binomial — the first appearance of what we now call the Gaussian or normal distribution, published 76 years before Gauss.
To derive this, de Moivre needed an approximation for n! — which led him to discover what is now (ironically) called Stirling's approximation.
The binomial distribution (bars) is approximated by the continuous normal curve — de Moivre's central limit theorem in action.
The Doctrine of Chances (1st edition 1718, greatly expanded 1738 and 1756) was the first comprehensive textbook of probability theory. It went far beyond the gambling problems of Pascal and Fermat.
Key contributions in the Doctrine:
De Moivre invented generating functions to solve the Fibonacci-like recurrence relations arising in probability. This technique became one of the most powerful tools in combinatorics.
His Annuities upon Lives (1725) applied probability to life insurance and pension calculations — the birth of actuarial science.
De Moivre first derived n! ≈ C · nn+1/2e−n. Stirling later found the constant C = √(2π), but the approximation itself is de Moivre's work.
The Doctrine influenced Bayes, Laplace, and Gauss. It established probability as a legitimate branch of mathematics, not mere gambling arithmetic.
De Moivre introduced generating functions as a systematic method for solving linear recurrence relations. Given a sequence a0, a1, a2, ..., the generating function is:
G(x) = a0 + a1x + a2x² + a3x³ + …
By encoding a recurrence relation as an equation in G(x), de Moivre could use algebraic methods (partial fractions, etc.) to find closed-form solutions for sequences defined recursively.
He applied this to the Fibonacci sequence and related recurrences arising in probability problems, obtaining explicit formulas where direct computation would be impractical.
De Moivre derived the closed form Fn = (φn − ψn)/√5 where φ = (1+√5)/2 is the golden ratio. This is sometimes called the Binet formula but was known to de Moivre first.
Generating functions are now fundamental in combinatorics, computer science (analysis of algorithms), probability theory, and analytic number theory.
De Moivre's technique of decomposing rational generating functions into partial fractions to extract sequence terms became a standard technique taught in every discrete mathematics course.
While de Moivre developed generating functions for probability, they proved universal. Euler, Laplace, and Pólya extended them far beyond their origins.
De Moivre combined algebraic virtuosity with practical problem-solving, always driven by concrete applications.
A gambling, insurance, or mathematical question
Translate into generating functions or series
Apply algebraic and analytic techniques
Use asymptotic methods for large values
De Moivre was one of the supreme algebraists of his era. His facility with series, infinite products, and complex numbers allowed him to solve problems that defeated others. Newton himself deferred to de Moivre on questions of algebraic technique.
As a tutor who made his living solving problems for gamblers and insurance brokers, de Moivre was always motivated by applications. This kept his work grounded and useful, even as the mathematics reached deep levels of abstraction.
De Moivre first derived the factorial approximation n! ≈ C · nn+1/2e−n in 1733. Stirling then determined the constant C = √(2π). Yet the result is universally called "Stirling's approximation" — one of mathematics' great misattributions.
The normal distribution is called "Gaussian" despite de Moivre discovering it 76 years before Gauss. Stigler's law strikes again: the most important probability distribution bears the wrong name.
Pierre Rémond de Montmort published Essay d'analyse sur les jeux de hazard in 1708, before de Moivre's Doctrine. A bitter priority dispute followed, with accusations of plagiarism on both sides. Montmort had the earlier publication; de Moivre had the deeper results.
Despite being acknowledged as one of Europe's greatest mathematicians, de Moivre was never offered a professorship in England or France. His status as a Huguenot refugee and his lack of independent wealth kept him permanently on the margins of academic life.
De Moivre's influence permeates modern mathematics far more deeply than his name recognition suggests.
De Moivre's formula, de Moivre's theorem, de Moivre-Laplace theorem (CLT), de Moivre numbers. Far less than he deserves.
De Moivre is perhaps the most underrated mathematician of the 18th century. His work on probability, complex numbers, and asymptotic analysis were decades ahead of their time.
Despite lifelong poverty and marginalization, de Moivre produced work of the highest order. His story is a testament to intellectual resilience in the face of systemic exclusion.
AC circuit analysis uses complex phasors. De Moivre's formula is how engineers compute powers and roots of phasors, essential for multi-phase power systems and impedance calculations.
The wave function is complex-valued. De Moivre's formula underlies the mathematics of quantum state evolution, interference, and measurement — the rotation of state vectors in Hilbert space.
The normal distribution he discovered is the foundation of statistical inference, hypothesis testing, confidence intervals, and the central limit theorem that justifies most of applied statistics.
Roots of unity and cyclotomic polynomials (direct consequences of de Moivre's formula) appear in number-theoretic cryptographic protocols and error-correcting codes.
The FFT (Fast Fourier Transform) relies on roots of unity — direct applications of de Moivre's formula. Generating functions analyze algorithm complexity.
The Black-Scholes model, Value-at-Risk, and portfolio theory all rest on the normal distribution. Modern quantitative finance descends directly from de Moivre's Doctrine of Chances.
The Doctrine of Chances (3rd ed., 1756) — de Moivre's magnum opus. Available in modern reprint.
Miscellanea Analytica (1730) — contains de Moivre's formula, generating functions, and the factorial approximation.
I. Schneider, "Abraham de Moivre," in Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (1994).
H.M. Walker, "Abraham de Moivre," Scripta Mathematica (1934) — classic biographical study.
A. Hald, A History of Probability and Statistics and Their Applications before 1750 (2003) — comprehensive treatment of de Moivre's probability work in context.
S. Stigler, The History of Statistics (1986) — essential for understanding de Moivre's statistical legacy.
P. Nahin, An Imaginary Tale: The Story of √(−1) (1998) — accessible account of how de Moivre's formula fits into the history of complex numbers.
"Chance very often discovers to us that which reason could not, and it is an effect of the highest wisdom to be able to manage and direct it."
— Abraham de Moivre, The Doctrine of Chances, 1718Abraham de Moivre (1667–1754)