Abraham de Moivre
Abraham de Moivre (26 May 1667 in
VitryleFrançoisVitryleFrançois is a commune in the Marne department in northeastern France. It is located on the Marne River and is the western terminus of the MarneRhine Canal. History :In 1142, Louis VII invaded Champagne and seized VitryleFrançois...
,
ChampagneThe Champagne wine region is a historic province within the Champagne administrative province in the northeast of France. The area is best known for the production of the sparkling white wine that bears the region's name...
,
FranceThe French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semipresidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
– 27 November 1754 in
LondonLondon is the capital city of :England and the :United Kingdom, the largest metropolitan area in the United Kingdom, and the largest urban zone in the European Union by most measures. Located on the River Thames, London has been a major settlement for two millennia, its history going back to its...
,
EnglandEngland is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...
; abʁaam də mwavʁ) was a
FrenchThe French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semipresidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
mathematicianA mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
famous for
de Moivre's formulaIn mathematics, de Moivre's formula , named after Abraham de Moivre, states that for any complex number x and integer n it holds that...
, which links
complex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the onedimensional number line to the twodimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s and
trigonometryTrigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
, and for his work on the
normal distribution and
probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of nondeterministic events or measured quantities that may either be single...
. He was a friend of
Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
, Edmund Halley, and
James StirlingJames Stirling was a Scottish mathematician. The Stirling numbers and Stirling's approximation are named after him.Biography:...
. Among his fellow
HuguenotThe Huguenots were members of the Protestant Reformed Church of France during the 16th and 17th centuries. Since the 17th century, people who formerly would have been called Huguenots have instead simply been called French Protestants, a title suggested by their German coreligionists, the...
exiles in
EnglandEngland is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...
, he was a colleague of the editor and translator
Pierre des MaizeauxPierre des Maizeaux, also spelled Desmaizeaux was a French Huguenot writer exiled in London, best known as the translator and biographer of Pierre Bayle....
.
De Moivre wrote a book on
probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of nondeterministic events or measured quantities that may either be single...
,
The Doctrine of ChancesThe Doctrine of Chances was the first textbook on probability theory, written by 18thcentury French mathematician Abraham de Moivre and first published in 1718. De Moivre wrote in English because he resided in England at the time, having fled France to escape the persecution of Huguenots...
, said to have been prized by gamblers. De Moivre first discovered Binet's formula, the
closedformMaths:* Closedform expression, a finitary expression* Closed differential form, a differential form \alpha with the property that d\alpha = 0Poetry:* In poetry analysis, a type of poetry that exhibits regular structure, such as meter or a rhyming pattern;...
expression for Fibonacci numbers linking the
nth power of
φIn mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
to the
nth Fibonacci number.
Early years
Abraham de Moivre was born in Vitry in Champagne on May 26, 1667. His father, Daniel de Moivre, was a surgeon who, though middle class, believed in the value of education. Though Abraham de Moivre's parents were Protestant, he first attended Christian Brothers' Catholic school in Vitry, which was unusually tolerant given religious tensions in France at the time. When he was eleven, his parents sent him to the Protestant Academy at Sedan, where he spent four years studying Greek under Jacques du Rondel. The Protestant
Academy of SedanThe Academy of Sedan was a Huguenot academy in Sedan, founded in 1579 and suppressed in 1681. It was one of the main centres for the production of Reformed pastors in France for a hundred years.History:...
had been founded in 1579 at the initiative of Françoise de Bourbon, widow of HenriRobert de la Marck; in 1682 the Protestant Academy at Sedan was suppressed and de Moivre enrolled to study logic at Saumur for two years. Although mathematics was not part of his course work, de Moivre read several mathematical works on his own including Elements de mathematiques by Father Prestet and a short treatise on games of chance, De Ratiociniis in Ludo Aleae, by Christiaan Huygens. In 1684 he moved to Paris to study physics and for the first time had formal mathematics training with private lessons from
Jacques OzanamJacques Ozanam was a French mathematician.Biography:Jacques Ozanam was born in SainteOlive, Ain, France....
.
Religious persecution in France became severe when King Louis XIV issued the
Edict of FontainebleauThe Edict of Fontainebleau was an edict issued by Louis XIV of France, also known as the Revocation of the Edict of Nantes. The Edict of Nantes of 1598, had granted the Huguenots the right to practice their religion without persecution from the state...
in 1685, which revoked the
Edict of NantesThe Edict of Nantes, issued on 13 April 1598, by Henry IV of France, granted the Calvinist Protestants of France substantial rights in a nation still considered essentially Catholic. In the Edict, Henry aimed primarily to promote civil unity...
, that had given substantial rights to French Protestants. It forbade Protestant worship and required that all children be baptized by Catholic priests. De Moivre was sent to the Prieure de SaintMartin, a school the authorities sent Protestant children to for indoctrination into Catholicism. It is unclear when de Moivre left the Prieure de SaintMartin and moved to England, as the records of the Prieure de SaintMartin indicate that he left the school in 1688, but de Moivre and his brother presented themselves as Huguenots admitted to the Savoy Church in London on August 28, 1687.
Middle years
By the time he arrived in London, de Moivre was a competent mathematician with a good knowledge of many of the standard texts. To make a living, de Moivre became a private tutor of mathematics, visiting his pupils or teaching in the coffee houses of London. De Moivre continued his studies of mathematics after visiting the Earl of Devonshire and seeing Newton’s recent book, Principia. Looking through the book, he realized it was far deeper than books he had studied previously, and was determined to read and understand it. However, as he was required to take extended walks around London to travel between his students, de Moivre had little time for study so he would tear pages from the book and carry them around in his pocket to read between lessons. Eventually de Moivre become so knowledgeable about the material that Newton referred questions to him, saying, “Go to Mr. de Moivre; he knows these things better than I do.”
By 1692, de Moivre became friends with
Edmond HalleyEdmond Halley FRS was an English astronomer, geophysicist, mathematician, meteorologist, and physicist who is best known for computing the orbit of the eponymous Halley's Comet. He was the second Astronomer Royal in Britain, following in the footsteps of John Flamsteed.Biography and career:Halley...
and soon after with
Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
himself. In 1695, Halley communicated de Moivre’s first mathematics paper, which arose from his study of fluxions in the Principia, to the
Royal SocietyThe Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...
. This paper was published in the Philosophical Transactions that same year. Shortly after publishing this paper de Moivre also generalized Newton’s famous
Binomial TheoremIn elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...
into the
Multinomial theoremIn mathematics, the multinomial theorem says how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem to polynomials.Theorem:...
. The
Royal SocietyThe Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...
became apprised of this method in 1697 and made de Moivre a member two months later.
After de Moivre had been accepted, Halley encouraged him to turn his attention to astronomy. In 1705, de Moivre discovered, intuitively, that “the centripetal force of any planet is directly related to its distance from the centre of the forces and reciprocally related to the product of the diameter of the evolute and the cube of the perpendicular on the tangent”. In other words, if a planet, M, follows an elliptical orbit around a focus F and has a point P where PM is tangent to the curve and FPM is a right angle so that FP is the perpendicular to the tangent, then the centripetal force at point P is proportional to F*M/(R*(F*P)
^{3}) where R is the radius of the curvature at M.
Johann BernoulliJohann Bernoulli was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family...
proved this formula in 1710.
Despite these successes, de Moivre was unable to obtain an appointment to a Chair of Mathematics at a university, which would have released him from his dependence on timeconsuming tutoring that burdened him more than it did most other mathematicians of the time. At least a part of the reason was a bias against his French origins.
In November 1697 he was elected a Fellow of the Royal Society and in 1712 was appointed to a commission set up by the society, alongside MM. Arbuthnot, Hill, Halley, Jones, Machin, Burnet, Robarts, Bonet, Aston and Taylor to review the claims of Newton and Leibniz as to who discovered calculus. The full details of the controversy can be found in the Leibniz and Newton calculus controversy article.
Throughout his life de Moivre remained poor. It is reported that he was a regular customer of Slaughter's Coffee House, St. Martin's Lane at Cranbourn Street, where he earned a little money from playing
chessChess is a twoplayer board game played on a chessboard, a squarecheckered board with 64 squares arranged in an eightbyeight grid. It is one of the world's most popular games, played by millions of people worldwide at home, in clubs, online, by correspondence, and in tournaments.Each player...
.
Later years
De Moivre continued studying the fields of probability and mathematics until his death in 1754 and several additional papers were published after his death. As he grew older, he became increasingly lethargic and needed longer sleeping hours. He noted that he was sleeping an extra 15 minutes each night and correctly calculated the date of his death on the day when the additional sleep time accumulated to 24 hours, November 27, 1754. He died in
LondonLondon is the capital city of :England and the :United Kingdom, the largest metropolitan area in the United Kingdom, and the largest urban zone in the European Union by most measures. Located on the River Thames, London has been a major settlement for two millennia, its history going back to its...
and was buried at
St MartinintheFieldsSt MartinintheFields is an Anglican church at the northeast corner of Trafalgar Square in the City of Westminster, London. Its patron is Saint Martin of Tours.Roman era:Excavations at the site in 2006 led to the discovery of a grave dated about 410...
, although his body was later moved.
Probability
De Moivre pioneered the development of analytic geometry and the theory of probability by expanding upon the work of his predecessors, particularly Christiaan Huygens and several members of the Bernoulli family. He also produced the second textbook on probability theory,
The Doctrine of Chances: a method of calculating the probabilities of events in playThe Doctrine of Chances was the first textbook on probability theory, written by 18thcentury French mathematician Abraham de Moivre and first published in 1718. De Moivre wrote in English because he resided in England at the time, having fled France to escape the persecution of Huguenots...
. (The first book about games of chance, Liber de ludo aleae ("On Casting the Die") , was written by Girolamo Cardano in the 1560s, but not published until 1663.) This book came out in four editions, 1711 in Latin, and 1718, 1738 and 1756 in English. In the later editions of his book, de Moivre gives the first statement of the formula for the
normal distribution curve, the first method of finding the probability of the occurrence of an error of a given size when that error is expressed in terms of the variability of the distribution as a unit, and the first identification of the
probable errorStatistics:In statistics, the probable error of a quantity is a value describing the probability distribution of that quantity. It defines the halfrange of an interval about a cental point for the distribution, such that half of the values from the distribution will lie within the interval and...
calculation. Additionally, he applied these theories to gambling problems and actuarial tables.
An expression commonly found in probability is n! but before the days of calculators calculating n! for a large n was time consuming. In 1733 de Moivre proposed the formula for estimating a factorial as
n! =
cn^{n+1/2}e
^{−n}. He obtained an expression for the constant
c but it was
James StirlingJames Stirling was a Scottish mathematician. The Stirling numbers and Stirling's approximation are named after him.Biography:...
who found that c was √(2
π)
. Therefore,
Stirling's approximationIn mathematics, Stirling's approximation is an approximation for large factorials. It is named after James Stirling.The formula as typically used in applications is\ln n! = n\ln n  n +O\...
is as much due to de Moivre as it is to Stirling.
De Moivre also published an article called Annuities upon Lives, in which he revealed the normal distribution of the mortality rate over a person’s age. From this he produced a simple formula for approximating the revenue produced by annual payments based on a person’s age. This is similar to the types of formulas used by insurance companies today. See also
de Moivre–Laplace theoremIn probability theory, the de Moivre–Laplace theorem is a normal approximation to the binomial distribution. It is a special case of the central limit theorem...
De Moivre’s formula
In 1707 de Moivre derived:

which he was able to prove for all positive integral values of
n. In 1722 he suggested it in the more well known form of
de Moivre's FormulaIn mathematics, de Moivre's formula , named after Abraham de Moivre, states that for any complex number x and integer n it holds that...
:

In 1749 Euler proved this formula for any real n using
Euler's formulaEuler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
, which makes the proof quite straightforward. This formula is important because it relates complex numbers and
trigonometryTrigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
. Additionally, this formula allows the derivation of useful expressions for cos(
nx) and sin(
nx) in terms of cos(
x) and sin(
x).
External links