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Abraham De Moivre

Abraham de Moivre (26 May 1667 in Vitry-le-François, Champagne, France – 27 November 1754 in London, England) was a French mathematician famous for de Moivre's formula, which links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He was a friend of Isaac Newton, Edmund Halley, and James Stirling.

De Moivre wrote a book on probability theory, The Doctrine of Chances, said to have been prized by gamblers. De Moivre first discovered Binet's formula, the closed-form expression for Fibonacci numbers linking the nth power of ? to the nth Fibonacci number.

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Abraham De Moivre was born in Vitry in the province of Champagne, France, on May 26, 1667, to a Protestant family. He showed an early interest in mathematics and studied it— secretly—at the various religious schools he was attending. In 1685 Louis XIV revoked the Edict of Nantes—a decree issued in 1598 granting religious freedom to French Protestants—and a period of repression followed. By one account De Moivre was imprisoned for two years before leaving for London, where he would spend the rest of his life. He studied mathematics on his own and became very proficient in it. By sheer luck he happened to be at the house of the Earl of Devonshire, where he worked as a tutor, at the very moment when Isaac Newton stepped out with a copy of the Principia, his great work on the theory of gravitation. De Moivre took up the book, studied it on his own, and found it far more demanding than he had expected (it is a difficult text even for a modern reader). But by assiduous study—he used to tear pages out of the huge volume so he could study them between his tutoring sessions—he not only mastered the work but became an expert on it, so much so that Newton, in later years, would refer to De Moivre questions addressed to himself, saying, “Go to Mr. De Moivre; he knows these things better than I do.”

In 1692 he met Edmond Halley (of comet fame), who was so impressed by his mathematical ability that he communicated to the Royal Society De Moivre’s first paper, on Newton’s “method of fluxions” (i.e., the differential calculus). Through Halley, De Moivre became a member of Newton’s circle of friends that also included John Wallis and Roger Cotes. In 1697 he was elected to the Royal Society and in 1712 was appointed as member of the Society’s commission to settle the bitter priority dispute between Newton and Leibniz over the invention of the calculus. He was also elected to the academies of Paris and Berlin.

Despite these successes, De Moivre was unable to secure himself a university position—his French origin was one reason— and even Leibniz’s attempts on his behalf were unsuccessful. He made a meager living as a tutor of mathematics, and for the rest of his life would lament having to waste his time walking between the homes of his students. His free time was spent in the coffeehouses and taverns on St. Martin’s Lane in London, where he answered all kinds of mathematical questions addressed to him by rich patrons, especially about their chances of winning in gambling.

When he grew old he became lethargic and needed longer sleeping hours. According to one account, he declared that beginning on a certain day he would need twenty more minutes of sleep on each subsequent day. On the seventy-third day— November 27, 1754—when the additional sleeping time accumulated to 24 hours, he died; the official cause was recorded as “somnolence” (sleepiness). He was eighty-seven years old, joining a long line of distinguished English mathematicians who lived well past their eighties: William Oughtred, who died in 1660 at the age of 86, John Wallis (d. 1703 at 87), Isaac Newton (d. 1727 at 85), Edmond Halley (d. 1742 at 86), and in our time, Alfred North Whitehead (d. 1947 at 86) and Bertrand Russell, who died in 1970 at 98. The poet Alexander Pope paid him tribute in An Essay on Man:

Who made the spider parallels design,

Sure as Demoivre, without rule or line?

De Moivre’s mathematical work covered mainly two areas: the theory of probability, and algebra and trigonometry (considered as a unified field). In probability he extended the work of his predecessors, particularly Christiaan Huygens and several members of the Bernoulli family. A generalization of a problem first posed by Huygens is known as De Moivre’s problem: Given n dice, each having f faces, find the probability of throwing any given number of points. His many investigations in this field appeared in his work The Doctrine of Chances: or, a Method of Calculating the Probability of Events in Play (London, 1718, with expanded editions in 1738 and 1756); it contains numerous problems about throwing dice, drawing balls of different colors from a bag, and questions related to life annuities. Here also is stated (though he was not the first to discover it) the law for finding the probability of a compound event. A second work, A Treatise of Annuities upon Lives (London, 1725 and 1743), deals with the analysis of mortality statistics (which Halley had begun some years earlier), the division of annuities among several heirs, and other questions of interest to financial institutes and insurance companies.

In the theory of probability one constantly encounters the expression n! (read “n factorial”), defined as 1.2.3... n. The value of n! grows very rapidly with increasing n; for example, 10!=3,628,800 while 20! =2,432,902,008,176,640,000. To find n! one must first find (n-1)!, which in turn requires finding(n-2)! and so on, making a direct calculation of n! for large n extremely time consuming. It is therefore desirable to have an approximation formula that would estimate n! for large n by a single calculation. In a paper written in 1733 and presented privately to some friends, De Moivre developed the formula

n! = Cn^(n+1/2)e^(-n);

where c is a constant and e the base of natural logarithms.  He was unable, however, to determine the numerical value of this constant; this task befell a Scot, James Stirling (1692–1770), who found that C = (2.pi)^(1/2) . Stirling’s formula, as it is known today, is thus as much due to De Moivre; it is usually written in the form

n! =(2.pi.n)n^n.e^-n.

As an example, for n =20 the formula gives 2,422,786,847 ×10^18, compared to the correct, rounded value 2,32,902,008 ×10^18.

De Moivre’s third major work, Miscellanea Analytica (London, 1730), deals, in addition to probability, with algebra and analytic trigonometry. A major problem at the time was how to factor a polynomial such as x^2n + Px^n +1 into quadratic factors. This problem arose in connection with Cotes’ work on the decomposition of rational expressions into partial fractions (then knownas “recurring series”). De Moivre completed Cotes’s work, left unfinished by the latter’s early death. Among his many results we find the following formula, sometimes known as “Cotes’ property of the circle”:

Capture001

To obtain this factorization, we only need to find (using De Moivre’s theorem) the 2n different roots of the equation x^2n  +1 =0, that is, the 2n complex values of  (-1)^(1/2) , and then multiply the corresponding linear factors in conjugate pairs. The fact that trigonometric expressions appear in the factorization of a purely algebraic expression such as x^2n  +1 amazes any student who encounters such a formula for the first time; in De Moivre’s time it amazed even professional mathematicians.

De Moivre’s famous theorem,

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was suggested by him in 1722 but was never explicitly stated in his work; that he knew it, however, is clear from the related formula

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which he had already found in 1707 (De Moivre derived it for positive integral values of n; Euler in 1749 proved it for any real n).He frequently used it in Miscellanea Analytica and in numerous papers he published in the Philosophical Transactions, the official journal of the Royal Society.

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 London and was buried at St Martin-in-the-Fields, although his body was later moved.

Courtesy: Princeton University.


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