## Abraham De Moivre (1667 - 1754)

De Moivre was born to a Huguenot (French Protestant) family in France in 1667, but lived in England from the age of 18 when his family fled to England after the revocation of the Edict of Nantes when all Huguenots were expelled from Catholic France.

A natural mathematician, De Moivre was hugely inspired after reading Newton’s book Principia Mathematica, and in his later life the two became great friends.

De Moivre became a maths tutor, hoping to get a teaching post at one of the universities. Unfortunately, at this time foreigners were not accepted into the academic establishment, and despite his mathematical achievements he had to rely on his tutoring to support himself.

He was eventually admitted to the Royal Society in 1697, and presided over the rival claims of Newton and Leibniz as to who had discovered calculus. However, as he was appointed on the recommendation of Newton, he was not entirely without bias!

## De Moivre’s mathematics

De Moivre led the way for the development of analytical geometry, and the theory of probability.

He published his Doctrine of Chances, or Method of Calculating the Probabilities of Events at Play (not a very snappy title!) in 1618, dedicating it to his friend Newton.

The book includes sections on statistical independence as well as discussions on various games.
Despite this, his name is linked to a result called De Moivre’s Theorem which is that,

(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta.

This important result is a close relative to a fundamental result that Euler discovered concerning the complex exponential.

De Moivre’s Theorem links trigonometry - through the sine and cosine - to complex numbers, those numbers involving the square root of -1, denoted by i.

It is easy to give an illustration of the power of this result. Consider a particular value of n:

(\cos\theta+i\sin\theta)^3=\cos 3\theta+i\sin 3\theta

If we expand the left hand side of this expression, we obtain by the Binomial Theorem:

\cos^3\theta+3i\cos^2\theta\sin\theta-3\cos\theta\sin^2\theta-i\sin^3\theta=\cos3\theta+i\sin3\theta

Equating the real parts and the imaginary parts on either side we then have,

\cos3\theta=\cos^3\theta-3\cos\theta\sin^2\theta

\sin3\theta=3\cos^2\theta\sin\theta-\cos^3\theta

De Moivre even managed to bring mathematics into his own death. On studying his sleep patterns, he realised that he was sleeping 15 minutes ({1\over{4}} hour) extra each night; he surmised that he would die when he slept for 24 hours.

So using the idea of an arithmetic progression and assuming that he slept the standard 8 hours when he started this morbid line of thought, he examined the sequence:

8,\ 8+{1\over{4}},\ 8+2.{1\over{4}},\ 8+3.{1\over{4}},\ \ldots

and sought the term that represented 24 hours. He found it and concluded that he would die in 64 years. And now for the spooky bit - his prediction came true and he died in poverty in London, 1754!