## History of complex numbers: De Moivre's formula

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- Published: Monday, 21 March 2016 12:08

Besides a purely theoretical value - as a kind of "evidence" for consistency and therefore the "lawfulness" of complex numbers, the stated geometric interpretation also has an extremely important practical significance. With its help, the complex numbers can be provided in the so-called trigonometric form that is convenient in many applications - in particular, for the geometrical interpretation of operations of multiplication and division of complex numbers, as well as solving the problem of extracting the roots of a complex number.

De Moivre's formula is the best known, but not the only or even the most important achievement of the British mathematician Abraham de Moivre (1667-1754). His true vocation was the theory of probability, in the history of which Moivre’s name stands next to the names of two other classics of the science - Blaise Pascal and Pierre Simon Laplace. But this is only one manifestation of the ironic twist of fate for this great mind of the XVII - XVIII centuries.

The greatest irony was the scientist’s life journey. In 1688 at age 21, Abraham, an ardent supporter of the Huguenots, Protestant by faith, honest and principled in his convictions, was forced to permanently leave the Catholic France. He settled in London, where he soon won the affection and respect of the great Newton. Newton so highly valued Moivre’s mathematical knowledge and talent that he used to send to him everybody who turned to him on issues related to math. At the end of the century Moivre’s scientific achievements became so great that he was elected a member of the Royal Society (British Academy of Sciences). In 1735 de Moivre was elected a member of the Berlin Academy of Sciences, and in 1754 (the year of his death) – a foreign (!) Member of the Paris Academy. Is it not ironic: a frenchman by birth, born and raised in France and at the same time ... a foreign member of his National Academy of Sciences?

Intrigued by the problem of expressing cosφ through cosnφ, Moivre established the following relationship, which was, in fact, a primary form of the formula which was to bear his name.

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Moivre felt that there were great possibilities for application of complex numbers in trigonometry. Particularly interesting results are obtained, when the Moivre formula is used alongside the formula for the Newton binomial.

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