Sin 4 theta de moivre biography
Abraham de Moivre
French mathematician (1667–1754)
Abraham de MoivreFRS (French pronunciation:[abʁaamdəmwavʁ]; 26 May 1667 – 27 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory.
He moved to England at a young age due to the religious persecution of Huguenots in France which reached a climax in 1685 with the Edict of Fontainebleau. He was a friend of Isaac Newton, Edmond Halley, and James Stirling. Among his fellow Huguenot exiles in England, he was a colleague of the editor and translator Pierre des Maizeaux.
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 the golden ratioφ to the nth Fibonacci number. He also was the first to postulate the central limit theorem, a cornerstone of probability theory.
Life
Early years
Abraham de Moivre was born in Vitry-le-François in Champagne on 26 May 1667. His father, Daniel de Moivre, was a surgeon who believed in the value of education. Though Abraham de Moivre's parents were Protestant, he first attended the 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 Sedan had been founded in 1579 at the initiative of Françoise de Bourbon, the widow of Henri-Robert 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 works on mathematics on his own, including The \(n^\text{th}\) roots of unity are the complex solutions to the equation \[ z^n = 1.\] Suppose complex number \(z = a + bi\) is a solution to this equation, and consider the polar representation \(z = r e^{i\theta}\), where \(r = \sqrt{a^2 + b^2}\) and \(\tan \theta = \frac{b}{a}, 0 \leq \theta < 2\pi \). Then, by De Moivre's theorem, we have \[ 1 = z^n = \big(r e^{i\theta} \big) ^n = r^n (\cos \theta + i \sin \theta)^n = r^n (\cos n \theta + i \sin n \theta).\] This implies \(r^n = 1\) and, since \(r\) is a real, non-negative number, we have \( r = 1.\) Also, \(n \theta = 2k \pi\) or \( \theta = \frac{2k \pi}{n}\) for some integer \(k\). Now, the values \(k = 0, 1, 2, \ldots, n-1\) give distinct values of \(\theta\) and, for any other value of \(k\), we can add or subtract an integer multiple of \(n\) to reduce to one of these values of \(\theta\). Therefore, the \(n^\text{th}\) roots of unity are the complex numbers \[ e^{\frac{2k\pi }{ n} i} = \cos \left( \frac{2k\pi }{ n } \right) + i \sin \left( \frac{2k\pi }{ n } \right) \text{ for } k = 0, 1, 2, \ldots, n-1. \] Observe that this gives \(n\) complex \(n^\text{th}\) roots of unity, as we know from the fundamental theorem of algebra. Since all of the complex roots of unity have absolute value 1, these points all lie on the unit circle. Furthermore, since the angle between any two consecutive roots is \(\frac{2\pi}{n}\), the complex roots of unity are evenly spaced around the unit circle. What are the complex solutions to the equation \(z = \sqrt[3]{1}?\) Cubing both sides gives \(z^3 = 1,\) implying \(z\) is a \(3^\text{rd}\) root of unity. By the above, the \(3^\text{rd}\) roots of unity are \[ e^{ \frac{2k\pi }{ 3 } i} = \cos \left( \frac{2k\pi }{ 3} \right) + i \sin \left( \frac{2k\pi }{ 3 } \right) \text{ for } k = 0,1,2.\] This gives the roots of unity \(1, e^{\frac{2\pi}{3} i}, e^{\frac{4\pi}{3} i}\), or \[ 1,\quad -\frac{1}{2} + \frac{\sqrt{3}}{2} i,\quad -\frac{1}{2} - \frac Theorem: (cos x + i sin x)^n = cos nx + i sin nx Not to be confused with De Moivre–Laplace theorem. In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity) states that for any real numberx and integern it is the case that where i is the imaginary unit (i = −1). The formula is named after Abraham de Moivre, although he never stated it in his works. The expression cos x + i sin x is sometimes abbreviated to cisx. The formula is important because it connects complex numbers and trigonometry. By expanding the left hand side and then comparing the real and imaginary parts under the assumption that x is real, it is possible to derive useful expressions for cos nx and sin nx in terms of cos x and sin x. As written, the formula is not valid for non-integer powers n. However, there are generalizations of this formula valid for other exponents. These can be used to give explicit expressions for the nth roots of unity, that is, complex numbersz such that z = 1. Using the standard extensions of the sine and cosine functions to complex numbers, the formula is valid even when x is an arbitrary complex number. For and , de Moivre's formula asserts that or equivalently that In this example, it is easy to check the validity of the equation by multiplying out the left side. De Moivre's formula is a precursor to Euler's formula with x expressed in radians rather than degrees, which establishes the fundamental relationship between the trigonometric functions and the complex exponential function. One can derive de Moivre's formula using Euler's formula and the exponential law for integer powers since Euler's formula implies that the left side is equal to while the right side is equal to The truth of de Moivre's theorem can be established by using mathematical induction for natura .De Moivre's formula
Example
Relation to Euler's formula
Proof by induction