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Demoivre's Theorem: A Historical Overview 
 

Introduction 

 In mathematics, Demoivre's theorem is a statement about integer roots of complex numbers. It is named after Abraham de Moivre, who proved it in 1707. The theorem is as follows: If z is any complex number such that z^n = 1 for some positive integer n, then z is an nth root of unity. This theorem has a number of applications in number theory, algebra, and complex analysis. It is also the basis for a number of results in these fields, including the arithmetic-geometric mean and the fundamental theorem of algebra. 

 William Demoivre: A Life in Mathematics 

 Abraham De Moivre was born in France in 1667. He studied at the Royal Academy in Paris and later moved to England, where he worked as an astronomer, actuary, and mathematician. He is best known for his work on probability, but he also made significant contributions to the fields of geometry, trigonometry, and algebra. The most significant result of De Moivre's life was the discovery of Demoivre's theorem. Before he arrived at this statement, he had to experiment with a variety of mathematical concepts and ideas. He was particularly interested in complex numbers and their use in trigonometry and calculus. He studied the polar form of complex numbers, deriving a number of important results, including the theorem of addition, subtraction, multiplication, and division for complex numbers. He was also the first to consider complex exponentials, which would later become a part of the core of the complex analysis. Early Work on Complex Numbers Before De Moivre arrived at his theorem, he had to answer the fundamental problem of how to make sense of negative numbers. For centuries, physicists and mathematicians alike had posed the same question. It was De Moivre's insight and ingenuity which allowed him to finally give a satisfactory answer. His approach was to treat negative numbers as points on a line or circle and to use an analogous trigonometric representation. This meant that a negative number could be represented as a point on the 360-degree circle, with the position on the circle determined by the magnitude of the number. This allowed De Moivre to represent the operations on negative numbers as rotations and translations on the circle. This was a breakthrough in mathematics, as it allowed complex numbers and their operations to be viewed geometrically. It was this insight that formed the basis for his subsequent work on complex numbers and which eventually led to the statement of his theorem. 

 

Making Sense of Negative Numbers 

 Once De Moivre had gained an understanding of how to treat negative numbers, he was able to begin to formulate a statement that would eventually become known as Demoivre's Theorem. This statement included the concept of an “nth root” of a complex number and the idea of a “cycle” of such roots. De Moivre's theorem stated that if “z” is any complex number such that z^n = 1 for some positive integer “n”, then z is an “nth” root of unity. This theorem was remarkable at the time, as it gave a description of how the root of any given complex number could be determined. The theorem also provided an algebraic solution to the problem of finding the roots of a polynomial with a negative coefficient. The Significance of Euler's Formula De Moivre's theorem was also significant for its implications for Leonhard Euler's famous formula, e^iθ = cosθ + i sinθ. This equation states that for any real angle θ, the complex number e^iθ is equal to cos θ + i sin θ. Therefore, by combining Euler's formula and De Moivre's theorem, it is possible to express the roots of a complex number in terms of its angle. This discovery was revolutionary at the time, as it allowed complex numbers and complex exponentials to be formulated in terms of trigonometric functions. This opened up a new range of possibilities, as it allowed complex numbers to be manipulated more easily. 

 Demoivre's Theorem and the Binomial Theorem 

 De Moivre's theorem is closely related to the binomial theorem, as the roots of a polynomial equation can be expressed in terms of the binomial coefficients. This can be done using the formula (a+b)^n = a^n + binomial coefficients. The theorem can also be used to prove the arithmetic-geometric mean, which states that the mean of two numbers is equal to their geometric mean multiplied by their harmonic mean. This result is closely related to the binomial theorem, as it is possible to express the geometric mean and harmonic mean in terms of binomial coefficients. A final thought Demoivre's theorem has been of immense benefit to mathematics, as it provided a way to treat complex numbers with greater ease and accuracy. This theorem is closely related to other fundamental mathematical results, such as the fundamental theorem of algebra, the binomial theorem, and the arithmetic-geometric mean. Therefore, Demoivre's theorem is arguably one of the most significant tragedies of mathematics ever formulated. 

 Example

(√ 3 + i)^5

To use DeMoivre's Theorem, we need to convert 

 z = − √ 3 + i = x + i y into Polar Form, i.e., r ( cos θ + i sin θ ), where, r > 0, &, θ ∈ ( − π, π ]. z = x + i y = r ( cos θ + i sin θ ) = r c i s θ ⇒ x = r cos θ , y = r sin θ , r = √ x 2 + y 2 ∴ r = √ ( − √ 3 ) 2 + 1 2 = 2 Hence, from x = r cos θ , cos θ = − √ 3 2 , &, similarly, sin θ = 1 2 clearly , θ = 5 π 6 Thus, − √ 3 + i = 2 c i s ( 5 π 6 ) . Now, by DeMoivre's Theorem, ( r c i s θ ) n = r n c i s ( n θ ) In our case, n = 5, r = 2, θ = 5 π 6 ∴ ( − √ 3 + i ) 5 = ( 2 c i s 5 π 6 ) 5 = ( 2 5 ) ( c i s ( 5 ⋅ 5 π 6 ) ) = 32 ( c i s 25 π 6 ) = 32 ( c i s ( 4 π + π 6 ) ) = 32 ( cos ( 4 π + π 6 ) + i sin ( 4 π + π 6 ) ) = 32 ( cos ( π 6 ) + i sin ( π 6 ) ) = 32 ( √ 3 2 + 1 2 i ) = 16 ( √ 3 + i ).