Demoivre's Theorem

Abraham DeMoivre, who lived from 1667 to 1754, was a famous French mathematician known for many mathematical discoveries. However, he is most known for his DeMoivre's theorem, which holds that (cos x + i sin x)n = cos (nx) + i sin (nx). This is true for any complex number (and more specifically for any real number) x and any integer n. The formula is important because it creates a bridge between complex numbers (i is used as a representation for the imaginary unit) and trigonometry. For instance, one can derive many trigonometric identities from this theorem. Also, by expanding the left hand side, then eventually comparing the imaginary and real parts, it is possible to derive useful expressions for cos(nx) and sin(nx) in terms of both cos(x) and sin(x). Moreover, one is able to use this formula in order to find explicit expressions for the nth roots of unity, which are basically complex numbers z such that zn = 1.

In order to use the theorem to find the nth roots of a complex number, one must convert the complex number a + bi into its polar form: r (cos x + sin x), where r is the modulus of a + bi and r = Ð'ЃÐ"Јa2 + b2 .

If z = r (cos Ð'Ñ"Ð"† + i sin Ð'Ñ"Ð"†), then it will have n diverse nth roots, shown as:

(for k = 0,1,2,Ð'Ѓc,n-1). The n distinct roots will have the same modulus, r1/n, but will have different arguments determined by for radians or for degrees, where k = 0,1,2,Ð'Ѓc, n-1.

...