45. De Moivre’s Theorem

The following statements follow immediately from the definitions of addition and multiplication.

(1) The real (or imaginary) part of the sum of two complex numbers is equal to the sum of their real (or imaginary) parts.

(2) The modulus of the product of two complex numbers is equal to the product of their moduli.

(3) The amplitude of the product of two complex numbers is either equal to the sum of their amplitudes, or differs from it by $2\pi$.

It should be observed that it is not always true that the principal value of $\mathrm{am}(z{z}^{\prime })$ is the sum of the principal values of $\mathrm{am}z$ and $\mathrm{am}{z}^{\prime }$. For example, if $z=z^{\prime }=-1+i$, then the principal values of the amplitudes of $z$ and $z^{\prime }$ are each $\frac{3}{4}\pi$. But $z{z}^{\prime }=-2i$, and the principal value of $operatornameam(z{z}^{\prime })$ is $-\frac{1}{2}\pi$ and not $\frac{3}{2}\pi$.

The two last theorems may be expressed in the equation $$r(\cos \theta +i\sin \theta )\times \rho (\cos \varphi +i\sin \varphi )=r\rho \{\cos (\theta +\varphi )+i\sin (\theta +\varphi )\},$$ which may be proved at once by multiplying out and using the ordinary trigonometrical formulae for $\cos (\theta +\varphi )$ and $\sin (\theta +\varphi )$. More generally $$\begin{array}{c}{r}_1(\cos {\theta }_1+i\sin {\theta }_1)\times r_2(\cos {\theta }_2+i\sin {\theta }_2)\times \cdots \times r_n(\cos {\theta }_n+i\sin {\theta }_n)\\ =r_1{r}_2\dots r_n\{\cos ({\theta }_1+{\theta }_2+\cdots +{\theta }_n)+i\sin ({\theta }_1+{\theta }_2+\cdots +{\theta }_n)\}.\end{array}$$

A particularly interesting case is that in which $$r_1=r_2=\cdots =r_n=1,{\theta }_1={\theta }_2=\cdots ={\theta }_n=\theta .$$

We then obtain the equation $$(\cos \theta +i\sin \theta {)}^n=\cos n\theta +i\sin n\theta ,$$ where $n$ is any positive integer: a result known as De Moivre’s Theorem.¹

Again, if $$z=r(\cos \theta +i\sin \theta )$$ then $$1/z=(\cos \theta –i\sin \theta )/r.$$ Thus the modulus of the reciprocal of $z$ is the reciprocal of the modulus of $z$, and the amplitude of the reciprocal is the negative of the amplitude of $z$. We can now state the theorems for quotients which correspond to (2) and (3).

(4) The modulus of the quotient of two complex numbers is equal to the quotient of their moduli.

(5) The amplitude of the quotient of two complex numbers either is equal to the difference of their amplitudes, or differs from it by $2\pi$.

Again $$\begin{array}{rl}(\cos \theta +i\sin \theta {)}^{-n}& =(\cos \theta –i\sin \theta {)}^n\\ & =\{\cos (-\theta )+i\sin (-\theta ){\}}^n\\ & =\cos (-n\theta )+i\sin (-n\theta ).\end{array}$$ Hence De Moivre’s Theorem holds for all integral values of $n$, positive or negative.

To the theorems (1)-(5) we may add the following theorem, which is also of very great importance.

(6) The modulus of the sum of any number of complex numbers is not greater than the sum of their moduli.

Let $\overline{OP}$, $\overline{O{P}^{\prime }}$, … be the displacements corresponding to the various complex numbers. Draw $PQ$ equal and parallel to $O{P}^{\prime }$, $QR$ equal and parallel to $OP”$, and so on. Finally we reach a point $U$, such that $$\overline{OU}=\overline{OP}+\overline{O{P}^{\prime }}+\overline{OP”}+\dots .$$ The length $OU$ is the modulus of the sum of the complex numbers, whereas the sum of their moduli is the total length of the broken line $OPQR\dots U$, which is not less than $OU$.

A purely arithmetical proof of this theorem is outlined in Exs. xxi. 1.

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1. It will sometimes be convenient, for the sake of brevity, to denote $\cos \theta +i\sin \theta$ by $\mathrm{Cis}\theta$: in this notation, suggested by Profs. Harkness and Morley, De Moivre’s theorem is expressed by the equation $(\mathrm{Cis}\theta {)}^n=\mathrm{Cis}n\theta$.↩︎

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