# De Moivre's formula.

A biography of Abraham DeMoivre can be found in The MacTutor History of Mathematics archive, at the following URL:

$zz' = (ac-bd)+(ad+bc)i$


Proposition 1.4.1   If and , then .

Another formulation of this proposition follows:

Proof. Let and . Then:

Corollary 1.4.2

Proof. For any and , we have:

The result follows.

Corollary 1.4.3 (De Moivre's formula)   If and is any natural number, then .

Another formulation of this proposition follows:

Proof. We prove it by induction on .
(i)
For , we have and
(ii)
Suppose that for some natural number , the equation holds. Then, by Proposition 4.1 we have:

Example 1.4.4   Let .

Then: .

This formula can be generalized to any integer .

Corollary 1.4.5

Proof. For non negative , this is exactly De Moivre's formula. We consider only the case where is a negative integer (whence ).

Example 1.4.6   Let . Then

Example 1.4.7   Let . Find all the integers such that is real.

We have:

By De Moivre's formula (v.s. 4.3), we have:

By 3.8, we have:

i.e.

Noah Dana-Picard 2007-12-24