de Moivre’s theorem states that for any integer  and any real number ,

de Moivre’s theorem can be prove using Euler’s formula, which is a easier way or by using trigonometry function and the results we showed in the previous lessons. I shall show you both ways.

This result follows naturally from Euler’s Formula.

While this is all good, I’m sure many of you mathematicians out there, well at least for me, would like a challenge of proving it without Euler’s Formula. Read on.

Formally stated, we are to prove

Clearly the proposed equation is true if n=0. Now let n be any positive integer, i.e. . Let . Then

and

And so the polar form of  is . Therefore,

This completes the proof if n is a nonnegative integer.

Now suppose that n is a negative integer. Then, by introducing a new term, we can say  is a positive integer.

In the second line, we have used the first half of the proof since we know that m is nonnegative.

It is good to note that we have used two techniques here. One, we rationalized the denominator multiplying through with a term that removes the imaginary part. Two, we used a mixture of trigonometry identities such as  and  to get the intended result.

It pays to notice how other branches of mathematics are used in this proof. We will now move to a direct application of de Moivre’s theorem, find the nth roots of a complex number.