De Moivre's Theorem

Statement of DeMoivre's Theorem

If 'n' be any rational number, positive or negative, then

Proof:

Part I:

When 'n' is any integer, positive or negative then

Part II:

When 'n' is a fraction.

Part I:

Case 1:

When n is a positive integer.

By actual multiplication, we have

Similarly by the method of induction,

Case 2:

When 'n' is a negative integer.

Let n = -m and m > 0

Part II:

Case 3:

When 'n' is a fraction.

where 'p' is any positive integer and 'q' is any integer.

From Part I,

Now taking q^th root of both sides,

Hence the theorem.

Corollary 1

Corollary 2

Corollary 3