De Moivre's Theorem: does it apply only to rationals?

By DHChandler (t448) on July 22, 1998 :

I understand that DeMoivre's theorem only applies to rational numbers.

My students are not convinced by this and wonder why it cannot apply to any real number ?

By Adam Wood (ajpw2) on July 26, 1998 :

Your students are right -- in fact it applies to any complex number, which include the reals.

However I believe de Moivre only proved it for rationals, and many (school) text books only seem to go that far.

The result for complex numbers follows from Euler's Formula:
e^ix = cos(x) + i sin(x)

Hope this helps,
Adam

By Eva on July 31, 1998 :

Dear David,

De Moivre's theorem says that

$(\cos x + i \sin x)^{n} = \cos n x + i \sin n x$ .

This is true for all integers $n$ , and all complex numbers $x$ .

The problem when $n$ is not an integer is that $\cos n x$ is single-valued, whereas $(\cos x + i \sin x)^{n}$ is many-valued.

Example: $x = π / 3$ , $n = 1 / 2$ . In this case

$\cos n x + i \sin n x = \cos (π / 6) + i \sin (π / 6) = \sqrt{3} / 2 + i / 2$

whereas $(\cos x + i \sin x)^{1 / 2} = [\cos (π / 3) + i \sin (π / 3 {)]}^{1 / 2} = [1 / 2 + i \sqrt{3} / 2]^{1 / 2}$ .

Now $[\pm (\sqrt{3} / 2 + i / 2 {)]}^{2} = (\sqrt{3} / 2 + i / 2)^{2} = (1 / 4) (\sqrt{3} + i)^{2} = 1 / 2 + i \sqrt{3} / 2$ .

So we see that

$(\cos x + i \sin x)^{1 / 2} = \pm [\cos (x / 2) + i \sin (x / 2)]$ .

We have seen this in the case $x = π / 3$ , but it is true in general. Indeed, for any $x$ (even complex $x$ ),

$(\pm [\cos (x / 2) + i \sin (x / 2 {)])}^{2} = (\cos x / 2 + i \sin x / 2)^{2} = \cos^{2} (x / 2) - \sin^{2} (x / 2) + 2 i \sin (x / 2) . \cos (x / 2) = \cos x + i \sin x$ .

Thus in general $(\cos x + i \sin x)^{1 / 2} = \pm [\cos (x / 2) + i \sin (x / 2)]$ .

A similar statement holds for all complex $n$ , and all complex $x$ : if $n$ is not an integer, then $(\cos x + i \sin x)^{n}$ is many-valued (it has infinitely many values if $n$ is irrational), and one of these values is $\cos n x + i \sin n x$ . This is all that can be said.