# Raise complex number using De Moivre - integer only?

This is probably a silly question, but it is not really clear to me whether De Moivre's theorem of raising a complex number to the nth power only work if n is an integer value?
E.g. if I try to raise (2-2i) to the power of 3.01 then my manual calculation get a different result than my calculator, but for all integer values it works.

## Answers and Replies

cepheid
Staff Emeritus
Science Advisor
Gold Member
This is probably a silly question, but it is not really clear to me whether De Moivre's theorem of raising a complex number to the nth power only work if n is an integer value?
E.g. if I try to raise (2-2i) to the power of 3.01 then my manual calculation get a different result than my calculator, but for all integer values it works.

Well, at least according to the Wikipedia article on the subject, 'n' must be an integer:

http://en.wikipedia.org/wiki/De_Moivre's_formula

>>Well, at least according to the Wikipedia article on the subject, 'n' must be an integer:
argh, I missed that. Thanks
Do you know how I can raise a complex number by a real number?

I may be wrong but in mathematics, n is used to denote an integer by convention.

>>I may be wrong but in mathematics, n is used to denote an integer by convention.
I think you are right.

Do you know how I can raise a complex number by a real number?

You can raise it to any power by using this method:

$z^w = e^{ln|z^w|} = e ^{w \cdot ln|z|}$ and simplify from there. From the top of my head, I believe this formula gives an infinite number of solutions depending on how you choose your branch of the logarithm.

HallsofIvy
Science Advisor
Homework Helper
No, the power does not have to be an integer. DeMoivre's formula is very commonly used, for example, to find roots, the "1/n" power.

After some more Bing'ing I found this
http://www.suitcaseofdreams.net/De_Moivre_formula.htm

For r = 1 we obtain De Moivre’s formula for fractional powers:
(cos+i sin)p/q = cos((p/q))+i sin((p/q)). (1.27)

So the decimal power should work as well, right?. I will try and convert the decimal to a fraction, but I dont see why it would make a difference.

When I tried it worked correct for quadrant I and II, but I could not get it right for III and IV.

I will keep playing with this.

I played some more with this and it was all just a matter of adjusting θ depending of the quadrant.
For quadrant II
θ=(atan(im(z)/re(z))+pi)*n, where n is any number
And for III
θ=(atan(im(z)/re(z))-pi)*n