Raise complex number using De Moivre - integer only?

• jkristia
In summary, the conversation discussed De Moivre's theorem for raising a complex number to the nth power and whether it only works for integer values of n. It was mentioned that according to the Wikipedia article, n must be an integer. However, it was also suggested that n may be used to denote an integer in mathematics by convention. The conversation also explored how to raise a complex number by a real number, with the formula z^w = e^{w \cdot ln|z|} being mentioned. It was noted that this formula can give an infinite number of solutions depending on the chosen branch of the logarithm. The conversation also mentioned using De Moivre's formula to find roots and discussed its application
jkristia
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.

jkristia said:
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.

jkristia said:
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.

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 don't 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.
θ=(atan(im(z)/re(z))+pi)*n, where n is any number
And for III
θ=(atan(im(z)/re(z))-pi)*n

1. What is the De Moivre's theorem?

The De Moivre's theorem states that for any complex number z and any positive integer n, the nth power of z can be expressed as the product of n copies of z in polar form.

2. How do I raise a complex number to an integer power using De Moivre's theorem?

To raise a complex number z to an integer power n, simply convert z to polar form (rcisθ) and apply the De Moivre's theorem formula: zn = rncis().

3. Can De Moivre's theorem be used to raise a complex number to a non-integer power?

No, De Moivre's theorem only applies to integer powers. To raise a complex number to a non-integer power, you can use the polar form of the number and apply the general exponentiation rule for complex numbers: za+bi = racis() where r and θ are the modulus and argument of z respectively.

4. Can De Moivre's theorem be applied to any complex number?

Yes, De Moivre's theorem can be applied to any complex number, as long as the power is an integer.

5. Are there any limitations to using De Moivre's theorem to raise a complex number to an integer power?

One limitation is that the result of raising a complex number to an integer power using De Moivre's theorem may not always be a real number. This can be avoided by using the absolute value of the complex number when converting it to polar form.

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