de Moivre's theorem for non-integer powers

[aldld]

So, we just recently received textbooks (Haese & Harris) for our math HL class, and just flipping through it, I came across what I believe to be a (rather major) inaccuracy with regards to the statement of de Moivre's theorem.

The textbook states that

(|z| cis θ)ⁿ = |z|ⁿ cis nθ for all rational n.

Rational being the key word here. However, if we assume that this is true for rational powers then the formula is no longer well-defined. For example,

cis(0) = cis(2pi)

cis(0)^1/2 = cis(2pi)^1/2

cis(0/2) = cis(2pi/2)

cis(0) = cis(pi)

1 = -1, a contradiction.

The books "proof" of de Moivre's theorem for rational powers is as follows:

[cis(θ/n)]ⁿ = cis(n(θ/n)) = cis θ and so [cis θ]1/n = cis(θ/n)

However, this doesn't seem like a valid deduction to make. Sure, it's a useful way of thinking about it when finding the roots of complex numbers, but really it just seems sloppy, at least as far as I can tell. Or perhaps someone could shed some light on this?

[flinquinnster]

Ha, we're doing De Moivre's Theorem in class as well, and I tend to just nod blankly when given a statement - I just tend to believe it I was looking at the proof they had in my copy of the H+H textbook, and the proof for 1/n does look a bit dodgy. I would assume that it usually works, but clearly, with your example of 0 and 2pi it seems to fall down. I sense that it may have something to do with your choice of 0 - maybe it does weird things... I was under the impression (or at least my teacher is under the impression) that De Moivre's Theorem is technically only defined for n = positive integers, but in practical applications we can assume that n can be any real number (I'm assuming that would include irrational, but I'm really not sure).

Anyway, I don't think I'm of much use in bringing some clarity to the issue. Hopefully someone else more knowledgeable can explain I'll definitely try and ask my teacher to explain this to our class at least.

[HiggsHunter]

So, we just recently received textbooks (Haese & Harris) for our math HL class, and just flipping through it, I came across what I believe to be a (rather major) inaccuracy with regards to the statement of de Moivre's theorem.

The textbook states that

(|z| cis θ)ⁿ = |z|ⁿ cis nθ for all rational n.

Yes, since (|z| cis θ)ⁿ can have multiple values when n is non-integer, your textbook should have explained that |z|ⁿ cis nθ is ONE possible value.

[aldld]

So, we just recently received textbooks (Haese & Harris) for our math HL class, and just flipping through it, I came across what I believe to be a (rather major) inaccuracy with regards to the statement of de Moivre's theorem.

The textbook states that

(|z| cis θ)ⁿ = |z|ⁿ cis nθ for all rational n.

Yes, since (|z| cis θ)ⁿ can have multiple values when n is non-integer, your textbook should have explained that |z|ⁿ cis nθ is ONE possible value.

Ah, that's what I'd've figured. And no, the textbook does not mention anything about that.