Expressing geometrically the nth roots of a complex number on a circle

[5hassay]

Homework Statement

Let z \in \mathbb{C}. Prove that z^{1/n} can be expressed geometrically as n equally spaced points on the circle x^2 + y^2 = |z|^2, where |z|=|a+bi|=\sqrt{a^2 + b^2}, the modulus of z.

Homework Equations

//

The Attempt at a Solution

My problem is that I am confused with the question.

I understand that

z^{1/n} = |z|^{1/n} (\cos(\frac{\theta}{n} + \frac{2k\pi}{n}) + \sin(\frac{\theta}{n} + \frac{2k\pi}{n}) i)

for k=0,1,\ldots,n-1, where \theta=\mathrm{Arg}(z).

Yet, my confusion is this: Each above complex number we get for each k will produce a complex number with modulus |z|^{1/n}. Because the given circle is of radius |z|, I cannot see how any of these complex numbers can lie on it for a arbitrary complex number z.

Where is my confusion lying, or is there a mistake in this question? I had asked the question giver, but they explained that the modulus didn't matter for plotting on such circle, which didn't make sense...

Thanks you.

 

[Dick]

Homework Statement

Let z \in \mathbb{C}. Prove that z^{1/n} can be expressed geometrically as n equally spaced points on the circle x^2 + y^2 = |z|^2, where |z|=|a+bi|=\sqrt{a^2 + b^2}, the modulus of z.

Homework Equations

//

The Attempt at a Solution

My problem is that I am confused with the question.

I understand that

z^{1/n} = |z|^{1/n} (\cos(\frac{\theta}{n} + \frac{2k\pi}{n}) + \sin(\frac{\theta}{n} + \frac{2k\pi}{n}) i)

for k=0,1,\ldots,n-1, where \theta=\mathrm{Arg}(z).

Yet, my confusion is this: Each above complex number we get for each k will produce a complex number with modulus |z|^{1/n}. Because the given circle is of radius |z|, I cannot see how any of these complex numbers can lie on it for a arbitrary complex number z.

Where is my confusion lying, or is there a mistake in this question? I had asked the question giver, but they explained that the modulus didn't matter for plotting on such circle, which didn't make sense...

Thanks you.

I agree with you. E.g. If you put z=2 and n=2 then the values of 2^(1/2) are sqrt(2) and -sqrt(2). They lie on the circle of radius sqrt(2) not the circle of radius 2. The question is bad.

 

[5hassay]

I agree with you. E.g. If you put z=2 and n=2 then the values of 2^(1/2) are sqrt(2) and -sqrt(2). They lie on the circle of radius sqrt(2) not the circle of radius 2. The question is bad.

Yeah, that's what I am thinking. Thanks for the reply