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

## 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 $z$.

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## 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 Helper

## 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 $z$.

//

## 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.

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