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

In summary, the homework question asks to prove that z^{1/n} can be expressed as n equally spaced points on a circle of radius |z| in the complex plane. However, this seems to be incorrect as the complex numbers obtained for different values of k will have different moduli and therefore cannot lie on the given circle. The question may be flawed.
  • #1
5hassay
82
0

Homework Statement



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

Homework Equations



//

The Attempt at a Solution



My problem is that I am confused with the question.

I understand that

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

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

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

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.
 
Physics news on Phys.org
  • #2
5hassay said:

Homework Statement



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

Homework Equations



//

The Attempt at a Solution



My problem is that I am confused with the question.

I understand that

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

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

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

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.
 
  • #3
Dick said:
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
 

1. What is the geometric representation of the nth root of a complex number on a circle?

The geometric representation of the nth root of a complex number on a circle is a point on the unit circle, also known as the complex plane. The distance of this point from the origin represents the magnitude of the root, while the angle it forms with the positive real axis represents the argument or phase of the root.

2. How can the nth root of a complex number be expressed geometrically?

The nth root of a complex number can be expressed geometrically by finding the point on the unit circle that corresponds to the root's magnitude and argument. This can be done by using the formula r^(1/n) * (cos(θ/n) + i*sin(θ/n)), where r is the magnitude of the original complex number and θ is its argument.

3. Can the geometric representation of the nth root of a complex number be visualized?

Yes, the geometric representation of the nth root of a complex number can be visualized on the complex plane. The point on the unit circle corresponding to the root's magnitude and argument can be plotted and seen as a point on the plane. This can help in understanding the relationship between the original complex number and its nth root.

4. How does the location of the nth root on the unit circle relate to its argument?

The location of the nth root on the unit circle is determined by its argument or phase. The angle formed by the root with the positive real axis is equal to the original root's argument divided by n. This means that the nth root will be located at a point on the unit circle that is n times closer to the positive real axis than the original root.

5. What is the significance of expressing the nth root of a complex number geometrically on a circle?

Expressing the nth root of a complex number geometrically on a circle can provide a visual representation of the root's magnitude and argument. This can aid in understanding the properties of complex numbers and their roots, such as the relationship between the original number and its nth root, as well as the symmetry of roots on the complex plane.

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
220
  • Calculus and Beyond Homework Help
Replies
8
Views
809
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
20
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
Back
Top