How to find the intersection of complex roots, Im(z) and Re(z)?

In summary, the homework statement is that z^6=2(cos(π/6)+isin(π/6)) and r^6=2, r=2^1/6. Find A ∩ B ∩ C which is the intersection of all three sets.
  • #1
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Homework Statement


(a) Find the 6th complex roots of √3 + i.
(b) Let A={z|z^6 =√3+i} and B={z|Im(z)>0} and C={z|Re(z)>0}. Find A ∩ B ∩ C.

Homework Equations


z^6=2(cos(π/6)+isin(π/6))
r^6=2, r=2^1/6
6θ=π/6+2kπ, θ=π/36+kπ/3

The Attempt at a Solution


I've done part (a):
When k=0, z = 2^1/6(cos(π/36)+isin(π/36)),
When k=1, z = 2^1/6(cos(13π/36)+isin(13π/36)),
When k=2, z = 2^1/6(cos(25π/36)+isin(25π/36)),
When k=3, z = 2^1/6(cos(37π/36)+isin(37π/36)),
When k=4, z = 2^1/6(cos(49π/36)+isin(49π/36)),
When k=5, z = 2^1/6(cos(61π/36)+isin(61π/36)).
As for part (b) though, I am unsure for what the answer is. I understand that A={z|z^6 =√3+i} relates to part (a) however, I don't understand how you would find the intersection of A, B and C.
Please help? Thanks.
 
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  • #2
Ha, they're throwing a little set theory your way. Set A is every answer in a), however set B is the set of all elements that have a positive imaginary part (the ones with sin>0) and C is the set of all elements that have a positive real part (cos>0) now A ∩ B ∩ C is the intersection of all these sets and is thus the set of those in part a) (basically every element in your universe), has sin>0 AND has cos>0. Find out which ones belong to it and you've got your answer.
 
  • #3
So, is it simply then just doing sinπ/36=0.0871557425... cosπ/36=0.9961946981... to find the answers to see whether they belong to the set or not?
I did that, and got that When k=0, z = 2^1/6(cos(π/36)+isin(π/36)),
When k=1, z = 2^1/6(cos(13π/36)+isin(13π/36) belong to the set of all three.
So, is the intersection then z = 2^1/6(cos(π/36)+isin(π/36)) and z = 2^1/6(cos(13π/36)+isin(13π/36)?
Moreover, could you then write the answer as A ∩ B ∩ C = z=2^1/6(cos(π/36)+isin(π/36)), z=2^1/6(cos(13π/36)+isin(13π/36)?
 
  • #4
Yes that should be the answer, since both of those equations meet the criterion of the set.
 
  • #5
Okay, great. Thank you very much for you help!
 

FAQ: How to find the intersection of complex roots, Im(z) and Re(z)?

How do I find the intersection of complex roots?

To find the intersection of complex roots, you will first need to list out all the roots of the given equation. Then, plot these points on a complex plane and look for any points where the roots cross each other. These points represent the intersection of complex roots.

What is Im(z) and Re(z)?

Im(z) and Re(z) are the imaginary and real parts of a complex number, respectively. The imaginary part is the coefficient of the imaginary unit, i, and the real part is the coefficient of the real unit, 1. For example, in the complex number z = 3 + 4i, the real part (Re(z)) is 3 and the imaginary part (Im(z)) is 4.

How do I plot complex numbers on a complex plane?

To plot complex numbers on a complex plane, you will need to use a real axis (x-axis) and an imaginary axis (y-axis). The real axis represents the real part of the complex number, while the imaginary axis represents the imaginary part. Plot the real and imaginary parts as coordinates on the complex plane to get the location of the complex number.

Can the intersection of complex roots have an imaginary part?

Yes, the intersection of complex roots can have an imaginary part. This means that the roots intersect at a point that has both a real and imaginary component. This is why it is important to plot complex roots on a complex plane to accurately identify their intersection.

Are there any formulas or methods to find the intersection of complex roots?

There are several formulas and methods that can be used to find the intersection of complex roots, such as the quadratic formula, the cubic formula, and the method of substitution. However, the best method will depend on the specific equation and its complexity. It is always helpful to plot the roots on a complex plane to visually identify their intersection.

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