- #1
bosox097
- 1
- 0
How do you do these two problems?
1. Find the sum of the 6th roots of unity.
2. Find the product of the 6th roots of unity.
1. Find the sum of the 6th roots of unity.
2. Find the product of the 6th roots of unity.
In summary, the 6th roots of unity are -1 and 1, and their product is -1. The sum of the 6th roots of unity is 0, and this can be seen by looking at the vertices of a regular hexagon and using group theory. This pattern can also be observed by considering the coefficients of a polynomial with roots at the 6th roots of unity. Additionally, symmetry can be used to determine the sum of the 6th roots of unity.
- #2
Data
- 998
- 0
The real 2nth roots of unity, for any natural n, are -1 and 1. If a complex number is an mth root of unity (for any m) then its complex conjugate is as well. If [itex]z \in \mathbb{C}[/itex] then [itex]z\overline{z} = |z|^2[/itex]. Thus the product of the 2nth roots of unity (for any n) is -1.
Furthermore, -z is a 2nth root of unity whenever z is. Thus the sum of the 2nth roots of unity (for any n) is 0.
I hope this wasn't a homework problem!
Furthermore, -z is a 2nth root of unity whenever z is. Thus the sum of the 2nth roots of unity (for any n) is 0.
I hope this wasn't a homework problem!
Last edited:
- #3
robert Ihnot
- 1,059
- 1
You should look at symetric functions. Take the cubic: (x-a)(x-b)(x-c)=0. Then if this is multiplied out, we get
[tex]X^3-X^2(a+b+c)+X(ab+ac+bc)-(abc) = 0.[/tex]
[tex]X^3-X^2(a+b+c)+X(ab+ac+bc)-(abc) = 0.[/tex]
- #4
mathwonk
Science Advisor
Homework Helper
- 11,635
- 1,930
look at the vertices of a regular hexgon and think of vector addition, and then use group theory.
- #5
HallsofIvy
Science Advisor
Homework Helper
- 42,989
- 975
(x-a)(x-b)= x2- (a+ b)x+ ab
(x-a)(x-b)(x-c)= x3- (a+ b+ c)x2+ (ab+bc+ ac)x- abc
(x-a)(x-b)(x-c)(x-d)= x4- (a+ b+ c+ d)x3+ (ab+ac+ ad+ bc+ bd+ cd)x2- (abc+ acd+ bcd)x+ abcd
Do you see the pattern?
Even more simply: the nth roots of unity are equally spaced around the unit circle in the complex plane. What does symmetry tell you about their sum?
(x-a)(x-b)(x-c)= x3- (a+ b+ c)x2+ (ab+bc+ ac)x- abc
(x-a)(x-b)(x-c)(x-d)= x4- (a+ b+ c+ d)x3+ (ab+ac+ ad+ bc+ bd+ cd)x2- (abc+ acd+ bcd)x+ abcd
Do you see the pattern?
Even more simply: the nth roots of unity are equally spaced around the unit circle in the complex plane. What does symmetry tell you about their sum?
1. What are the 6th roots of unity?
The 6th roots of unity are the six complex numbers that, when raised to the power of 6, equal 1. These numbers are 1, -1, 0.5 + 0.866i, 0.5 - 0.866i, -0.5 + 0.866i, and -0.5 - 0.866i.
2. How do I solve a 6th root of unity problem?
To solve a 6th root of unity problem, you can use the formula e^(2πik/6), where k is an integer from 0 to 5. This will give you the 6th roots of unity listed above.
3. What is the significance of 6th roots of unity in mathematics?
The 6th roots of unity have several important applications in mathematics, including in solving polynomial equations, finding values of trigonometric functions, and in Fourier analysis. They also have connections to number theory and complex analysis.
4. Can the 6th roots of unity be graphed on the complex plane?
Yes, the 6th roots of unity can be graphed on the complex plane. They will form a regular hexagon with vertices at (1,0), (-1,0), (0.5,0.866), (0.5,-0.866), (-0.5,0.866), and (-0.5,-0.866).
5. How are 6th roots of unity related to other roots of unity?
The 6th roots of unity are a subset of the larger group of nth roots of unity, which are numbers that, when raised to the power of n, equal 1. The 6th roots of unity are also related to the 3rd roots of unity, as the 3rd roots of unity are the squares of the 6th roots of unity.
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