Solving a Constructible Identity: 1+e to the 2pi/7i and Beyond

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In summary, the conversation discusses an identity involving a geometric progression and how it was obtained. The identity is 1+e^{\frac{2\pi}{7}i}+e^{\frac{4\pi}{7}i}+e^{\frac{6\pi}{7}i}+e^{\frac{8\pi}{7}i}+e^{\frac{10\pi}{7}i}+e^{\frac{12\pi}{7}i}=\frac{e^(\frac{12\pi}{7}i)^7-1}{e^{\frac{12\pi}{7}i}-1}, and it was derived using the formula
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Homework Statement


hi,
I'm working on constructible things again and in one of the proofs our prof threw out this identity and I just don't know where it came from:
[tex]1+e^{\frac{2\pi}{7}i}+e^{\frac{4\pi}{7}i}+e^{\frac{6\pi}{7}i}+e^{\frac{8\pi}{7}i}+e^{\frac{10\pi}{7}i}+e^{\frac{12\pi}{7}i}=\frac{e^(\frac{12\pi}{7}i)^7-1}{e^{\frac{12\pi}{7}i}-1}[/tex]
HOW did he get that?


Edit: I can't tell if the final term is supposed to be 2pi of 12 pi. I dunno.

Homework Equations





The Attempt at a Solution

 
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  • #2
Hint : geometric progression.

The exponents on the Right Hand Side should be 2pi, not 12 pi.
 
  • #3
[tex]x^n-1=(x-1)(1+x+x^2+x^3+x^4+x^5+...x^{n-1})[/tex]

for n=natural number.
 
  • #4
Thanks. I see it now.
CC
 

1. How do you solve an identity with a complex number?

Solving an identity with a complex number involves using the properties of complex numbers, such as the distributive property and the properties of exponents. It also involves using the knowledge of the unit circle and trigonometric identities.

2. What is the significance of the number 2pi/7i in this constructible identity?

The number 2pi/7i represents a complex number with a magnitude of 2pi and an imaginary unit of 7i. In this constructible identity, it is used to represent the angle of rotation in the complex plane.

3. Can you explain how to construct a geometric solution for this identity?

To construct a geometric solution for this identity, you need to use a compass and straight edge to construct a regular heptagon (a seven-sided polygon). The angle of rotation in the complex plane is equal to 2pi/7, which is the angle formed by two consecutive vertices of the heptagon.

4. Are there any other ways to solve this constructible identity?

Yes, there are other ways to solve this constructible identity, such as using algebraic manipulation and trigonometric identities. However, the geometric solution is the most visually intuitive and can provide a deeper understanding of the relationship between complex numbers and geometrical constructions.

5. Why is this constructible identity significant in mathematics?

This constructible identity is significant because it demonstrates the connection between complex numbers and geometry. It also showcases the power of using geometric constructions to solve mathematical problems. Additionally, this identity has historical significance as it was first solved by ancient Greek mathematicians using only a compass and straight edge.

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