Simple Complex Number Question (Roots of 1)

In summary, the conversation discusses finding the complex fifth roots of one using Euler's equation and the mistake made in expressing the solution as 1^(1/5) instead of (e^(i2pi/5))^5. The person is unsure if they should ask for a regrade but it is clear that their solution was correct.
  • #1
bmb2009
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0

Homework Statement


So on one of my homework assignments I had to find the complex fifth roots of one. Because of Euler's equation the arguments are simply 0, 2pi/5,4pi/5,6pi/5,and 8pi/5. It is easy to see that (e^i2pi/5)^5 = (e^i2pi) = cos(2pi) + isin(2pi) = 1 + 0 = 1 but on my paper I wrote e^(i2pi/5) = 1^(1/5) and similar expressions for all the arguments of the fifth roots and lost 10 point out of 15. I wanted to ask the TA to regrade it but I don't want to bother him if I'm flat out wrong expressing the way I did? Any input would be great.. Thanks!


Homework Equations





The Attempt at a Solution

 
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  • #2
bmb2009 said:

Homework Statement


So on one of my homework assignments I had to find the complex fifth roots of one. Because of Euler's equation the arguments are simply 0, 2pi/5,4pi/5,6pi/5,and 8pi/5. It is easy to see that (e^i2pi/5)^5 = (e^i2pi) = cos(2pi) + isin(2pi) = 1 + 0 = 1 but on my paper I wrote e^(i2pi/5) = 1^(1/5) and similar expressions for all the arguments of the fifth roots and lost 10 point out of 15. I wanted to ask the TA to regrade it but I don't want to bother him if I'm flat out wrong expressing the way I did? Any input would be great.. Thanks!

Homework Equations


The Attempt at a Solution


Saying e^(i2pi/5) = 1^(1/5) is flat out wrong. 1^(1/5) doesn't mean ANY fifth root of 1. It means 1. Saying (e^(i2pi/5))^5=1 is really what you meant. Still you got the roots. Up to you whether to appeal.
 
Last edited:

1. What are the roots of 1 in complex numbers?

The roots of 1 in complex numbers are 1 and -1. In complex numbers, the imaginary unit i is defined as the square root of -1. Therefore, the two roots of 1 are 1 and -1 multiplied by i, which can also be written as 1i and -1i.

2. How do you find the roots of 1 in complex numbers?

To find the roots of 1 in complex numbers, we can use the general formula for finding the nth root of a complex number:
z1/n = r1/n(cos((θ+2kπ)/n) + isin((θ+2kπ)/n)), where z = 1, r = 1, θ = 0, and k is any integer. Plugging in these values, we get two roots: 1(cos0 + isin0) = 1 and 1(cos2π + isin2π) = 1(cos0 + isin0) = -1.

3. Why are there two roots of 1 in complex numbers?

In complex numbers, the square root of any number can have two solutions: one positive and one negative. This is because when we square a complex number, we get a negative result. Therefore, when we take the square root, we need to take into account both the positive and negative solutions.

4. What is the relationship between the roots of 1 and the unit circle?

The roots of 1, which are 1 and -1, are also known as the first and second roots of unity. These roots correspond to the two points where the line x=1 intersects the unit circle. This relationship can be seen by graphing the complex number z=1 on the complex plane, which results in a point at (1,0) on the unit circle.

5. How are the roots of 1 used in mathematics and science?

The roots of 1 have many important applications in mathematics and science. They are used in trigonometry to represent the solutions to certain equations, in geometry to represent points on the unit circle, and in physics to describe periodic functions. They also play a role in electrical engineering and signal processing. Additionally, the concept of roots of unity has connections to number theory and abstract algebra.

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