lanedance said:how about writing it as
[tex] cos(\frac{2 \pi}{n}} ) + i sin(\frac{2 \pi}{n}} ) = e^{\frac{2 \pi}{n}} [/tex]
Driessen12 said:no because sin(pi) is also zero and cos(pi) is 1 giving us 1 which is exactly what I have to prove cannot happen. For example choose k to be 2 and n to be 4, then we have e^(i(pi)) which is 1. I am not sure how to prove that this cannot happen though. I need to show that it is impossible to have any multiple of pi as my argument. any ideas?
Primitive roots of unity are complex numbers that, when raised to certain powers, result in all the other complex roots of unity. They are also referred to as primitive nth roots of unity, where n is a positive integer.
All primitive roots of unity are also roots of unity, but not all roots of unity are primitive roots of unity. The primitive roots of unity form a subset of the roots of unity.
Primitive roots of unity have important applications in mathematics, particularly in number theory and algebra. They are also used in signal processing and cryptography.
The formula for finding primitive roots of unity is given by ωn = e2πi/n, where n is the order of the root. Alternatively, you can also use a table or calculator to find the primitive roots of unity for a given order.
Yes, there can be multiple primitive roots of unity for the same order. For example, there are two primitive fifth roots of unity, ω5 and ω5^2, both of which have an order of 5.