Is e^i recognized as a root of unity? Squares of real and imaginary part sum to 1
Favorites|Homepage
Subscriptions | sitemap
HOME > Mathematics > Is e^i recognized as a root of unity? Squares of real and imaginary part sum to 1

Is e^i recognized as a root of unity? Squares of real and imaginary part sum to 1

[From: ] [author: ] [Date: 11-10-02] [Hit: ]
Suppose e^(ix) = 1, where x is real.===> x = 2pi * k ...Now,......
Is e^i recognized as a root of unity? This is a complex number, and the square of the real part and the square of the imaginary part equal 1 when added together.

-
No. Roots of unity are complex number z such that z^n = 1 for some positive integer n. Here's a proof that e^i is not a root of unity:

Suppose e^(ix) = 1, where x is real. Then:

e^(ix) = cos(x) + i sin(x) = 1
===> cos(x) = 1 AND sin(x) = 0
===> x = 2pi * k ... for some integer k

Now, suppose e^i is a root of unity. Then, for some integer n > 0:

(e^i)^n = e^(in) = 1
===> n = 2pi * k

Since n > 0, we cannot have k = 0. Thus:

pi = n / (2k)

which would imply pi is rational. But, pi is not rational, so e^i is not a root of unity.
1
keywords: Is,root,of,sum,real,imaginary,part,and,as,unity,to,Squares,recognized,Is e^i recognized as a root of unity? Squares of real and imaginary part sum to 1
New
Hot
© 2008-2010 http://www.science-mathematics.com . Program by zplan cms. Theme by wukong .