Essentially, I need to show that e^inx is dense in the complex unit circle for irrational angle x. I'm not sure how to go about this but I think I need to show that between any two points on the unit circle, we can find some e^inx.
Asked
Active
Viewed 335 times
0
-
see here and probably lots of other places as well: https://math.stackexchange.com/questions/282102/prove-that-the-orbit-of-an-iterated-rotation-of-0-by-api-a-irrational-aro – operatorerror Feb 24 '18 at 21:52
-
Full discussions in either of two inexpensive book by Ivan Niven. In Irrational Numbers, page 72 Theorem 6.3, he shows that the multiples of an irrational are dense and uniformly distributed n the unit interval ( we are taking the fractional parts). In Diophantine Approximations, an entire chapter, pages 24-25 has this thing, Theorem 3.2 – Will Jagy Feb 24 '18 at 21:57
-
Welcome to the site! You may not have realized that this is a completely common homework problem (and a well known fact about irrational rotations) that has been discussed many times before. Separately, we have some written advice about "How to ask a good question" https://math.meta.stackexchange.com/questions/9959/how-to-ask-a-good-question . – Carl Mummert Feb 24 '18 at 22:15
1 Answers
-3
Its so simple.$$exp(jx)=cos(x)+jsin(x)$$. Since $$sin^2(x)+cos^2(x)=1$$ this complex number always is on the circumference of a circle with radius 1
user528935
- 141
- 4
-
This doesn't price that irrational rotations are dense on the circle. – Michael McGovern Feb 24 '18 at 22:45