Thanks Andy, Charles and Henry, but I guess I wasn't clear. I'm aware of the Fundamental Theorem of Algebra and what happens when n is a real integer. My question is about what happens when n is a complex number with integer parts. I've determined (and had confirmed in offline discussion) that the number of roots depends on which branch one uses for the angle (t or theta). I'm wondering if there's any standard definition of the roots of 1 when n is complex (not purely real). Kerry On Thu, May 9, 2013 at 6:28 AM, Andy Latto <andy.latto@pobox.com> wrote:
On Thu, May 9, 2013 at 2:25 AM, Kerry Mitchell <lkmitch@gmail.com> wrote:
Hi all,
I am playing with a problem that has boiled down to this: for positive integer n, there are n complex roots of 1 (or any complex number). What happens when n is a Gaussian integer? How many roots are there and what are they like?
Write your complex number z as r e^(i * t), where r and t are real. (I use t instead of the standard theta for ease of expressing in ASCII)
Then the nth roots of z are r^(1/n) e^(i (2pi k + t)/n), for k = 0, 1, 2, ... n - 1.
Geometrically, the n roots are evenly spaced on a circle centered at the origin.
Andy
I've done some preliminary work on this; can someone point me to a reference so I can see if I'm on the right track?
Thanks, Kerry _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
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