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December 30th, 2013, 06:31 AM  #21 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: Cube roots of 1
I don't want to post anything there because I would prefer to keep that post clean for any wandering transcendental number theorist that may happen to pass by. For your first bullet (by first bullet, I am referring to the top bullet), you are asking about the kobayashi maru style cheats (which I like all of them BTW) presented by [color=#0000FF]crom[/color], [color=#00FF00]mathbalarka[/color] and [color=#00FF00]Hoempa[/color]? If yes, then I think there should be many (perhaps infinite) ways to represent root 3. I don't think I understand the second bullet. Is an algebraic number? Too tired to look it up... If not, then But like I said, I'm not sure about what you are asking, so bear with me. 
December 30th, 2013, 08:17 AM  #22 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Cube roots of 1 Also, note that is also transcendental by LindemannWeierstrass theorem. But that does not prove whether or not is algebraic or not. And is transcendental by Gelfond–Schneider, unfortunately, so I cannot prove or disprove whether your expression holds or not. PS : That should have been but similar consideration holds and we have nothing much. PPS : You can post this on that topic, as I believe it is non trivial enough to get considerations from those transcendental passer by s... Oops, I mean transcendental number theorist passerby s... 
December 30th, 2013, 10:34 PM  #23  
Global Moderator Joined: Dec 2006 Posts: 20,826 Thanks: 2160  When using latex, type \ln rather than just ln for the ln function. Quote:
 
December 30th, 2013, 11:37 PM  #24  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Cube roots of 1 Quote:
 
December 31st, 2013, 02:57 AM  #25  
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: Quote:
And that's what I used. Second, if you know one root you can get the others by using symmetry and rotations. Multiplication by rotates a point counterclockwise in the complex plane by 90°. Multiplication by rotates a point counterclockwise in the complex plane by 45°. ...etc. In general, if you want to rotate counterclockwise by N°, multiply by So, I know one root of is x = 1. I know the roots are symmetrically located at intervals of 120° = 360°/3. I can use as a rotation machine. To rotate x = 1, multiply it like so, There's another root, now do it again, And there's the other root. Anyway, hope you see how to play around with multiplications additions and rotations.  

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