Re: Magic number 23
- To: mathgroup at smc.vnet.net
- Subject: [mg41508] Re: Magic number 23
- From: "Christopher J. Henrich" <chenrich at monmouth.com>
- Date: Fri, 23 May 2003 03:28:34 -0400 (EDT)
- References: <baclk4$r54$1@smc.vnet.net> <200305211201.IAA06871@smc.vnet.net> <baiahc$djl$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
In article <baiahc$djl$1 at smc.vnet.net>, David Terr <dterr at wolfram.com> wrote: > Dave Langers wrote: > > >>Mathematica knows the exact values of the trigonometric functions for some > >>special angles. I was curious how many such values there are. > >> > > > >Take a look at: > >http://mathworld.wolfram.com/TrigonometricAngles.html > > > >It doesn't explain what might be special about sin(pi/23), except that > >it cannot be written as a simple exact value. > > > >BTW: This is interestingly enough related to constructions with compass > >and straightedge: > >http://mathworld.wolfram.com/ConstructibleNumber.html > >http://mathworld.wolfram.com/ConstructiblePolygon.html > > > >Greetings, > >Dave > > > > > 23 is magic in the sense it's the smallest positive number n such that > sin(pi/n) is not solvable with radicals. To get radical expressions for > smaller values of n, use FunctionExpand. I have difficulty understanding this statement. I think my problem is that "solvable with radicals" is a slightly slippery concept. As someone on this thread has pointed out, Sin[Pi/n] is trivially expressible in radicals: (1^[1/2n] - 1^[-1/2n])/(2 * 1^[1/2]). and indeed, Mathematica sometimes gives you answers like this. Older books sometimes used "n-th root of 1" notation for Exp[ (2 Pi I)/n]. When I ask for FunctionExpand[Sin[Pi/7]] I get a complicated expression which is, to be sure, algebraic in that it is made up of square roots and cube roots. But, once again, it includes cube roots of complex numbers. I think that Mathematica uses Cardano's formula for the solution of cubic equations. FunctionExpand[Sin[Pi/11]] is a truly impressive expression with some fifth roots in it. I really wonder how Mathematica did that. Is there a body of lore on solutions of equations using the extraction of roots of *real* numbers? -- Chris Henrich
- References:
- Re: Magic number 23
- From: Dave Langers <RemoveThisPart.d.langers@wxs.nl>
- Re: Magic number 23