
Re: Can a fraction have none noneending and nonerepeating decimal representation?
Posted:
Aug 3, 2013 10:43 PM


On 08/03/2013 06:54 PM, David Bernier wrote: > On 08/02/2013 01:38 PM, David Bernier wrote: >> On 08/02/2013 06:49 AM, Richard Tobin wrote: >>> In article <85ffb478968f4de69284b1e53fc19f0d@googlegroups.com>, >>> <jonas.thornvall@gmail.com> wrote: >>> >>>> But one 1/phi is not a rational ratio, but the 1/6 hexagon (center to >>>> vertex/sum of sides) is and so is all polygons derived from multiplying >>>> the vertices. >>> >>> You were claiming that pi is rational. It is not. >>> >>>  Richard >>> >> >> This jogged my memory about Lagrange and irrational numbers. >> Lambert was the first to prove the irrationality of pi: >> in 1761, >> >> < http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational > . >> >> Lagrange showed that if x > 0 is rational, then sin(x) >> is irrational. >> >> Ref.: (from sci.math in 2011) >> ====================================================================== >> In JeanGuillaume Garnier's >> "Analyse algebrique, faisant suite a la premiere section de l'algebre", >> 1814, >> >> pages 538 and 539, he mentions Lagrange and C. Haros. >> He writes that Lagrange showed that, if x > 0 is a rational >> number, say x = p/q with gcd(p, q) = 1, then >> sin(x) is irrational. >> >> This uses the Taylor series: >> <snip> >> ======================================================================= >> d.b. in sci.math, http://mathforum.org/kb/message.jspa?messageID=7364740 >> >> >> sin(pi) = 0 is rational. pi>0. >> >> If we assume Lagrange's result: >> x>0, x rational ==> sin(x) is irrational, >> then we can conclude that pi must be irrational. > > [...] > JeanGuillaume Garnier's argument is very sketchy. > > It relies on a modification of the "greedy algorithm", where > (a) The signs alternate in the representation of a rational > p/q with 0 < p/q < 1 as an alternating sum of > Egyptian fractions. > > (b) If m_k is the k'th summand (up to +/), and > m_{k+1} is the k+1'st summand, then > m_k divides m_{k+1}. > > Garnier gives the algorithm as applied to p/q = 887/1103 . > > The algorithm is simple, and an explanation is given > for why it terminates for rational numbers p/q . > I haven't checked this through, however. > > Then, 887/1103 is represented as: > > ? 1  1/5 + 1/(5*47)  1/(5*47*50) > + 1/(5*47*50*367)  1/(5*47*50*367*551) + 1/(5*47*50*367*551*1103) > > > %67 = 887/1103 [ PARI/gp ]. > > Garnier then agues in general about sin(x), so > in particular sin(1), assuming x is rational.
If we look at the proof by contradiction by Fourier that e is irrational at Wikiepedia, < http://en.wikipedia.org/wiki/Proof_that_e_is_irrational#Proof >
It uses the Taylor series of exp(x) at x=1; similarly, I think we can get a proof by contradiction that sin(1) is irrational.
But then, trying to do the same with sin(2), sin(3) etc. doesn't seem so easy.
Dave L. Renfro wrote in sci.math:
<< [A] Joseph Liouville, "Sur l'irrationnalité du nombre e = 2,718...", Journal de Mathématiques Pures et Appliquées (= Liouville's Journal) (1) 5 (May 1840), 192.
Liouville's paper is online at http://gallica.bnf.fr/Catalogue/noticesInd/FRBNF34348784.htm
This paper modifies Fourier's method of proving e is irrational (see below) to prove that e is not quadratically irrational. (This doesn't follow from the already known fact that e^r is irrational for every nonzero rational number, by the way.)
[B] Joseph Liouville, "Addition a la note sur l'irrationnalité du nombre e", Journal de Mathématiques Pures et Appliquées (= Liouville's Journal) (1) 5 (June 1840), 193194.
Liouville's paper is online at http://gallica.bnf.fr/Catalogue/noticesInd/FRBNF34348784.htm
This paper extends the proof of Liouville [A] to prove that e^2 is not quadratically irrational. >> cf.: < http://mathforum.org/kb/message.jspa?messageID=3853756 > .
So in 1840, Liouville showed that e isn't the root of a degree two integer polynomial.
So, e^2 is irrational (and more).
So, maybe proving sin(2) is irrational isn't too hard ...
> The Taylor series for sin(x) gives: > > sin(1) = > 1  1/3! + 1/5!  1/7! + 1/9!  ... (***) > > He writes: > "The series for sin(1) [ more generally: sin(p/q) ] > is also an alternating modified greedy algorithm expansion > series , as we had for 887/1103. But we know that the > alternating modified greedy algorithm expansion terminates > for rationals. Since (***) is nonterminating, > sin(1) is irrational [ more generally sin(p/q) ]. " > > I accept the part: > "the alternating modified greedy algorithm expansion terminates > for rationals". > > But, > > (I) The "alternating modified greedy algorithm expansion" > is not explained rigorously for irrationals. > > It's Chapter xxvii, "Transformations des fractions", > pp. 528540, > Analyse algébrique, faisant suite a la première section de l'algèbre, > 1814. On the web: > > http://books.google.ca/books?id=iS4PAAAAQAAJ > > So, in view of (I), it's sort of like some handwaiving > is going on. But neverthelss, Garnier says that Haros > told him about this and that Lagrange had a Memoire > in cinquieme cahier du Journal de l'Ecole Polytechnique.
 new: http://sci.math.narkive.com/

