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Topic: Can a fraction have none noneending and nonerepeating decimal representation?
Replies: 108   Last Post: Aug 16, 2013 5:22 PM

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David Bernier

Posts: 3,317
Registered: 12/13/04
Re: Can a fraction have none noneending and nonerepeating decimal
representation?

Posted: Aug 3, 2013 10:43 PM
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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 <85ffb478-968f-4de6-9284-b1e53fc19f0d@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 Jean-Guillaume 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.

>
> [...]
> Jean-Guillaume 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 on-line 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), 193-194.

Liouville's paper is on-line 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 non-terminating,
> 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. 528-540,
> 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 hand-waiving
> 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/


Date Subject Author
7/30/13
Read Can a fraction have none noneending and nonerepeating decimal representation?
JT
7/30/13
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7/31/13
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JT
7/31/13
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Virgil
7/31/13
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JT
7/31/13
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Tucsondrew@me.com
7/31/13
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JT
7/31/13
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Tucsondrew@me.com
7/31/13
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JT
7/31/13
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7/31/13
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JT
7/31/13
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Virgil
7/31/13
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Virgil
7/31/13
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JT
7/31/13
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7/31/13
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JT
7/31/13
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7/31/13
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JT
7/31/13
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JT
7/31/13
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JT
7/31/13
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JT
7/31/13
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JT
7/31/13
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LudovicoVan
7/31/13
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7/31/13
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7/31/13
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7/31/13
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JT
7/31/13
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JT
7/31/13
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Virgil
8/1/13
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JT
8/1/13
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JT
8/1/13
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Virgil
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Virgil
8/1/13
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JT
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8/3/13
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grei
7/31/13
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JT
8/4/13
Read the surfer's value (Nicholas de Cusa proved that pi was
transcendental in the 15th cue (http://21stcenturysciencetech.com
Brian Q. Hutchings
8/4/13
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JT
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8/2/13
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8/2/13
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7/31/13
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7/31/13
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7/30/13
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