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Deep
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 Posted: Tue Jul 08, 2008 2:15 am Post subject: Rational or Irrational |
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Consider the following equation under the given conditions.
Q = (cos kD)^(2/k) + (sin kD)^(2/k) (1)
Conditions: 0 < D < pi/2, odd k > 7 such that cos kD > 0, sin kD > 0
Question: Is Q rational or irrational ?
Any helpful reply will be gratefully appreciated.
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The World Wide Wade
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| Posted: Tue Jul 08, 2008 7:33 am Post subject: Re: Rational or Irrational |
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In article
<5a3c2cc1-d336-42e0-acdf-e783ee2dfacc@m73g2000hsh.googlegroups.com>,
Deep <deepkdeb@yahoo.com> wrote:
| Quote: |
Consider the following equation under the given conditions.
Q = (cos kD)^(2/k) + (sin kD)^(2/k) (1)
Conditions: 0 < D < pi/2, odd k > 7 such that cos kD > 0, sin kD > 0
Question: Is Q rational or irrational ?
Any helpful reply will be gratefully appreciated.
Deep
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Looks to me like Q can be any real number in [0, 2^(1-1/k)] with the
given conditions. |
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Deep
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| Posted: Wed Jul 09, 2008 2:01 am Post subject: Re: Rational or Irrational |
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On Jul 7, 10:33 pm, The World Wide Wade <aderamey.a...@comcast.net>
wrote:
| Quote: |
In article
5a3c2cc1-d336-42e0-acdf-e783ee2df...@m73g2000hsh.googlegroups.com>,
Deep <deepk...@yahoo.com> wrote:
Consider the following equation under the given conditions.
Q = (cos kD)^(2/k) + (sin kD)^(2/k) (1)
Conditions: 0 < D < pi/2, odd k > 7 such that cos kD > 0, sin kD > 0
Question: Is Q rational or irrational ?
Any helpful reply will be gratefully appreciated.
Deep
Looks to me like Q can be any real number in [0, 2^(1-1/k)] with the
given conditions.
***** |
One simple numerical example showing Q rational will close the
chapter.
***** |
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James Waldby
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| Posted: Wed Jul 09, 2008 10:44 am Post subject: Re: Rational or Irrational |
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On Tue, 08 Jul 2008 19:01:19 -0700, Deep wrote:
| Quote: |
On Jul 7, 10:33Â pm, The World Wide Wade ... wrote...
 Deep <deepk...@yahoo.com> wrote:
Consider the following equation under the given conditions.
Q = (cos kD)^(2/k) + (sin kD)^(2/k) Â Â Â (1)
Conditions: 0 < D < pi/2, odd k > 7 such that cos kD > 0, sin kD > 0
Question: Is Q rational or irrational ?
....
Looks to me like Q can be any real number in [0, 2^(1-1/k)] with the
given conditions.
*****
One simple numerical example showing Q rational will close the chapter.
*****
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For any fixed k>2, in each interval a < d < b where
cos k*d > 0 and sin k*d > 0, Q_k(d) is a continuous
function. If, as appears to be the case, Q(d) isn't
constant, then (a,b) contains a subinterval (a',b')
where Q(a') and Q(b') differ. Then the open interval
between Q(a') and Q(b') contains infinitely many rationals.
Simple numerical example: k=9, a=.004921717291, b=a+10^-12.
Q(a) < 3/2 < Q(b). By continuity there exists some c,
a<c<b, such that Q(c) = 3/2, a rational number. |
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