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January 28th, 2014, 12:04 AM   #1
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irrational numbers

prove:


are all irrational numbers
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January 28th, 2014, 03:07 AM   #2
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Re: irrational numbers

Quote:
Originally Posted by Albert.Teng
prove:


are all irrational numbers
Hello.

I will realize the cosine:


















(*)





1║) If "q" It is divisible only once by "2", then:







Absurdity:



2║)












The rest of the show, leave it to someone else.

Regards.
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January 28th, 2014, 05:50 AM   #3
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Re: irrational numbers

In general, it is provably true that and are both rational only for finitely many values of , i.e, one of 0, 1/2, or 1 (+ or -).
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February 12th, 2014, 04:26 PM   #4
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Re: irrational numbers

[quote=mente oscura]
Quote:
Originally Posted by "Albert.Teng":337yooi9
prove:


are all irrational numbers
Hello.

I will realize the cosine:



[/quote:337yooi9]

Hello.

I'm sorry. I had a mistake at the beginning.

















Suppose, :







,

Absurdity:




If we had started by:



















(*)





1║) If "q" It is divisible only once by "2", then:







Absurdity:



2║)












Regards.
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February 12th, 2014, 05:55 PM   #5
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Re: irrational numbers





The equation immediately above, by the rational root theorem, has no rational roots, hence sin(10) is irrational. A similar proof can be constructed for cos(10).
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