Talk:Proof that π is irrational

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[edit] A nice proof

Thanks to Michael Hardy for writing a very nice article. I did notice one typo, which I've fixed:

(F'\cdot\sin + F\cdot\cos)' = F''\cdot\sin + F'\cdot\cos + F'\cdot\cos - F\cdot\sin = (F'' - F)\cdot\sin + 2F'\cdot\cos \neq f\cdot\sin

but (corrected version)

(F'\cdot\sin - F\cdot\cos)' = F''\cdot\sin + F'\cdot\cos - F'\cdot\cos + F\cdot\sin = (F'' + F)\cdot\sin = f\cdot\sin.

Notice that this also makes the integral work out as a positive integer, since cos π = −1.

I also think the argument about f (j)(0) and f (j)(π) should be phrased in terms of f (2j)(0) and f (2j)(π), in light of the way the function F is defined, but I thought I'd bring that one up on the talk page before I stick it in the article. DavidCBryant 12:16, 17 August 2007 (UTC)

OK, I see that Michael has been here and gone, without doing much to the article. So I'm putting the change about the (2j)th derivative of f into the article, plus a bit more explanatory language about why F(0) and F(π) are always positive integers – actually, F(0) = F(π), if you think about it. DavidCBryant 15:47, 17 August 2007 (UTC)
The change about the (2j)th derivative of f was wrong. By using Binomial theorem you can see that f is a polynomial like a2nx2n + a2n − 1x2n − 1 + ... + anxn. So for 0 ≤ j < n f (j)(π)=0; but it is not correct that for 0 ≤ j < n f (2j)(π)=0 (example: Let n be 2. Then f(x)={x^2(a-bx)^2 \over 2}={a^2x^2-2abx^3+b^2x^4\over 2} and f''(0) = a2 (in this example j=1). So I corrected this and the following sentence. Furthermore, I added why f (j)(0) is an integer (by using Binomial theorem). You will see that the proof is correct now. If you see anything that can be simplified, then let me know. Egndgf 15:28, 23 August 2007 (UTC)

[edit] A little help here

Sorry if this is wrong place to ask, but nobody could tell me. Why you can't just say that if pi is rational, then a circle has a finite number of sides? 24.218.46.235 17:59, 4 November 2007 (UTC)

Why could you say that? A rational number is an integer over another integer, e.g. 355/113.a It \scriptstyle\pi were exactly 355/113, how would that imply that a circle has a finite number of sides? Michael Hardy 00:13, 5 November 2007 (UTC)