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Originally posted by MaskedAvatar
I also liked the time you'd been smoking and you started hitting on the other women of ATS!
Good luck, it probably takes a 400-page treatise of your proofs sent by you to yourself in your handwriting by sealed registered mail and the recommendation of 2arctan(142857) to the power of 3 x academics in their various Secret Societies...
Originally posted by BlackJackal
I was looking at your findings while close to pi it is definetly off as shown by the numbers posted by amantine. A question for amantine did you use mathmatica to obtain your results?
Originally posted by Nans DESMICHELS
I'm trying to understand why when I do :
2*atan(142857142857142857)
I'm close to PI
and when I do 2*atan(142857142857142857142857)
I'm still closer to PI.
Originally posted by Nans DESMICHELS
4atan(1)-2atan(142857142857)
0.000000000014000000000014
One proof of irrationality of pi:
Look at the following integral
That iintegral has as values
I(2) = -2п^2 + 24
I(3) = -24п^2 + 240
I(4) = 2п^4 - 360п^2 + 3360
I(5) = 60п^4 - 6720п^2 + 60480
I(6) = -2п^6 + 1680п^4 - 151200п^2 + 1330560
All sums of powers of п. We will determine a maximum for the integral:
x(п-x) ≤ 0.5п(п-0.5п )
0.5п(п-0.5п ) = 0.25п^2
0.25п^2 ≤ 3
x(п-x) ≤ 3
Now the other part:
0 ≤ sin(x) ≤ 1
sin(x) ≤ 1
Filling this in and calculating the integral gives you an upperlimit for I(n) of п*3^n/n!
Now let's assume п is rational (and repeats): п=p/q, where p and q are two whole numbers and the expression has been simplified as much as possible.
That means for example with I(4):
(2p^4 - 360p^2q^2 + 3360q^4)/q^4
Because this is an positive numbers that has been simplified as much as possible we can say:
1/q^4 ≤ (2p^4 - 360p^2q^2 + 3360q^4)/q^4
This is true for every I(n), so you can say:
1/q^n ≤ I(n)
We determined a upper limit for I(n), so 1/q^n is always smaller than that:
1/q^n ≤ п*3^n/n!
1 ≤ п*(3q)^n/n!
But lim(x->∞ ) x^n/n! goes to 0. That would mean:
lim(x->∞ ) 1 ≤ п*(3q)^n/n!
gives
1 ≤ 0
This is ofcourse not true. That problem is our assumption that п is rational. Q.E.D.