I just got bored

pi_calculation.sm (13,47 KiB) scaricato 1308 volta(e).

Wrote

I just got bored

No need to get bored. Simon Plouffe [UQAM] is infinitely accurate wrt 'n'.
pi(29) Simon = normalized 40 decimals π in computing machinery.

Wrote

Pi has defeated the greatest Mathematicians including Legendre.
The final proof is from the French Geometre Schwab [1813].
Method known "Isoperimetres". However, it was known to may
true decimals from the Mesopotamians from continued fraction.
As well, the first IBM Mainframe pi was from continued fractions.

Jean

Maths Pi [Simon Plouffe, Fibonacci].sm (27,15 KiB) scaricato 1098 volta(e).

John Wallis' 1655 infinite product for pi. Imagine trying to calculate the convergence of this beast in 1655!

infinite_product_for_pi_by_wallis_1655.sm (16,94 KiB) scaricato 1231 volta(e).

Wrote

John Wallis' 1655 infinite product for pi. Imagine trying to calculate the convergence of this beast in 1655!

John Wallis was in fact Newton's teacher. We ow to John the special algorithm
to calculate continued fractions top down instead of the hand computed down-up.
We have that in Mathcad 11, but couldn't make it work Smath.
Fourier had a trick to calculate faster . I think the rats had breakfast with
that paper. In 1982, a Cray X-100 toke 32 hours 15 min to calculate the Mrsenne
primes numbers.
However, pi is only infinite wrt a circle of infinite diameter.
What is the largest known circle of manageable utility ? Hello Watson !

Welcome to Smath Community.