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Calculating PI |
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Introduction |
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The first attempts to calculate PI dated back to Greeks
and exploit the following inequalities:  where pn and Pn are perimeters of inscribed and circumscribed n side polygons respectively, while C is the circumference of a circle of radius r. It lead to knowledge of PI with about 35 digits. Then PI calculations adopted techniques independent from circle, with the introduction of calculus and analytical geometry (17th century) first, and of series (late 17th century) then. Research on further PI digits went on reaching > 100 digits. In 1761, the mathematician Lambert (1728-1777) proved that PI is an irrational number. In 1882, Lindenman proved that PI is a transcendental number With the introduction of computers PI could be calculated up to million of digits and another result was achieved: up to now, digits composing PI appear randomly distributed. |
 Johann Heinrich Lambert (1728-1777)  Ferdinand von Lindemann (1852-1939)  John Machin (1680-1751)
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Machin formula |
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method uses a series expansion as in 17th century mathematicians
attempts but exploits the obvious advantages of having a computer. Let's start writing the series expansion of arctg: 
but since  we have: 
The biggest defect of this series expansion is that it converges very slowly, to get 3 digits one must calculate up to k > 2000. To speed up calculations you can notice that the arctg series expansion converges rapidly if |x| < 1. Thus we arrive to the Machin formula:  which is expanded into series as follows:  to speed up calculation is better to use powers of form k+1 which are directly calculated from previous iterations. This can be achieved by multiplication of the two elements being subtracted by 5 / 5 and 239 / 239. This leads to  |
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When to stop |
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| How far
should we go calculating the series to reach n figures in PI
calculations? The idea is that if we want PI with a precision of n figures, then we must ensure that the kth element of the sum must not alter the PI digit n places to the right of the comma. Thus, since the kth element of the sum is: 
it is straightforward to understand that the most significant addendum is the first. Thus we can write our condition as 
and since the left term is always less than 100 we arrive at the following  (logarithm is base 10). |