I always thought the formula for Pi was simple, 22/7 but apparently that is not the case. There are multiple mathematicians who have spent a significant time coming up with a formula for calculating Pi precisely. For example, Madhava an Indian scholar, who lived from 1350 to 1425, found that pi equals 4 multiplied by a series that begins with 1 and then alternately subtracts or adds fractions in which 1 is placed over successively higher odd numbers (so 1/3, 1/5, and so on). One way to express this would be:
A formula presents how pi can be calculated using a series developed by the Indian scholar Madhava.
While the formula is quite simple to implement and calculate it takes a long time to get accurate results. There are other formulas as well to calculate Pi. The latest one was found when physicists Arnab Priya Saha and Aninda Sinha of the Indian Institute of Science were exploring the String Theory and instead found a completely new formula for calculating Pi. They published their findings in their Paper (Field Theory Expansions of String Theory Amplitudes)
Saha and Sinha discovered the following formula which shows that Madhava’s formula is only a special case of a much more general equation for calculating pi.
A formula presents a way of calculating pi that was identified by physicists Arnab Priya Saha and Aninda Sinha.
I tried understanding the math behind the formula but it didn’t really make much sense to me so I am just going to quote the explanation given by Scientific American here instead of trying to explain it myself. 🙂
This formula produces an infinitely long sum. What is striking is that it depends on the factor λ , a freely selectable parameter. No matter what value λ has, the formula will always result in pi. And because there are infinitely many numbers that can correspond to λ, Saha and Sinha have found an infinite number of pi formulas.
If λ is infinitely large, the equation corresponds to Madhava’s formula. That is, because λ only ever appears in the denominator of fractions, the corresponding fractions for λ = ∞ become zero (because fractions with large denominators are very small). For λ = ∞, the equation of Saha and Sinha therefore takes the following form:
Saha and Sinha’s formula can be adapted based on the assumption of an infinitely large parameter.The first part of the equation is already similar to Madhava’s formula: you sum fractions with odd denominators. The last part of the sum (–n)n – 1, however, is less familiar. The subscript n – 1 is the so-called Pochhammer symbol. In general, the expression (a)n corresponds to the product a x(a + 1) x (a + 2) x … x (a + n – 1). For example, (5)3 = 5 x 6 x 7 = 210. And the Pochhammer symbol in the above formula therefore results in: (–n)n – 1 = (–n) x (–n + 1) x (–n + 2) x … x (–n + n – 3) x (–n + n – 2).
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As the two string theorists report, however, pi can be calculated much faster for smaller values of λ. While Madhava’s result requires 100 terms to get within 0.01 of pi, Saha and Sinha’s formula for λ = 3 only requires the first four summands. “While [Madhava’s] series takes 5 billion terms to converge to 10 decimal places, the new representation with λ between 10 [and] 100 takes 30 terms,” the authors write in their paper.
Source: Hacker News: String Theorists Accidentally Find a New Formula for Pi
– Suramya
