Pi
One of the greatest mathematicians in history, a singular genius by the name of Srinivasa Ramanujan discovered an algorithm for computing the value of π. His algorithm, an infinite series which adds eight decimal places for each increment of k can be stated as follows:
$$\frac{1}{\text{\pi}}\text{}=\text{}\frac{2\sqrt{2}}{9801}\text{}\sum _{\text{k}=0}^{\infty}\text{}\frac{\text{(4k)!(1103}+\text{26390k)}}{\text{(k!}{\text{)}}^{4}\text{39}{6}^{\text{4k}}}$$This incredible algorithm, which was arrived at through pure insight was, for several decades, the fastest way at computing for the digits of π. It wasn’t until the late ’80’s that the brothers David and Gregory Chudnovsky made as significant a headway into computing for the value of π by building upon Ramanujan’s original algorithm. The Chudnovskys were able to compute the value of π to over a billion places (10^{10}) with this algorithm:
$$\frac{1}{\text{\pi}}\text{}=\text{12}\sum _{\text{k}=0}^{\infty}\text{}\frac{\text{(}-1{\text{)}}^{\text{k}}\text{(6k)! (13591409}+\text{545140134k)}}{\text{(3k)!(k!}{\text{)}}^{3}\text{64032}{0}^{\text{3k}+\text{3}/2}}$$If you’ll compare the two, the Chudnovsky series’ origin is readily apparent. They both begin the series with a computationally quick factorial (k!). The hallmark of the Chudnovskys’ method is the use of a very large coefficient (545140134) and the exponent on the last computational portion of the algorithm (3k+3/2).
- Tags: mathematics, pi, Srinivasa Ramanujan
Comments (5)
igniteous
Ramanujan is the shit!
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Solaris
What's with all the math posts? Bored?
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Balerion
The Chudnovskys should've won a Fields medal for their work. Unfortunately, Fields awards are only for mathematicians below 30 years of age.
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Raymond
A little bit. I really like math and I wanted to share this little tidbit of information.
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Raymond
I agree. So many others should’ve won for their work, if not only for the age at which they finally realized their great contributions to mathematics. :(
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