Eikonal Blog

2016.08.24

Approximate numerics

Filed under: Uncategorized — sandokan65 @ 21:22

~1700BCE Babylon: \sqrt{2} \approx 1 + \frac{24}{60} + \frac{51}{60^2} + \frac{10}{60^3} \approx 1.414,213

pi

Filed under: Uncategorized — sandokan65 @ 21:19

Seen somewhere recently:

  • \frac\pi4 = \frac11 - \frac13 + \frac15 - \frac17 + \cdots
  • \frac{\pi^2}6 = \frac1{1^1} + \frac1{2^2} + \frac1{3^2} + \cdots
  • \frac2\pi = \left(1-\frac13\right)\left(1+\frac15\right)\left(1-\frac17\right)\left(1-\frac1{11}\right)\left(1+\frac1{13}\right)\left(1+\frac1{17}\right)\cdots where fractional terms \frac1p are over all primes, and the sign in front of them is minus if p=4n-1 and plus otherwise.

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