Buffon's constant

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Buffon's constant is $\scriptstyle \frac{2}{\pi} \,$ . Its decimal expansion is 0.63661977236758... (see A060294) and its continued fraction is

${\cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{3 + \cfrac{1}{31 + \cfrac{1}{1 + \ddots}}}}}}}\,$

(see A053300).

Like many other numbers involving $π$ , Buffon's constant can be expressed as an infinite sum or product:

$\frac{2}{\pi} = \prod_{n = 2}^{\infty} \frac{p_n + 2 - (p_n \mod 4)}{p_n} = \frac{2}{3} \times \frac{6}{5} \times \frac{6}{7} \times \frac{10}{11} \times \ldots$

(where $p n$ is the $n$ th prime number)^[1];

$\frac{2}{\pi} = 1 + \sum_{n = 1}^{\infty} (-1)^{n}(4n + 1) \left(\frac{\prod_{i = 1}^n 2i - 1}{\prod_{j = 1}^n 2 j}\right)^3 = 1 - 5\left(\frac{1}{2}\right)^3 + 9\left(\frac{3}{8}\right)^3 - 13\left(\frac{5}{16}\right)^3 + \ldots$

It is also the limit $\lim_{n \to \infty} \frac{z(n)}{\log n}$ , where $z (n)$ is the expected number of real zeros of a random polynomial of degree $n$ with real coefficients chosen from a standard Gaussian distribution.^[2]

Theorem. The probability $\scriptstyle P(l,\, d) \,$ that a needle of length $\scriptstyle l \,$ will randomly land on a line, given a floor with equally spaced parallel lines at a distance $\scriptstyle d \,\geq\, l \,$ apart, assuming that the angle and the position of the fallen needle are independently and uniformly random, is $\scriptstyle P(l,\, d) \,=\, \frac{2}{\pi} \cdot \frac{l}{d} \,$ .

Proof. If the needle always fell perpendicular (angle $\scriptstyle \theta \,=\, \frac{\pi}{2} \,$ radians) to the parallel lines, we would have $\scriptstyle P_{\perp}(l,\, d) \,=\, \frac{l}{d} \,$ . So we have

$P(l, d) \,=\, \left( \int_{0}^{\pi} \sin \theta \cdot \frac{d\theta}{\pi} \right) \cdot P_{\perp}(l, d) = \left( \frac{- [\cos \theta]_{0}^{\pi}}{\pi} \right) \cdot \frac{l}{d} = \frac{2}{\pi} \cdot \frac{l}{d} \,$

as specified by the theorem. □

↑ David Blatner, The Joy of Pi. New York: Walker & Company (1997): 119, circle by upper right corner.
↑ S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 141

[0] David Blatner, The Joy of Pi. New York: Walker & Company (1997): 119, circle by upper right corner.

[1] S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 141

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Buffon's constant

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