Pi, π

In General > s.a. Euler's Equation; series.
* Value: The transcendental number

π = 3.1415 92653 58979 32384 62643 38327 95028 84197 16939 93751 05820 ....

* Approximations: Archimedes' approximation, 3 + 10/71 < π < 3 + 1/7; In 1655 the English mathematician John Wallis published a book in which he derived a formula for pi as the product of an infinite series of ratios,

π/2 = (2 · 2 / 1 · 3) (4 · 4 / 3 · 5) (6 · 6 / 5 · 7) · · · ,

where the terms are of the form (2j)(2j) / (2j–1)(2j+1) with j from 1 to infinity; Also, π equals 22/7 to within a 0.04% error!
* Properties: Related to the quadrature of the circle; Its digits pass tests for randomness; Thought to be a normal number; 2001, D Bailey & R Crandall showed connection to chaos theory.
* Integral representations: It appears in

$\def\dd{{\rm d}} \pi = \int_{-1}^1 {\dd x\over\sqrt{1-x^2}}\qquad {\rm or}\qquad \pi = \int_{-\infty}^{+\infty} {\dd x\over x^2+1}\;, \qquad{\rm and}\qquad \int_{-\infty}^{+\infty} \dd x\,{\rm e}^{-x^2} = \sqrt{\vphantom{l}\pi}\;.$

@ General references: issue SA(88)feb; Blatner 99; Posamentier & Lehmann 04; Adrian 06 [I].
@ Statistical estimation: Bloch & Dressler AJP(99)apr; > s.a. statistical geometry [Buffon's needle].
@ Wallis' formula: Friedmann & Hagen JMP(15) + news PhysOrg(15)nov, pt(15)nov [and quantum physics]; Chashchina & Silagadze PLA-a1704 [comments]; Cortese & García a1709 [from the harmonic oscillator].
@ Related topics: Jáuregui & Tsallis JMP(10)-a1004, Amdeberhan et al JMP(12) [representation in terms of q-exponential function, and generalizations].