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(17)-a1704 [comments]; Cortese & García JGP(17)-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].

History and Curiosities
* History: Until the V century, the best value was Archimedes' approximation; 1665, Wallis formula; 1706, Name given by William Jones (and 100 digits calculated), from the initial of "perimetron" – later used by Euler; 1844, Johann Dase of Hamburg calculated (in his head!) 205 digits; 1882, Proved to the transcendental; 1897, The Indiana state House of Representatives passed a bill decreeing that π = 4.0 – but the bill didn't make it past the Senate; 1949, ENIAC calculated 2037 digits in 70 hours; 1999, over $$6 \times 10^9$$ digits calculated by computer, using Ramanujan's work; 2001, $$5 \times 10^{11}$$ digits known.
* Memorization: J Conway memorized many digits; 2000, a ND kid memorized more than 5000; But the records are 40,000 for Hideaki Tomoyori (1987, in 17 hr 21 min), and 42,194 for Hiroyuki Goto (1995, in 9 hr 21 min).
* Related facts: March 14th is celebrated as Pi Day.
@ References: news ns(10)mar [interesting facts]; news ns(12)mar [US ruling that the idea of making music based on π cannot be copyrighted]; news sn(18)mar [in support of using τ = 2π instead].