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 (2*j*)(2*j*) / (2*j*–1)(2*j*+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].

> __Approximations__:
see Wikipedia page.

**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 × 10^{9}
digits calculated by computer, using Ramanujan's work; 2001, 5 × 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].

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send feedback and suggestions to bombelli at olemiss.edu – modified 15 mar 2018