# Pai

Duara yenye kipenyo cha 1 ina mzingo mwenye urefu wa π
Uwiano wa urefu wa mzingo na ule wa kipenyo ni 3 na kitu. Uwiano kamili unaitwa π, pai.
**Pai** (jina la herufi ya Kigiriki **π**) ni namba ya duara kwa maana ya uwiano wa urefu wa mzingo na ule wa kipenyo.

Jinsi ilivyo kawaida kwa herufi mbalimbali za Kigiriki, pai pia inatumika kama kifupisho kwa ajili ya maarifa na dhana za hesabu na fisikia.

Imejulikana hasa kama namba ya duara. Ikiandikwa inanaza 3.141592653589793238462643... lakini haiwezi kuandikwa kamili kwa kuongeza tarakimu baada ya nukta maana hakuna mwisho. Namba za aina hii zisizo sehemu ya namba nyingine au ambazo haziwezi kuonyeshwa kuwa wianisho safi baina namba kamili huitwa namba zisizowiana.

Chamkano cha 22/7 ni karibu zaidi na pai na 355/113 ni karibu zaidi tena.

Wanahisabati duniani husheherekea sikukuu ya pai tarehe 14 Machi au pia 22 Julai.

## Marejeo

* Arndt, Jörg; Haenel, Christoph (2006). *Pi Unleashed*. Springer-Verlag. ISBN 978-3-540-66572-4. Iliwekwa mnamo 2013-06-05. `{{cite book}}`: Invalid `|ref=harv` (help) English translation by Catriona and David Lischka.
* Ayers, Frank (1964). *Calculus*. McGraw-Hill. ISBN 978-0-070-02653-7. `{{cite book}}`: Invalid `|ref=harv` (help)
* Berggren, Lennart; Borwein, Jonathan; Borwein, Peter (1997). *Pi: a Source Book*. Springer-Verlag. ISBN 978-0-387-20571-7. `{{cite book}}`: Invalid `|ref=harv` (help)
* Beckmann, Peter (1989) [1974]. *History of Pi*. St. Martin's Press. ISBN 978-0-88029-418-8. `{{cite book}}`: Invalid `|ref=harv` (help)
* Borwein, Jonathan; Borwein, Peter (1987). *Pi and the AGM: a Study in Analytic Number Theory and Computational Complexity*. Wiley. ISBN 978-0-471-31515-5. `{{cite book}}`: Invalid `|ref=harv` (help)
* Boyer, Carl B.; Merzbach, Uta C. (1991). *A History of Mathematics* (tol. la 2). Wiley. ISBN 978-0-471-54397-8. `{{cite book}}`: Invalid `|ref=harv` (help)
* Bronshteĭn, Ilia; Semendiaev, K. A. (1971). *A Guide Book to Mathematics*. H. Deutsch. ISBN 978-3-871-44095-3. `{{cite book}}`: Invalid `|ref=harv` (help)
* Eymard, Pierre; Lafon, Jean Pierre (1999). *The Number Pi*. American Mathematical Society. ISBN 978-0-8218-3246-2. `{{cite book}}`: Invalid `|ref=harv` (help), English translation by Stephen Wilson.
* Joseph, George Gheverghese (1991). *The Crest of the Peacock: Non-European Roots of Mathematics*. Princeton University Press. ISBN 978-0-691-13526-7. Iliwekwa mnamo 2013-06-05. `{{cite book}}`: Invalid `|ref=harv` (help)
* Posamentier, Alfred S.; Lehmann, Ingmar (2004). *Pi: A Biography of the World's Most Mysterious Number*. Prometheus Books. ISBN 978-1-59102-200-8. `{{cite book}}`: Invalid `|ref=harv` (help)
* Reitwiesner, George (1950). "An ENIAC Determination of pi and e to 2000 Decimal Places". *Mathematical Tables and Other Aids to Computation*. **4** (29): 11–15. doi:10.2307/2002695. `{{cite journal}}`: Invalid `|ref=harv` (help)
* Roy, Ranjan (1990). "The Discovery of the Series Formula for pi by Leibniz, Gregory, and Nilakantha". *Mathematics Magazine*. **63** (5): 291–306. doi:10.2307/2690896. `{{cite journal}}`: Invalid `|ref=harv` (help)
* Schepler, H. C. (1950). "The Chronology of Pi". *Mathematics Magazine*. **23** (3). Mathematical Association of America: 165–170 (Jan/Feb), 216–228 (Mar/Apr), and 279–283 (May/Jun). doi:10.2307/3029284. `{{cite journal}}`: Invalid `|ref=harv` (help). issue 3 Jan/Feb, issue 4 Mar/Apr, issue 5 May/Jun
* Blatner, David (1999). *The Joy of Pi*. Walker & Company. ISBN 978-0-8027-7562-7.
* Borwein, Jonathan and Borwein, Peter, "The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions", *SIAM Review*, **26**(1984) 351–365
* Borwein, Jonathan, Borwein, Peter, and Bailey, David H., *Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi",* The American Mathematical Monthly*, **96**(1989) 201–219*
* Chudnovsky, David V. and Chudnovsky, Gregory V., "Approximations and Complex Multiplication According to Ramanujan", in *Ramanujan Revisited* (G.E. Andrews et al. Eds), Academic Press, 1988, pp 375–396, 468–472
* Cox, David A., "The Arithmetic-Geometric Mean of Gauss", *L' Ensignement Mathematique*, **30**(1984) 275–330
* Delahaye, Jean-Paul, "Le Fascinant Nombre Pi", Paris: Bibliothèque Pour la Science (1997) ISBN 2-902918-25-9
* Engels, Hermann, "Quadrature of the Circle in Ancient Egypt", *Historia Mathematica* **4**(1977) 137–140
* Euler, Leonhard, "On the Use of the Discovered Fractions to Sum Infinite Series", in *Introduction to Analysis of the Infinite. Book I*, translated from the Latin by J. D. Blanton, Springer-Verlag, 1964, pp 137–153
* Heath, T. L., *The Works of Archimedes*, Cambridge, 1897; reprinted in *The Works of Archimedes with The Method of Archimedes*, Dover, 1953, pp 91–98
* Huygens, Christiaan, "De Circuli Magnitudine Inventa", *Christiani Hugenii Opera Varia I*, Leiden 1724, pp 384–388
* Lay-Yong, Lam and Tian-Se, Ang, "Circle Measurements in Ancient China", *Historia Mathematica* **13**(1986) 325–340
* Lindemann, Ferdinand, "Ueber die Zahl pi" Ilihifadhiwa 22 Januari 2015 kwenye Wayback Machine., *Mathematische Annalen* **20**(1882) 213–225
* Matar, K. Mukunda, and Rajagonal, C., "On the Hindu Quadrature of the Circle" (Appendix by K. Balagangadharan). *Journal of the Bombay Branch of the Royal Asiatic Society* **20**(1944) 77–82
* Niven, Ivan, "A Simple Proof that pi Is Irrational", *Bulletin of the American Mathematical Society*, **53**:7 (July 1947), 507
* Ramanujan, Srinivasa, "Modular Equations and Approximations to π", *Quarterly Journal of Pure and Applied Mathematics*, **XLV**, 1914, 350–372. Reprinted in G.H. Hardy, P.V. Seshu Aiyar, and B. M. Wilson (eds), *Srinivasa Ramanujan: Collected Papers*, 1927 (reprinted 2000), pp 23–29
* Shanks, William, *Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals*, 1853, pp. i–xvi, 10
* Shanks, Daniel and Wrench, John William, "Calculation of pi to 100,000 Decimals", *Mathematics of Computation* **16**(1962) 76–99
* Tropfke, Johannes, *Geschichte Der Elementar-Mathematik in Systematischer Darstellung* (*The history of elementary mathematics*), BiblioBazaar, 2009 (reprint), ISBN 978-1-113-08573-3
* Viete, Francois, *Variorum de Rebus Mathematicis Reponsorum Liber VII. F. Viete, Opera Mathematica* (reprint), Georg Olms Verlag, 1970, pp 398–401, 436–446
* Wagon, Stan, "Is Pi Normal?", *The Mathematical Intelligencer*, **7**:3(1985) 65–67
* Wallis, John, *Arithmetica Infinitorum, sive Nova Methodus Inquirendi in Curvilineorum Quadratum, aliaque difficiliora Matheseos Problemata*, Oxford 1655–6. Reprinted in vol. 1 (pp 357–478) of *Opera Mathematica*, Oxford 1693
* Zebrowski, Ernest, *A History of the Circle: Mathematical Reasoning and the Physical Universe*, Rutgers Univ Press, 1999, ISBN 978-0-8135-2898-4

## Viungo vya nje

Wikimedia Commons ina media kuhusu:
 Pai Digits of Pi katika Open Directory Project
* "pai" at Wolfram Mathworld
* Representations of pai at Wolfram Alpha
* pai Search Engine 2 billion searchable digits of Kigezo:Pi, WIKINIUM\_MATH\_PH\_0 , and WIKINIUM\_MATH\_PH\_1
* Eaves, Laurence (2009). "Pi". *Sixty Symbols*. Brady Haran for the University of Nottingham. Ilihifadhiwa kwenye nyaraka kutoka chanzo mnamo 2020-07-06. Iliwekwa mnamo 2014-06-17.
* Grime, Dr. James (2014). "Pi is Beautiful – Numberphile". *Numberphile*. Brady Haran.

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|  | Makala hii kuhusu mambo ya hisabati bado ni **mbegu**. Je, unajua kitu kuhusu **Pai** kama historia yake au mahusiano yake na mada nyingine? Labda unaona habari katika Wikipedia ya Kiingereza au lugha nyingine zinazofaa kutafsiriwa? Basi unaweza kuisaidia Wikipedia kwa kuihariri na kuongeza habari*.* |