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The American Mathematical Monthly Volume 96, 1989 - Issue 3
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Original Articles

Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi

J. M. BorweinMathematics Department, Dalhousie University, Halifax, N.S. B3H 3J5Canada
,
P. B. BorweinMathematics Department, Dalhousie University, Halifax, N.S. B3H 3J5Canada
&
D. H. BaileyNASA Ames Research Center. Moffett Field, CA94035
Pages 201-219 | Published online: 02 Feb 2018

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David H. Bailey. (2021) Jonathan Borwein: Renaissance Mathematician. The American Mathematical Monthly 128:9, pages 773-779.
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David H. Bailey. (2017) Jonathan Borwein: Experimental Mathematician. Experimental Mathematics 26:2, pages 125-129.
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Scott T. Chapman. (2016) A Letter From the Editor: Jonathan M. Borwein (1951–2016). The American Mathematical Monthly 123:9, pages 847-848.
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Jesús Guillera. (2016) New proofs of Borwein-type algorithms for Pi. Integral Transforms and Special Functions 27:10, pages 775-782.
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David H. Bailey & Jonathan Borwein. (2014) Pi Day Is Upon Us Again and We Still Do Not Know if Pi Is Normal. The American Mathematical Monthly 121:3, pages 191-206.
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Nayandeep Deka Baruah, Bruce C. Berndt & Heng Huat Chan. (2009) Ramanujan's Series for 1/π: A Survey. The American Mathematical Monthly 116:7, pages 567-587.
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Ira Rosenholtz. (1999) Tangent Sequences, World Records, π, and the Meaning of Life: Some Applications of Number Theory to Calculus. Mathematics Magazine 72:5, pages 367-376.
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Jeffrey Nunemacher. (1992) On Computing Euler's Constant. Mathematics Magazine 65:5, pages 313-322.
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Harry A. Mavromatis. (1992) The wallis formula for PI, including two new expressions involving irrationals. International Journal of Computer Mathematics 43:3-4, pages 197-203.
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Harry A. Mavromatis. (1991) Some comprehensive expressions for π and 1/π . International Journal of Computer Mathematics 38:1-2, pages 91-100.
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Anchalee Sripattanet & Atid Kangtunyakarn. (2023) Approximation of G-variational inequality problems and fixed-point problems of G-κ-strictly pseudocontractive mappings by an intermixed method endowed with a graph. Journal of Inequalities and Applications 2023:1.
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Christian Elsholtz, Clemens Heuberger & Daniel Krenn. (2023) Algorithmic counting of nonequivalent compact Huffman codes. Applicable Algebra in Engineering, Communication and Computing.
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Martin Knapmeyer & Michaela Walterová. 2022. Geophysical Exploration of the Solar System. Geophysical Exploration of the Solar System 65 178 .
Jong Min Kim, Ahrae JoKyung Ah Lee, Hyeuk Jin Han, Ye Ji Kim, Ho Young KimGyu Rac LeeMinjoon Kim, Yemin ParkYun Sik KangJuhae JungKeun Hwa Chae, Eoyoon Lee, Hyung Chul Ham, Hyunchul JuYeon Sik Jung & Jin Young Kim. (2021) Conformation-modulated three-dimensional electrocatalysts for high-performance fuel cell electrodes. Science Advances 7:30.
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Kartik Sondhi, Sai Guruva Reddy Avuthu, Jörg Richstein, Z Hugh Fan & Toshikazu Nishida. (2021) Fabrication and non-destructive characterization of through-plastic-via (TPV) in flexible hybrid electronics. Flexible and Printed Electronics 6:2, pages 025001.
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Naina Gupta, Arpan Jati & Anupam Chattopadhyay. (2021) MemEnc: A Lightweight, Low-Power, and Transparent Memory Encryption Engine for IoT. IEEE Internet of Things Journal 8:9, pages 7182-7191.
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Wenchang Chu. (2020) Infinite series identities derived from the very well-poised $$\Omega $$-sum. The Ramanujan Journal 55:1, pages 239-270.
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Gal Raayoni, Shahar Gottlieb, Yahel Manor, George Pisha, Yoav Harris, Uri Mendlovic, Doron Haviv, Yaron Hadad & Ido Kaminer. (2021) Generating conjectures on fundamental constants with the Ramanujan Machine. Nature 590:7844, pages 67-73.
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Wongvisarut Khuangsatung & Atid Kangtunyakarn. (2020) The Method for Solving Fixed Point Problem of G-Nonexpansive Mapping in Hilbert Spaces Endowed with Graphs and Numerical Example. Indian Journal of Pure and Applied Mathematics 51:1, pages 155-170.
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Richard P. Brent. 2020. From Analysis to Visualization. From Analysis to Visualization 323 347 .
Armengol Gasull. 2019. Difference Equations, Discrete Dynamical Systems and Applications. Difference Equations, Discrete Dynamical Systems and Applications 129 167 .
Jesús Guillera. (2017) Self-replication and Borwein-like algorithms. The Ramanujan Journal 47:2, pages 447-455.
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Vito Lampret. (2017) Wallis' sequence estimated accurately using an alternating series. Journal of Number Theory 172, pages 256-269.
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Dongxi Ye. (2016) Level 16 analogue of Ramanujan's theories of elliptic functions to alternative bases. Journal of Number Theory 164, pages 191-207.
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Jonathan M. Borwein. 2014. From Alexandria, Through Baghdad. From Alexandria, Through Baghdad 531 561 .
N.D. Bagis & M.L. Glasser. (2013) Ramanujan type approximation formulas . Journal of Number Theory 133:10, pages 3453-3469.
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HENG HUAT CHAN & SHAUN COOPER. (2012) Rational analogues of Ramanujan's series for 1/π. Mathematical Proceedings of the Cambridge Philosophical Society 153:2, pages 361-383.
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Nayandeep Deka Baruah & Bruce C. Berndt. (2009) Eisenstein series and Ramanujan-type series for 1/π. The Ramanujan Journal 23:1-3, pages 17-44.
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Nayandeep Deka Baruah & Bruce C. Berndt. (2009) Ramanujan's Eisenstein series and new hypergeometric-like series for . Journal of Approximation Theory 160:1-2, pages 135-153.
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S.K. Sen, Ravi P. Agarwal & Raffaella Pavani. (2009) Best -digit rational bounds for irrational numbers: Pre- and super-computer era . Mathematical and Computer Modelling 49:7-8, pages 1465-1482.
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Michael TrottMichael Trott. 2006. The Mathematica GuideBook for Numerics. The Mathematica GuideBook for Numerics 1 967 .
K. Karamanos & I.S. Kotsireas. (2005) Statistical analysis of the first digits of the binary expansion of Feigenbaum constants and . Journal of the Franklin Institute 342:3, pages 329-340.
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David H. bailey, Simon M. plouffe, Peter B. borwein & Jonathan M. borwein. (1997) The quest for PI. The Mathematical Intelligencer 19:1, pages 50-56.
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Nick Lord. (2016) Recent calculations of π : the Gauss-Salamin algorithm. The Mathematical Gazette 76:476, pages 231-242.
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