Selected Pi References

That the ratio of circumference to diameter is the same (and roughly equal to 3) for all circles has been accepted as "fact" for centuries (at least 4000 years, as far as I can determine), but knowing why this is true, as well as knowing the exact value of this ratio, is another story.

Any good book on the history of mathematics will supply you with dozens of interesting facts about the evolution of our understanding of ; a couple of these are listed below. In addition, literally hundreds of technical and semi-technical articles have been written about over the years; I've included a few of these, too.

A trip to any good library will easily yield more information than I could ever hope to catalogue here. (And please consider visiting a library before emailing me with your questions. I am neither a historian nor an expert on ; I'm just a well-informed fan!)


P. Beckmann, A History of Pi, 4th. ed., Golem Press, 1970.

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley-Interscience, 1987.

C. B. Boyer and U. Merzbach, A History of Mathematics, 2nd. ed., Wiley, 1991.

F. Cajori, A History of Mathematics, 3rd. ed., Chelsea, 1980.

H. Dörrie, 100 Great Problems of Elementary Mathematics: Their History and Solution, translated by D. Antin, Dover, 1965.

Expository Articles

H. C. Schepler, "The Chronology of Pi", Mathematics Magazine, Vol. 23, 1950, pp. 165 -- 170, 216 -- 228, 279 -- 283.

Stan Wagon, "Is Normal?", The Mathematical Intelligencer, Vol. 7, No. 3, 1985, pp. 65 -- 67.

J. W. Wrench, Jr., "The Evolution of Extended Decimal Approximations to ", The Mathematics Teacher, Vol. 53, 1960, pp. 644 -- 650.

Technical and Semi-Technical Articles

G. Almkvist and B. Berndt, "Gauss, Landen, Ramanujan, the Arithmetic-Geometric Mean, Ellipses, , and the Ladies Diary", The American Mathematical Monthly, Vol. 95, No. 1, 1988, pp. 585 -- 608.

E. F. Assmus, "Pi", The American Mathematical Monthly, Vol. 92, 1985, pp. 213 -- 214.

David H. Bailey, "The Computation of to 29,360,000 Decimal Digits Using Borweins' Quartically Convergent Algorithm", Mathematics of Computation, Vol. 50, No. 181, 1988, pp. 283 -- 296.

J. M. Borwein and P. B. Borwein, "The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions", SIAM Review, Vol. 26, No. 3, 1984, pp. 351 -- 366.

J. M. Borwein, P. B. Borwein, and D. H. Bailey, "Ramanujan, Modular Equations, and Approximations to or How to Compute One Billion Digits of ", The American Mathematical Monthly, Vol. 96, No. 3, 1989, pp. 201 -- 219.

Dario Castellanos, "The Ubiquitous ", Mathematics Magazine, Vol. 61, 1988, pp. 67 -- 98, 148 -- 163.

G. Miel, "Of Calculations Past and Present: The Archimedean Algorithm", The American Mathematical Monthly, Vol. 90, No. 1, 1983, pp. 17 -- 35.

S. Rabinowitz and S. Wagon, "A spigot algorithm for the digits of ", The American Mathematical Monthly, Vol. 102, 1995, pp. 195 -- 203.

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Neal Carothers -