Pi: The Next Generation: A Sourcebook on the Recent History of Pi and Its Computation

509 Pages·2016·64.15 MB·English

by  David H. Bailey

David H. Bailey
Jonathan M. Borwein
Pi : The Next
Generation
A Sourcebook
on the Recent History of Pi
and Its Computation
Pi: The Next Generation

David H. Bailey Jonathan M. Borwein
•
.
Pi: The Next Generation

A Sourcebook on the Recent History
of Pi and Its Computation
David H. Bailey Jonathan M. Borwein
Lawrence Berkeley National Laboratory Centre for Computer Assisted Research
Berkeley, CA, USA Mathematics and its Applications,
School of Mathematical and Physical Sciences
University of Newcastle
Newcastle, Australia
ISBN 978-3-319-32375-6 ISBN 978-3-319-32377-0 (eBook)
DOI 10.1007/978-3-319-32377-0
Library of Congress Control Number: 2016942560

Mathematics Subject Classification (2010): 01-00, 01-08, 01A05, 01A75, 11-00, 11-03, 26-03, 65-03, 68-03

© Springer International Publishing Switzerland 2016
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To Our Grandchildren

Contents
Foreword ix
Preface xiii
1. Computation of (cid:25) using arithmetic-geometric mean (1976) 1
2. Fast multiple-precision evaluation of elementary functions (1976) 9
3. The arithmetic-geometric mean of Gauss (1984) 21
4. The arithmetic-geometric mean and fast computation of elementary
functions (1984) 79
5. A simpli(cid:12)ed version of the fast algorithms of Brent and Salamin (1985) 97
6. Is pi normal? (1985) 103
7. The computation of (cid:25) to 29,360,000 decimal digits using Borweins’
quartically convergent algorithm (1988) 109
8. Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, (cid:25), and
the Ladies Diary (1988) 125
9. Vectorization of multiple-precision arithmetic program and 201,326,000
decimal digits of pi calculation (1988) 151
10. Ramanujan and pi (1988) 165
11. Ramanujan, modular equations, and approximations to pi or how to
compute one billion digits of pi (1989) 175
12. Pi, Euler numbers, and asymptotic expansions (1989) 197
13. A spigot algorithm for the digits of (cid:25) (1995) 207
14. On the rapid computation of various polylogarithmic constants (1997) 219
15. Similarities in irrationality proofs for (cid:25), ln2, (cid:16)(2), and (cid:16)(3) (2001) 233
16. Unbounded spigot algorithms for the digits of pi (2006) 245
17. Mathematics by experiment: Plausible reasoning in the 21st Century
(2008) 259
18. Approximations to (cid:25) derived from integrals with nonnegative integrands
(2009) 293
19. Ramanujan’s series for 1/(cid:25): A survey (2009) 303
vii
viii CONTENTS
20. The computation of previously inaccessible digits of (cid:25)2 and Catalan’s
constant (2013) 327
21. Walking on real numbers (2013) 341
22. Birth, growth and computation of pi to ten trillion digits (2013) 363
23. Pi day is upon us again and we still do not know if pi is normal (2014) 425
24. The Life of (cid:25) (2014) 443
25. I prefer pi: A brief history and anthology of articles in the American
Mathematical Monthly (2015) 475
Index 501
Foreword
The fundamental constant π has played an indispensable role in mathematics, sci-
ence, and most world cultures for over 4000 years. One would like to calculate
the circumference of a circle without actually physically measuring it. The ancient
Babylonians and Hebrews recognized that it was considerably easier to determine
the radius of a circle than its circumference, and so the more physically demand-
ing task of calculating the circumference could be replaced by simply multiplying
the radius by 3. However, before 2000 BCE, the Babylonians replaced this ini-
tial approximation of 3 for π by 31. On the Egyptian Rhind Papyrus from about
8
1650 BCE, we find the slightly better approximation 4(8/9)2 = 3.16049.... We
have come a long way since the heydays of the Babylonian, Hebrew, and Egyptian
cultures. With the calculation of “houkouonchi” and Alexander J. Yee, we now
“know” 13.3 trillion digits of π. It might be pointed out that π was not always
denoted by π; William Jones first used the notation π in 1706. Euler adopted the
symbol by 1737, and since that time, the notation π has been universally used.
The value π arises in many contexts in mathematics in surprising ways. One
of these is the Buffon needle problem, which asks for the probability that a needle
of length ℓ will land on one of the vertical, equally spaced parallel lines on a floor,
where the distance between each two adjacent lines is equal to d. If d = ℓ, this
probability is 2/π. Any calculus student will testify that the value of π often arises
in the calculation of a definite integral or of an infinite series in closed form. The
greatIndianmathematician,SrinivasaRamanujan,evaluatedaplethoraofintegrals
andinfiniteseriesinclosedform,andhealsofoundmanybeautifulseriesidentities.
Chapter14inhissecondnotebookcontains87suchresults, andoftheseseriesand
integral evaluations in closed form and series identities, I counted 70 of them in
which π appears.
Unusual or surprising formulas for π are a source of delight for many of us.
When we calculate the simple continued fraction
1 1 1 1
π =3+ ,
7 + 15 + 1 + 293 +···
ofπ, weareperhapssurprisedbythe“large”denominator293. Thisindicatesthat
ifwetruncatethecontinuedfractionjustpriortothislargedenominator,weshould
obtain a good approximation to π. Indeed, the associated approximation 355 gives
113
thefirstsixdigitsofπ. Ontheotherhand,examiningthesimplecontinuedfraction
1 1 1 1 1
π4 =97+ ,
2 + 2 + 3 + 1 + 16539 +···
for π4, we find the “huge” denominator 16539. This leads to the approximations
9
π4 ≈97 =97.40909090909..., π ≈3.14159265262...,
22
while
π4 =97.409091034002..., π =3.14159265358....
ix