# Jonathan Borwein's books

We list below 20 books written or edited by Jonathan Borwein. We have included the books which give Borwein as an editor because in the cases we give these contain much material written by Borwein. We give extracts from different sources: the publisher's description, the preface to the book, and reviews of the book.

We list below 20 books written or edited by Jonathan Borwein. We have included the books which give Borwein as an editor because in the cases we give these contain much material written by Borwein. We give extracts from different sources: the publisher's description, the preface to the book, and reviews of the book.

Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (1987) with Peter B Borwein

Collins Dictionary of Mathematics (1989) with Peter B Borwein

A Dictionary of Real Numbers (1990) with Peter B Borwein

Pi: A Source Book (1997) with Lennart Berggren and Peter B Borwein

Convex Analysis and Nonlinear Optimization. Theory and Examples (2000) with Adrian S Lewis

Mathematics by Experiment: Plausible Reasoning in the 21st Century (2004) with David H Bailey

Experimentation in Mathematics: Computational Paths to Discovery (2004) with David H Bailey and Ronald Girgensohn

Techniques of Variational Analysis (2005) with Qiji J Zhu

Experimental Mathematics in Action (2007) with David Bailey, Neil J Calkin, Roland Girgensohn, D Russell Luke, and Victor H Moll

Communicating Mathematics in the Digital Era (2008) with Eugenio M Rocha and Jose Francisco Rodrigues

The Computer as Crucible: an Introduction to Experimental Mathematics (2008) with Keith Devlin

Convex Functions: Constructions, Characterizations and Counterexamples (2010) with Jon D Vanderwerff

Selected Writings on Experimental and Computational Mathematics (2010) with Peter B Borwein

Modern Mathematical Computation with Maple (2011) with Matthew Skerritt

An Introduction to Modern Mathematical Computing with Mathematica (2012) with Matthew Skerritt

Exploratory experimentation in mathematics: Selected works (2012) with David H Bailey

Lattice Sums Then and Now (2013) with M L Glasser, R C McPhedran, J G Wan and I J Zucker

Neverending Fractions. An Introduction to Continued Fractions (2014) with Alf van der Poorten, Jeffrey Shallit and Wadim Zudilin

Tools and mathematics: Instruments for learning (2016) with John Monaghan, Luc Troche

Pi: The Next Generation. A Sourcebook on the Recent History of Pi and its Computation (2016) with David H Bailey

**Click on a link below to go to that book**Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (1987) with Peter B Borwein

Collins Dictionary of Mathematics (1989) with Peter B Borwein

A Dictionary of Real Numbers (1990) with Peter B Borwein

Pi: A Source Book (1997) with Lennart Berggren and Peter B Borwein

Convex Analysis and Nonlinear Optimization. Theory and Examples (2000) with Adrian S Lewis

Mathematics by Experiment: Plausible Reasoning in the 21st Century (2004) with David H Bailey

Experimentation in Mathematics: Computational Paths to Discovery (2004) with David H Bailey and Ronald Girgensohn

Techniques of Variational Analysis (2005) with Qiji J Zhu

Experimental Mathematics in Action (2007) with David Bailey, Neil J Calkin, Roland Girgensohn, D Russell Luke, and Victor H Moll

Communicating Mathematics in the Digital Era (2008) with Eugenio M Rocha and Jose Francisco Rodrigues

The Computer as Crucible: an Introduction to Experimental Mathematics (2008) with Keith Devlin

Convex Functions: Constructions, Characterizations and Counterexamples (2010) with Jon D Vanderwerff

Selected Writings on Experimental and Computational Mathematics (2010) with Peter B Borwein

Modern Mathematical Computation with Maple (2011) with Matthew Skerritt

An Introduction to Modern Mathematical Computing with Mathematica (2012) with Matthew Skerritt

Exploratory experimentation in mathematics: Selected works (2012) with David H Bailey

Lattice Sums Then and Now (2013) with M L Glasser, R C McPhedran, J G Wan and I J Zucker

Neverending Fractions. An Introduction to Continued Fractions (2014) with Alf van der Poorten, Jeffrey Shallit and Wadim Zudilin

Tools and mathematics: Instruments for learning (2016) with John Monaghan, Luc Troche

Pi: The Next Generation. A Sourcebook on the Recent History of Pi and its Computation (2016) with David H Bailey

**1. Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (1987), by Jonathan M Borwein and Peter B Borwein.**

**1.1. Note.**

Reprinted 1988, 1996, Chinese edition 1995, paperback 1998.

**1.2. On the Back of the Book.**

Critical Acclaim for

*Pi and the AGM*:

"Fortunately we have the Borweins' beautiful book ... explores in the first five chapters the glorious world so dear to Ramanujan ... would be a marvellous text book for a graduate course." - Bulletin of the American Mathematical Society.

"What am I to say about this quilt of a book? One is reminded of Debussy who, on being asked by his harmony teacher to explain what rules he was following as he improvised at the piano, replied, "Mon plaisir." The authors are cultured mathematicians. They have selected what has amused and intrigued them in the hope that it will do the same for us. Frankly, I cannot think of a more provocative and generous recipe for writing a book ... (it) is cleanly, even beautifully written, and attractively printed and composed. The book is unique. I cannot think of any other book in print which contains more than a smidgen of the material these authors have included. - SIAM Review.

"If this subject begins to sound more interesting than it did in the last newspaper article on 130 million digits of Pi, I have partly succeeded. To succeed completely I will have gotten you interested enough to read the delightful and important book by the Borweins." - American Mathematical Monthly.

"The authors are to be commended for their careful presentation of much of the content of Ramanujan's famous paper, 'Modular Equations and Approximations to Pi'. This material has not heretofore appeared in book form. However, more importantly, Ramanujan provided no proofs for many of the claims that he made, and so the authors provided many of the missing details ... The Borweins, indeed have helped us find the right roads." - Mathematics of Computation.

**1.3. From the Preface.**

A central thread of this book is the arithmetic-geometric mean iteration of Gauss Lagrange and Legendre. A second thread is the calculation of

*π*. The two threads are intimately interwoven and provide a remarkable example of the application to twentieth-century computational concerns of the type of nineteenth-century analysis whose neglect Klein so deplores. The calculation of digits of π has had a fascination that has far exceeded utilitarian concerns - a fascination that has driven some to dedicate their lives to calculations we may now electronically effect in seconds. The methods that make the computation of hundreds of millions of digits of

*π*or any elementary function within our grasp are rooted in the AGM and this is where our interest in the material began. Making sense of this material took us in three directions and motivated our writing this book.

The first direction leads to nineteenth-century analysis and in particular the transformation theory of elliptic integrals. This necessitates at least a brief discussion of a number of topics including elliptic integrals and functions, theta functions, and modular functions. These attractive and once central concerns of analysis have been dropped from the standard curriculum - and much that is beautiful has become relatively inaccessible except to the expert or the archivist. In presenting this material we have not striven for generality. This is available in the specialty literature. At times we have settled for giving only a taste of the material and a few pointers on where it can be pursued.

We have found this excuse to consult the nineteenth-century masters a pleasurable and rewarding bonus - as Hermann points out in his introduction to Klein, "We are so used to thinking in terms of the 'progress' of science that it is hard for us to remember that certain matters were better understood one hundred years ago."

The second direction takes us into the domain of analytic complexity. How intrinsically difficult is it to calculate algebraic functions, elementary functions and constants, and the familiar functions of mathematical physics? Here part of the attraction is the surprising answers - the familiar methods are often far from optimal.

Finally, an honest treatment invites exploration of applications and ancillary material, particularly the rich and beautiful interconnections between the function theory and the number theory. Included, for example, are the Rogers-Ramanujan identities; algebraic series for π; results on sums of two and four squares; the transcendence of π and e and a discussion of Madelung's constant, lattice sums, and elliptic invariants.

Our primary concern throughout has been the interplay of analysis and mathematical application. We hope we have elucidated a variety of useful and attractive analytic techniques. This book should be accessible to any graduate student. Only rarely does it assume more than the content of undergraduate courses in real and complex analysis. It is, however, at times terse, at times computational, and some of the exercises are difficult. A fair amount of the material, particularly on the approximation and computation of π and the elementary functions, is new and only partially available in research papers.

**1.4. Review by: H London.**

*Mathematical Reviews*MR0877728

**(89a:11134)**.

This book reveals the close relationship between the algebraic-geometric mean iteration and the calculation of

*π*.

The topic of algebraic-geometric mean iteration leads to a discussion on the theory of elliptic integrals and functions, theta functions and modular functions. The calculation of

*π*leads into the area of calculating algebraic functions, elementary functions and constants plus the transcendence of

*π*and

*e*. The calculation of

*π*advanced with a frenzy with the advent of modern computers. Briefly, in 1706 the first 100 digits of

*π*were calculated. By 1844 the first 205 digits were known. In 1947 the first 808 digits of

*π*were computed using a desk calculator.

Then the modern computer came on the scene. Now the known first digits changed rapidly ...

**1.5. Review by: Richard Askey.**

*Amer. Math. Monthly*

**95**(9) (1988), 895-897.

The Mathematical Association of America, The American Mathematical Society and the Society for Industrial and Applied Mathematics have joined to try to get better media coverage of mathematics. One result of this is stories that say that computers have calculated 10 million, or now 130 million digits of π. When asked to comment on this by friends, most mathematicians wince and say they have no idea why anyone would want to do these calculations. Privately, they start to have serious doubts about more publicity for mathematics. If the computation of

*π*to millions of places is to be the image of a mathematician, then it might be better to keep mathematics out of the newspaper. However, ten million is so much larger then the 707 digits that was claimed to be known when I first heard of this problem that something interesting must lie behind these calculations. Part of the improvement comes from larger and faster machines, but much more comes from increased mathematical knowledge. Part of this is specific to certain numbers, and part is general and of wide applicability. All of it is very interesting.

...

... the next time you read a newspaper story about mathematics, read it the way political news needs to be read in many parts of the world, that is, read between the lines to get the story that is really there. It is probably very interesting, unlike the story that appeared in the paper. The problem of educating a few reporters so that they appreciate and understand some mathematics can probably be solved, but the problem of educating the editors to realise that not only does mathematics matter, but they should treat it seriously is much harder. However, an existence theorem exists at the Los Angeles Times, both for a reporter and an editor.

**1.6. Review by: Bruce C Berndt.**

*Mathematics of Computation*

**50**181 (1988), 352-354.

When the reviewer was a teenager, he and three friends, after a high school basketball game, would frequently get in a car and drive over the labyrinth of country roads in the rural area in which they lived. The game we played was to guess the name of the first village we would enter. Since there were many meandering roads and countless small hamlets that dotted the rural landscape, since it was dark, and since we were not blessed with keen senses of direction, we were often surprised when the signpost identified for us the town that we were entering.

For one not too familiar with the seemingly disparate topics examined by the Borweins in their book, one might surmise that the authors were travelling along mathematical byways with the same naiveté and lack of direction as the reviewer and his friends. However, the authors travel along well-lit roads that are marked by the road signs of elegance and usefulness and that lead to beautiful results. They do not take gravel-surfaced roads that lead to dead ends in cow pastures. But sometimes the destinations are surprising - at least to those not familiar with the landscape.

**1.7. Review by: Peter Cass.**

*The Mathematical Gazette*

**83**(497) (1999), 334-335.

This exciting and well-written book is the product of the sound scholarship and enormous enthusiasm of its authors. The subtitle, 'A study in analytic number theory and computational complexity', captures its central theme very well.

...

In exploring their principal themes, the arithmetic-geometric mean (AGM) iteration of Gauss, Lagrange, and Legendre, and the calculation of

*π*, the authors have gathered together a wealth of largely forgotten treasures from nineteenth-century analysis, drawn from the works of many masters, notably Gauss, Jacobi and Weierstrass. They have set them in a modern context, and directly related them to current research. For example, the transformation theory of elliptic functions, Jacobi's triple product, theta functions, and modular equations and algebraic approximations are discussed extensively. Chapter 3 contains welcome material elucidating some of Ramanujan's work, in particular the Rogers-Ramanujan identities.

...

I believe that most mathematicians will find much of interest in this engaging and literate book - a work of real enduring value.

**1.8. Review by: Jet Wimp.**

*SIAM Review*

**30**(3) (1988), 530-533.

What am I to say about this quilt of a book? One is reminded of Debussy who, on being asked by his harmony teacher to explain what rules he was following as he improvised at the piano, replied, "Mon plaisir." The authors are cultured mathematicians. They have selected what has amused and intrigued them in the hope that it will do the same for us. Frankly, I cannot think of a more provocative and generous recipe for writing a book. Intellectually, what can we really offer one another more genuine than our own excitement? - for, as much as some pretend otherwise, there is no trustworthy yardstick for the value of a piece of mathematics.

The book is cleanly, even beautifully, written, and attractively printed and composed. The book is unique. I cannot think of any other book in print which contains more than a smidgen of the material these authors have included. It is self-contained and copiously indexed. Each section concludes with exercises which invite the reader to share in the experience. These impressive exercises - really, excursions - are abundant, challenging, and selected so as to impinge on many subjects not discussed in the text. Can the book be used as a text? I recoil at such questions. But, yes. The book is a text if, for a little while, you are willing to be a student. And I suspect that if you purchase it you will not soon put it down.

**1.9. Review by: George E Andrews.**

*Bull. Amer. Math. Soc.*

**22**(1) (1990), 198-201.

P Beckmann wrote in April 1987 on the work of Y Kanada et al. calculating the first 140 million digits of

*π*:

It takes a lot of brains to program a computer to calculate (and verify!) 140 million decimal digits in a reasonable time, and the programmers have my respect and admiration - but they have it as sportsmen, not as scientists. The need for brains, or even the development of new techniques, is not what defines science. It also takes brains and the development of new techniques to stuff 21 people into a phone booth. And indeed, the calculation of the first 140 million digits is a feat worthy of ranking along with the other stunts in the Guinness Book of World Records. But I believe that when one applies the test of whether it can lead to a generalisation of knowledge, this feat will be found scientifically worthless.Surely if Beckmann's assertions are correct, we should be disturbed by the frivolous nature of such work ... Fortunately we have the Borweins' beautiful book to refute Beckmann's argument. There is both serious and beautiful mathematics underlying these massive calculations of the digits of

*π*.

...

It seems to me that this would be a marvellous text book for a graduate course in what has sometimes been called constructive mathematics, that never-never land between mathematics and computer science. This book contains important and currently neglected topics in classical analysis together with enough material on algorithms to make it a really exciting new course. The authors surely had something like this in mind for they provide ample exercises. I strongly hope that a number of people will be inspired to use the Borweins' book as the text for such a course.

**1.10. Review by: Allen Stenger.**

*Mathematical Association of America*(16 July 2018).

This is a very erudite book, and a large part of its charm is that it shows us many unexpected connections between different parts of mathematics. The AGM of title is the Arithmetic-Geometric Mean, a recurrence that was initially studied by Lagrange and Gauss in connection with numerical approximations to elliptic integrals. The book is inspired to some extent by Ramanujan's 1914 paper,

*Modular Equations and Approximations to Pi*. Like much of Ramanujan's work, this paper is full of interesting ideas but skimpy on proofs, and the present book is the first time that these ideas were worked out in detail and proved.

The publisher classifies this book as number theory, and there is some truth to that. It also deals with many of the topics of analysis from the late 1800s and the early 1900s, in particular theta functions and modular groups, and the last half of the book is concerned mostly with numerical approximations for various transcendental functions, and calculations for

*π*are spread throughout the book.

This summary makes the book sound like a hodgepodge, but in fact it hangs together very well. Roughly the first half of the book deals with elliptic integrals, theta functions, and the AGM. This part includes materials on partitions, the Rogers-Ramanujan identities, and representing numbers as sums of squares. It also includes the formulas and algorithms for calculating

*π*. The second half of the book is focused more on computer arithmetic and complexity of calculation, including results on fast Fourier transforms and fast multiplication. It also includes the history of π calculations and results on Diophantine approximations (irrationality measures) of

*π*. About $\large\frac{1}{4}\normalsize$ to $\large\frac{1}{3}\normalsize$ of the book is exercises, so it would work well as a textbook or problem book as well as a monograph (though there are no solutions; some exercises have hints).

...

Bottom line: the book is a tour de force. If you are interested in any of topics covered, it will lead you along somewhat-winding paths to other topics you'll like. Even if you're not interested, take a look and marvel at how closely connected its many diverse topics are.

**2. Collins Dictionary of Mathematics (1989), by Ephraim J Borowski and Jonathan M Borwein.**

**2.1. Note.**

Reprinted 1989, 1991; US editions (hard and soft) 1991; Edition with new publisher 1999, Second revised edition (11th and 12th printings) 2002; Third printing of revised edition 2003.

**2.2. From the Publisher of 1999 edition.**

This completely revised and updated edition is designed for anyone who needs to understand the basic terms of mathematics. Includes more than 9,000 definitions and 400 illustrations. Covers all major fields within mathematics, including real and complex analysis, abstract algebra, number theory, metamathematics, topology, vector calculus, differential equations, continuum mechanics, measure theory, graph theory, and logic. Now includes numerous useful links to authoritative Web sites to further expand research in the field. Contains biographical details of important mathematicians.

**2.3. From the Publisher of 2002 edition.**

A fully revised and expanded edition of the popular (over 50,000 copies sold) and authoritative Collins Dictionary of Mathematics. With over 9,000 definitions and 400 diagrams, this new edition has been fully revised to ensure it is fully congruent with mathematics as it is taught at A level and 1st year undergraduate level.

**2.4. Comment by: Jonathan M Borwein.**

*SIAM Review*

**48**(3) (2006), 585-594.

We started writing the Collins Dictionary in 1985 after a reader of the general Collins dictionary complained justifiably about certain of the mathematical and logical entries therein. Borowski and I were asked to revise the thousand or so mathematical terms, which we did. At the end we had a stack of handwritten file cards and a mild addiction which grew into the dictionary. This was typed on four Macintoshes (one a repentant Lisa), using the chalkboard as a database manager, with frequent airmailing of floppy disks across the Atlantic. We ended up having written a 9,000-or-so-term book which became the first text set from disk in Europe - an interesting if not a pretty process. Through ignorance on Collins' part, we had been left the "electronic and musical rights." By the mid-nineties this had resulted in an interactive CD version, the

*MathResource*, which embeds student

*Maple*. Ten years later the dictionary is sitting symmetrically inside

*Maple*.

After "finishing" the first edition of our dictionary in 1988, I found I could not enjoy a single colloquium or seminar for more than three years. I would constantly ask myself, "Did I define that term correctly, should I have included their result?" I felt like a giant hamster on a never-ending lexical treadwheel. Such is the life of a lexicographer or a compiler.

**2.4. Review of the Revised Reprint 1991 by: F J Papp.**

*Mathematical Reviews*MR1210061

**(94b:00008)**.

This dictionary is intended for students at levels from secondary school through master's degree. It is nicely designed to permit and encourage browsing. Thus, the general reader with some post-primary mathematical background will find much of interest. The authors' intent is to include "any term that an undergraduate might encounter not only within, but also in reading around, any course at college or university and [they] have also deliberately set out to tailor the explanation of each term to the mathematical knowledge of the reader who is likely to consult it". Overall, the authors have made reasonable choices with regard to the amount of detail to include with a given term. Where appropriate and possible, figures and formulas are included to illustrate a particular item. Since the dictionary is intended for such a broad audience, it comes as a very pleasant surprise to find that many advanced terms have been included. Dictionaries of this type frequently concentrate only on elementary terms and, if at all, give only a very brief mention of more advanced terms. Some entries, as might be expected, are very brief and others rather extensive. "Regular" has ten separate subentries, for example, ranging from "regular geometric figure" to "regular element of a ring". On the other hand, given the presence of some terms, the omission of others is somewhat unexpected. The terms "filter", "ultrafilter", and "net" are included; however, "subnet" and "universal net" are not. Within the dictionary, the spellings are generally the British standard spellings with U.S. variants sometimes indicated. The advertising copy on the covers is U.S. standard, for example "four-color theorem".

**2.5. Review by: Ros Herman.**

*New Scientist*(2 February 1991).

A dictionary of mathematics? Surely a contradiction in terms. Mathematics is about numbers, concepts, structures, abstract relationships. How could it make sense to organise such a matter as this by arranging mere words in the most accidental of orders, the alphabetical? And yet such is the frailty of the human mind that our mental maps of the elegant framework fail us from time to time, and we have to fall back on our childish verbalism and the sing-song sequence we learned parrot-fashion at our parents' knees.

So, rather than quibble over the function of such dictionaries, let us see what those on offer have to say about functions. The Borowski and Borwein account opens rather clumsily, not to say off-puttingly, with four bracketed, capital-lettered asides to hold up the eye (and brain) in the first clause alone. Yes, I am afraid the entry follows the classical, dictionary style of using semicolons to mark off alternative views, in a multidecker sandwich that has to be confined into one sentence. The next sentence builds on the first to exemplify the associated notation while the third is another multidecker that enlarges, in a illuminating way, on the subtle distinctions between function, mapping and transformation.

**3. A Dictionary of Real Numbers (1990)**,

**by Jonathan M Borwein and Peter B Borwein.**

**3.1. Note.**

Published by different publishers in each of 1991, 1992, 1993 and 2011.

**3.2. From the Preface.**

How do we recognise that the number .93371663 ... is actually $2 \log_{10} (e + \pi)/2$? Gauss observed that the number 1.85407467 ... is (essentially) a rational value of an elliptic integral - an observation that was critical in the development of nineteenth century analysis. How do we decide that such a number is actually a special value of a familiar function without the tools Gauss had at his disposal, which were, presumably, phenomenal insight and a prodigious memory? Part of the answer, we hope, lies in this volume.

This book is structured like a reverse telephone book, or more accurately, like a reverse handbook of special function values. It is a list of just over 100,000 eight-digit real numbers in the interval [0,1) that arise as the first eight digits of special values of familiar functions. It is designed for people, like ourselves, who encounter various numbers computationally and want to know if these numbers have some simple form. This is not a particularly well-defined endeavour - every eight-digit number is rational and this is not interesting. However, the chances of an eight digit number agreeing with a small rational, say with numerator and denominator less than twenty-five, is small. Thus the list is comprised primarily of special function evaluations at various algebraic and simple transcendental values. The exact numbers included are described below.

Each entry consists of the first eight digits after the decimal point of the number in question. The values are truncated not rounded. The next part of the entry specifies the function, and the final part of the entry is the value at which the function is evaluated. So -4.828313737... is entered as "8283 1373 ln : $5^{-3}$". The abbreviations are also described below. There are two exceptions to this format. One is to describe certain combinations of two functions, for instance "8064 9591 exp(√3) + exp(

*e*)" the meaning of which is self-evident. The other is for real roots of cubic polynomials, i.e. "2027 1481 rt: (5, 8, -8, -3)" means that the polynomial $5x^{3} + 8x^{2} - 8x - 3$ has a real root, the fractional part of which is .20271481 .... Repeats have in general been excluded except for cases where two genuinely different numbers agree through at least eight digits. These latter coincidences account for the fewer than fifty repeat entries.

If the number you are checking is not in the list try checking one divided by the number and perhaps a few other variants such as one minus the number. Of course, finding the number in the list, in general, only indicates that this is a good candidate for the number. Proving agreement or checking to further accuracy is then appropriate.

**3.3. Review by: Nick Lord.**

*The Mathematical Gazette*

**74**(470) (1990), 395-396.

This large book is precisely what its title suggests-an impressive list of some 100,000 numbers between 0 and 1 (in order, to 8 decimal places) which are of the form $x - [x]$ for interesting or significant real numbers $x$. By "interesting or significant" is meant either evaluations of standard functions at special points or simple combinations of significant numbers (including a sprinkling of physical constants).

...

The authors suggest that such a collection of numbers might be useful in research if you encounter a number whose value you know accurately but which you suspect to have deeper significance; agreement to 8 decimal places with an entry in the table reduces the risk of misappropriation and may reveal an unexpected connection. And, indeed, I was able to recapture something of the excitement caused by Euler's evaluation of $\Sigma \large\frac{1}{n^{2}\normalsize}$ by turning up (1) .6449 3406 = $\large\frac{\pi^2} 6$. However, except perhaps as an example of an enterprise which could not seriously have been contemplated before the advent of computers, I cannot really see a place for this book other than in university library.

**4. Pi: A Source Book (1997), by Lennart Berggren, Jonathan M Borwein and Peter B Borwein.**

**4.1. Note.**

Second Edition 2000. Third Edition, incorporating

*A Pamphlet on Pi*2003. Fourth edition 2011.

**4.2. Preface to the First Edition.**

Our intention in this collection is to provide, largely through original writings, an extended account of pi from the dawn of mathematical time to the present. The story of pi reflects the most seminal, the most serious, and sometimes the most whimsical aspects of mathematics. A surprising amount of the most important mathematics and a significant number of the most important mathematicians have contributed to its unfolding - directly or otherwise.

*π*is one of the few mathematical concepts whose mention evokes a response of recognition and interest in those not concerned professionally with the subject, It has been a part of human culture and the educated imagination for more than twenty-five hundred years. The computation of

*π*is virtually the only topic from the most ancient stratum of mathematics that is still of serious interest to modem mathematical research. To pursue this topic as it developed throughout the millennia is to follow a thread through the history of mathematics that winds through geometry, analysis and special functions, numerical analysis, algebra, and number theory. It offers a subject that provides mathematicians with examples of many current mathematical techniques as well as a palpable sense of their historical development.

**4.3. From the publisher of the Second Edition.**

Our intention in this collection is to provide, largely through original writings, an extended account of pi from the dawn of mathematical time to the present. The story of

*π*reflects the most seminal, the most serious, and sometimes the most whimsical aspects of mathematics. A surprising amount of the most important mathematics and a significant number of the most important mathematicians have contributed to its unfolding directly or otherwise.

*π*is one of the few mathematical concepts whose mention evokes a response of recognition and interest in those not concerned professionally with the subject. It has been a part of human culture and the educated imagination for more than twenty-five hundred years. The computation of pi is virtually the only topic from the most ancient stratum of mathematics that is still of serious interest to modern mathematical research. To pursue this topic as it developed throughout the millennia is to follow a thread through the history of mathematics that winds through geometry, analysis and special functions, numerical analysis, algebra, and number theory. It offers a subject that provides mathematicians with examples of many current mathematical techniques as well as a palpable sense of their historical development. Why a Source Book? Few books serve wider potential audiences than does a source book. To our knowledge, there is at present no easy access to the bulk of the material we have collected.

**4.4. Preface to the Second Edition.**

We are gratified that the first edition was sufficiently well received so as to merit a second. In addition to correcting a few minor infelicities, we have taken the opportunity to add an Appendix in which articles 9 and 12 by Viète and Huygens respectively are translated into English. While modern European languages are accessible to our full community - at least through colleagues - this is no longer true of Latin. Thus, following the suggestions of a reviewer of the first edition we have opted to provide a serviceable if fairly literal translation of three extended Latin excerpts. And in particular to make Viète's opinions and style known to a broader community.

We also record that in the last two years distributed computations have been made of the binary digits of

*π*using an enhancement due to Fabrice Bellard of the identity made in article 70. In particular the binary digits of

*π*starting at the 40 trillionth place are 0 0000 1111 1001 1111. Details of such ongoing computations, led by Colin Percival, are to be found at www.cecm.sfu.ca/projects/pihex.

Corresponding details of a billion ($2^{30}$) digit computation on a single Pentium Il PC, by Dominique Delande using Carey Bloodworth's desktop π program and taking under nine days, are lodged at www.cecm.sfu.ca/personal/jborwein/pi_cover.html. Here also are details of the computation of $2^{36}$ digits by Kanada et al. in April 1999.

We are grateful for the opportunity to thank Jen Chang for all her assistance with the cover design of the book. We also wish to thank Annie Marquis and Judith Borwein for their substantial help with the translated material.

**4.5. From the Publisher of Third Edition.**

This book documents the history of pi from the dawn of mathematical time to the present. One of the beauties of the literature on pi is that it allows for the inclusion of very modern, yet accessible, mathematics. The articles on pi collected herein fall into various classes. First and foremost there is a selection from the mathematical and computational literature of four millennia. There is also a variety of historical studies on the cultural significance of the number. Additionally, there is a selection of pieces that are anecdotal, fanciful, or simply amusing.

For this new edition, the authors have updated the original material while adding new material of historical and cultural interest. There is a substantial exposition of the recent history of the computation of digits of

*π*, a discussion of the normality of the distribution of the digits, and new translations of works by Viete and Huygen

**4.6. Preface to the Third Edition.**

Our aim in preparing this edition is to bring the material in the collection of papers in the second edition of this source book up to date. Moreover , several delightful pieces became available and are added.

This substantial supplement to the third edition serves as a stand-alone exposition of the recent history of the computation of digits of

*π*. It also includes a discussion of the thorny old question of normality of the distribution of the digits. Additional material of historical and cultural interest is included, the most notable being new translations of the two Latin pieces of Viète (translation of article 9 (Excerpt 1): Various Responses on Mathematical Matters: Book VII (1593) and (Excerpt 2): Defense for the New Cyclometry or "Anti-Axe"), and a thorough revision of the translation of Huygens's piece (article 12) published in the second edition.

We should like to thank Professor Marinus Taisbak of Copenhagen for grappling with Viète's idiosyncratic style to produce the new translation of his work. We should like to thank Karen Aardal for permission to use her photograph of Ludolph's new tombstone in the Pieterskerk in Leiden, the Smithsonian Institution for permission to reproduce a fine photo of ENIAC, and David and Gregory Chudnovsky for providing a

*Walk on the digits of π*. We should also like to thank Irving Kaplansky for his gracious permission to include his

*A Song about π*. Finally, our thanks go to our colleagues whose continued interest in

*π*has encouraged our publisher to produce this third edition, as well as for the comments and corrections to earlier editions that some of them have sent us.

**4.7. Review by: E J Barbeau.**

*Mathematical Review*s MR1467531

**(98f:01001)**.

These 70 facsimile papers and excerpts constitute a mixture of historical tracts and expository accounts in the long, fascinating and occasionally whimsical saga of the number pi. Early work of ancient non-European civilizations is sampled along with European research since the seventeenth century (for example, the transcendence papers of Hermite and Lindemann). Several recent contributions record progress in computing pi to an incredible degree of accuracy, reflecting the expertise of two of the editors in this area. In the body of the book, the papers follow one another chronologically without explanation. For context and sources, the reader must check through the table of contents, the bibliography, the list of credits and the first appendix describing the early history of

*π*A second appendix dates the progress in computing

*π*, while the third provides a list of formulae. The judicious representative selection makes this a useful addition to one's library as a reference book, an enjoyable survey of developments and a source of elegant and deep mathematics of different eras.

**5. Convex Analysis and Nonlinear Optimization. Theory and Examples (2000), by Jonathan M Borwein and Adrian S Lewis.**

**5.1. Note.**

Second extended edition, 2005. Paperback 2009.

**5.2. From the Publisher of the Second Edition.**

Optimisation is a rich and thriving mathematical discipline. The theory underlying current computational optimisation techniques grows ever more sophisticated. The powerful and elegant language of convex analysis unifies much of this theory. The aim of this book is to provide a concise, accessible account of convex analysis and its applications and extensions, for a broad audience. It can serve as a teaching text, at roughly the level of first year graduate students. While the main body of the text is self-contained, each section concludes with an often extensive set of optional exercises. The new edition adds material on semismooth optimisation, as well as several new proofs that will make this book even more self-contained.

**5.3. From the Preface.**

Optimization is a rich and thriving mathematical discipline. Properties of minimizers and maximizers of functions rely intimately on a wealth of techniques from mathematical analysis, including tools from calculus and its generalisations, topological notions, and more geometric ideas. The theory underlying current computational optimisation techniques grows ever more sophisticated - duality-based algorithms, interior point methods, and control-theoretic applications are typical examples. The powerful and elegant language of convex analysis unifies much of this theory. Hence our aim of writing a concise, accessible account of convex analysis and its applications and extensions, for a broad audience.

For students of optimisation and analysis, there is great benefit to blurring the distinction between the two disciplines. Many important analytic problems have illuminating optimisation formulations and hence can be approached through our main variational tools: subgradients and optimality conditions, the many guises of duality, metric regularity and so forth. More generally, the idea of convexity is central to the transition from classical analysis to various branches of modern analysis: from linear to nonlinear analysis, from smooth to non-smooth, and from the study of functions to multjfunctions. Thus, although we use certain optimisation models repeatedly to illustrate the main results (models such as linear and semidefinite programming duality and cone polarity), we constantly emphasise the power of abstract models and notation.

Good reference works on finite-dimensional convex analysis already exist. Rockafellar's classic Convex Analysis has been indispensable and ubiquitous since the 1970s, and a more general sequel with Wets, Variational Analysis, appeared recently. Hiriart-Urruty and Lemaréchal's Convex Analysis and Minimization Algorithms is a comprehensive but gentler introduction. Our goal is not to supplant these works, but on the contrary to promote them, and thereby to motivate future researchers. This book aims to make converts.

**5.4. Review by: John R Giles.**

*Mathematical Reviews*MR1757448

**(2001h:49001)**.

This text is an account of convex analysis as the unifying theory underlying current computational optimisation techniques. ...The present book gives a concise treatment of the area, aiming to show the relevance in particular of new developments in non-smooth analysis to optimisation theory. The material is set in Euclidean spaces but in a coordinate-free style which invites extension to Banach spaces. Such an extension is discussed in the last chapter of the book. Generally there is a progression in the study of optimality conditions from an assumption of differentiability towards non-differentiability through convexity and then local Lipschitz continuity.

...

There is a fascinating interweaving of theory and applications which themselves suggest new theory.

...

The book is written as a teaching text for beginning graduate students. It is intended for a wide audience; the student will need a certain level of mathematical maturity but no sophisticated knowledge of real analysis is assumed. Each of the 30 or so sections is about a lecture length and is followed by exercises classified as those which illustrate the ideas of the lecture and those which introduce additional theory of varying difficulty. A student working with the book as a lecture course would soon realise the importance of working the exercises; familiarity with manipulating concepts introduced in earlier sections will be necessary for full understanding and often theory refers to material introduced in earlier exercises.

The book is of manageable size and as such should appeal to the student. Further, the proofs are generally short and snappy, revealing the power of the abstract structural approach and fruitful interplay of geometrical and topological ideas. However, considerable ground is covered and, as a graduate text should, it develops the subject up to the frontiers of current research, giving an idea of areas for further exploration. Before each exercise section there is an informative guide to the relevant literature with references to a full bibliography. This text will give impetus to the teaching of analysis because it makes evident its significant applications in optimisation. But it will also bring added attraction to the study of optimisation because it reveals so much of its abstract structural base.

**6. Mathematics by Experiment: Plausible Reasoning in the 21st Century (2004), by Jonathan M Borwein and David H Bailey.**

**6.1. Note.**

Interactive CD version 2006. Expanded Second Edition 2008.

**6.2. From the Preface.**

The authors first met in 1985, when Bailey used the Borwein quartic algorithm for π as part of a suite of tests on the new Cray-2 then being installed at the NASA Ames Research Center in California. As our collaboration has grown over the past 18 years, we have became more and more convinced of the power of experimental techniques in mathematics. When we started our collaboration, relatively few mathematicians employed computations in serious research work. In fact, there appeared to be a widespread view in the field that "real mathematicians don't compute." In the ensuing years, computer hardware has skyrocketed in power and plummeted in cost, thanks to the remarkable phenomenon of Moore's Law. In addition, numerous powerful mathematical software products, both commercial and non-commercial, have become available. But just importantly, a new generation of mathematicians is eager to use these tools, and consequently numerous new results are being discovered.

The experimental methodology described in this book, as well as in the second volume of this work,

*Experimentation in Mathematics: Computational Paths to Discovery*, provides a compelling way to generate understanding and insight; to generate and confirm or confront conjectures; and generally to make mathematics more tangible, lively and fun for both the professional researcher and the novice. Furthermore, the experimental approach helps broaden the interdisciplinary nature or mathematical research: a chemist, physicist, engineer, and a mathematician may not understand each others' motivation or technical language, but they often share an underlying computational approach, usually to the benefit of all parties involved.

... Our goal in these books is to present a variety of accessible examples of modern mathematics where intelligent computing plays a significant role (along with a few examples showing the limitations of computing). We have concentrated primarily on examples from analysis and number theory, as this is where we have the most experience, but there are numerous excursions into other areas of mathematics as well. For the most part, we have contented ourselves with outlining reasons and exploring phenomena, leaving a more detailed investigation to the reader. There is, however, a substantial amount of new material, including numerous specific results that have not yet appeared in the mathematical literature, as far as we are aware.

This work is divided into two volumes, each of which can stand by itself. This volume, Mathematics by Experiment: Plausible Reasoning in the 21st Century, presents the rationale and historical context of experimental mathematics, and then presents a series of examples that exemplify the experimental methodology. We include in this volume a reprint of an article co-authored by one of us that complements this material. The second book, Experimentation in Mathematics: Computational Paths to Discovery, continues with several chapters of additional examples. Both volumes include a chapter on numerical techniques relevant to experimental mathematics.

Each volume is targeted to a fairly broad cross-section of mathematically trained readers. Most of this volume should be readable by anyone with solid undergraduate coursework in mathematics. Most of the second volume should be readable by persons with upper-division undergraduate or graduate-level coursework. None of this material involves highly abstract or esoteric mathematics.

The subtitle of this volume is taken from George Polya's well-known work,

*Mathematics and Plausible Reasoning*. This two-volume work has been enormously influential - if not uncontroversial - not only in the field of artificial intelligence, but also in the mathematical education and pedagogy community. ...

**6.3. Review by: John H Mason.**

*Mathematical Reviews*MR2033012

**(2005b:00012).**

Let me cut to the chase: every mathematics library requires a copy of this book (and its companion volume [J M Borwein, D H Bailey and R Girgensohn,

*Experimentation in mathematics*]). Every supervisor of higher degree students requires a copy on their shelf. Welcome to the rich world of computer-supported mathematics! This book advances the thesis that significant and difficult mathematics can and does emerge from experimentation with special cases, especially with computer support.

The introductory chapter quotes a gamut of mathematicians who found inspiration in specifics and who have done pages and pages of calculations not always revealed in the final write-up (Georg Riemann being a case in point with two pages from his scrap book reprinted). Doing mathematics has a strongly experimental, empirical aspect. Who has not followed Hilbert's advice and studied a particular but generic example in detail in order to find out 'what is going on'? Who has not at some time been struck by a similarity between two or more apparently disparate mathematical objects, and wondered if there was not something linking them together? The theme of experimentation as part of doing mathematics is comprehensively developed and illustrated in the book by several dozen pertinent and inspiring examples which draw on the historical as well as the current.

...

The book is also very well written. The text is friendly and supportive, yet the mathematics discussed is cutting edge. The tone is not patronising as often happens when the author feels the need to talk down to the reader. Rather the tone is, but collegial, expecting the reader to be mathematically confident, while providing enough detail for the argument to be plausible. The reader is invited to pause along the way and to experiment themselves, making use of websites for access to the more sophisticated software, and sometimes code provided for Maple or Mathematica, where suitable. The many diagrams and eight colour illustrations span mathematical sculptures and computer output.

The book obviously displays the authors' love of mathematics, and consummate mastery of a broad range of topics, ranging from the highly technical to the aesthetic and the historical. Topics mentioned include differential equations, differential geometry, integration theory, complex functions and Riemann surfaces, knot theory, quantum field theory, dynamical systems, number theory, Ramsey theory and Gödel, and of course power series. Chapter titles include: Pi and its friends, Normality of numbers, Constructive proofs and Numerical techniques. I suspect that a generation of mathematicians will be inspired to develop even more sophisticated software and mathematical ways of thinking by reading this book and its companion volume.

**6.4. Review by: Graham Hoare.**

*The Mathematical Gazette*

**89**(514) (2005), 143-144.

When Carl Siegel inspected a batch of Riemann's papers at Göttingen University he discovered several pages of numerical calculations including a number of lowest-order zeros of the zeta function each calculated to several decimal places. So the genesis of the Riemann hypothesis which appeared in Riemann's profound and beautiful memoir of 1859 may have depended as much, if not more, on these calculations as on some deep insight or intuition.

And how did Gauss at the age of 14 or 15, manage to conjecture correctly, as we now know, the Prime Number Theorem? We know he was adept at spotting patterns in numerical data but he was secretive about his methods, claiming that an architect never leaves the scaffolding for people to see how the building was constructed. So great mathematicians experimented; recall Archimedes and how he was led to his beautiful results in geometry. Others such as Newton, Euler, Ramanujan and von Neumann are cited for their extraordinary facility for numerical calculation and algebraic manipulation. So why do we need this book now?

The authors, with something approaching evangelical zeal, claim that in the last 15 years or so the astonishing increase in the power of computational artefacts heralds a paradigm shift in the way increasing numbers of mathematicians are changing their working procedures. In their view 'mathematics is not ultimately about formal proof; it is instead about secure mathematical knowledge'. The book is laced with a formidable set of examples, some quite astonishing, drawn principally from analysis and number theory, to support their thesis. Each chapter save the seventh ends with a commentary and further examples and challenges. Not surprisingly the virtues of such software packages as Mathematica and Maple are extolled. The text is interlaced with quotes, both ancient and modern, from great mathematicians who have exploited the experimental approach.

**6.5. Review of the Second Edition by: Michael Otto.**

*Mathematical Reviews*MR2473161

**(2010c:00001)**.

To anyone with solid undergraduate course work in mathematics the volume offers an overwhelming variety of instructive examples, information and experience. The book at hand is a rich work, written by two experts in the field. The amount of information as well as the manner in which it is presented makes it appear less a book than a laboratory manual or a Web site.

**7. Experimentation in Mathematics: Computational Paths to Discovery (2004), by Jonathan M Borwein, David H Bailey and Ronald Girgensohn.**

**7.1. Note.**

Interactive CD version 2006.

**7.2. From the Publisher.**

New mathematical insights and rigorous results are often gained through extensive experimentation using numerical examples or graphical images and analysing them. Today computer experiments are an integral part of doing mathematics. This allows for a more systematic approach to conducting and replicating experiments. The authors address the role of experimental research in the statement of new hypotheses and the discovery of new results that chart the road to future developments. Following the lead of

*Mathematics by Experiment: Plausible Reasoning in the 21st Century*this book gives numerous additional case studies of experimental mathematics in action, ranging from sequences, series, products, integrals, Fourier series, zeta functions, partitions, primes and polynomials. Some advanced numerical techniques are also presented. To get a taste of the material presented in both books view the condensed version.

**7.3. Review by: F Beukers.**

*Mathematical Reviews*MR2051473

**(2005h:11002)**.

The point of view of the authors is that mathematical discovery through experimentation and the use of increasingly intelligent software is going to play an indispensable role in mathematics. To support this point of view the authors present a very large variety of mathematical subjects where 'intelligent computing plays a role', as the authors say in the introduction. Although computer calculation does play a role in many parts of the book, it is my impression that the promise of a new approach to mathematical discovery has not been fulfilled, at least, not in the part under review. For example, Chapter 2, which is about Fourier theory, contains a very interesting and entertaining approach to this theory, which was at the basis of much of modern analysis. Of course, in teaching Fourier theory, the use of a computer can release us from much of the drudgery of calculation that the subject entails. The computer can also be used as a tool to explore examples which illustrate the pitfalls that beset Fourier theory. And it can be used to plot the many strange Fourier sums to the point where we can actually 'see' nowhere differentiable continuous functions. However, in this chapter I do not get the impression that a use of the computer has added anything to the theory of Fourier series and transforms. Most of the interesting phenomena were known in the pre-computer era, an exception being perhaps the sinc-integrals.

...

Much of the material in the book has arisen from the experiences of the authors while working on a computer based approach to different topics in mathematics. The variety obtained in this way is impressive, the authors have really touched and produced a treasure trove of lovely mathematical gems. Anyone who can appreciate such an attitude to mathematics is bound to enjoy it.

**8. Techniques of Variational Analysis (2005), by Jonathan M Borwein and Qiji J Zhu.**

**8.1. Note.**

Paperback 2010.

**8.2. From the Publisher.**

Variational arguments are classical techniques whose use can be traced back to the early development of the calculus of variations and further. Rooted in the physical principle of least action they have wide applications in diverse fields. This book provides a concise account of the essential tools of infinite-dimensional first-order variational analysis. These tools are illustrated by applications in many different parts of analysis, optimisation and approximation, dynamical systems, mathematical economics and elsewhere. Much of the material in the book grows out of talks and short lecture series given by the authors in the past several years. Thus, chapters in this book can easily be arranged to form material for a graduate level topics course. A sizeable collection of suitable exercises is provided for this purpose. In addition, this book is also a useful reference for researchers who use variational techniques - or just think they might like to.

**8.3. Review by: Richard B Vinter.**

*Mathematical Reviews*MR2144010

**(2006h:49002)**.

This book maps the progress that has been made since the publication of the Ekeland variational principle in 1974 in the development and application of the variational approach in nonlinear analysis. The authors are well equipped for their task. Jonathan Borwein was the co-discoverer (with David Preiss) of a landmark variational principle, with the novel feature that the perturbation term is taken to be a smooth function. Jim Zhu has worked extensively on the use of variational principles to the development of calculus rules in nonsmooth analysis and on optimal control applications.

...

This monograph is distinctive for bringing out the unifying role of variational principles across nonlinear analysis, the numerous examples of their application, and for the insights communicated by the authors, drawing on their experience as key participants in their development.

**9. Experimental Mathematics in Action (2007), by David Bailey, Jonathan Borwein, Neil J Calkin, Roland Girgensohn, D Russell Luke, and Victor H Moll.**

**9.1. From the Publisher.**

The last twenty years have been witness to a fundamental shift in the way mathematics is practiced. With the continued advance of computing power and accessibility, the view that `real mathematicians don't compute' no longer has any traction for a newer generation of mathematicians that can really take advantage of computer-aided research especially given the scope and availability of modern computational packages such as Maple, Mathematica, and MATLAB. The authors provide a coherent variety of accessible examples of modern mathematics subjects in which intelligent computing plays a significant role.

**9.2. Review by: Andrew Odlyzko.**

*The American Mathematical Monthly*

**118**(10) (2011), 946-951.

That computers are revolutionising mathematics (and almost everything else) is now a cliché. But clichés are often true, and sometimes bear re-examining to gain a deeper understanding. Experimental Mathematics in Action by David Bailey, Jonathan Borwein, Neil Calkin, Roland Girgensohn, Russell Luke, and Victor Moll provides an opportunity to do so. It is a very nice and useful book. However, one can hope it will become obsolete in a few years. We are in a transitional period in terms of acceptance of computers into mainstream mathematics, and the lessons of this book should soon become fully absorbed into standard books

...

Experimental Mathematics in Action is based on lectures given at a Mathematical Association of America short course on experimental mathematics given at the 2006 annual AMS-MAA meeting. Its aim is to facilitate the adoption of nontrivial computational techniques in mathematics. Chapter 1 has a brief philosophical introduction to the role of computing in mathematics, and then lists eight roles for it:

- Gaining insight and intuition, or just knowledge.

- Discovering new facts, patterns, and relationships.

- Graphing to expose mathematical facts, structures, or principles.

- Rigorously testing and especially falsifying conjectures.

- Exploring a possible result to see if it merits formal proof.

- Suggesting approaches for formal proof.

- Computing replacing lengthy hand derivations.

- Confirming analytically derived results.

There follow brief sketches of examples for each.

**9.3. Review by: MR.**

*European Mathematical Society Newsletter*(1 October 2011).

This book contains material originating from a course on

*Experimental Mathematics in Action*, organised by Jonathan Borwein in San Antonio in 2006. One of aims of the book is to defend an assertion that the statement "the real mathematicians don't compute" is no longer valid with a new generation of mathematicians. Computer-aided research, taking advantage of modern computational packages (such as Maple or Mathematica) has its importance nowadays. The book presents several examples and methodological ways to make clear what computational or experimental mathematics is or should be. Starting with some remarks of philosophical character, the authors present many algorithms for experimental mathematics, such as high-precision arithmetic, integer relation detection, prime number computations and finding the roots of polynomials. Moreover, their aim is also to present (the slightly controversial) idea that "mathematics is done more like physics in that you come about things experimentally". This does not mean that a mathematician should give up proving theorems; this is just a statement about how some of the mathematical facts can be discovered. The book is nicely written, with a special touch of mathematical poetry and beauty-behind-the-computation opinion. It will be appreciated not only by number theoreticians but also by anyone who does not prevent computers from entering the pure garden of mathematical delights.

**9.4. Review by: David P Roberts.**

*Mathematical Association of America*(13 August 2007).

[The book] began as a two-day course given by the six authors at the joint AMS-MAA meetings in San Antonio in January 2006. The eight lectures there became the eight main chapters of the book, each consisting of about twenty to thirty pages. The final chapter consists of exercises, roughly five to ten pages worth for each of the previous eight chapters, and then twenty pages of additional exercises.

...

The Borwein chapters: Secure mathematical knowledge. Chapter 1 is a "Philosophical introduction." It begins by acknowledging a general distaste for philosophy among scientists, but says "we are of the opinion that mathematical philosophy matters more now than it has in nearly a century." The moderation, as defined above, is captured in the sentence "while I appreciate fine proofs and aim to produce them when possible, I no longer view proof as the royal road to secure mathematical knowledge."

An example of unproved but secure mathematical knowledge is provided by a relation between multizeta functions ... A small part of the security comes from the fact that the relation is proved for $N = 1$. Most of the security comes from the fact that the relation has been numerically checked to 1000 decimal places for many $N$. As a rule, we should view relations such as this unproven zeta identity as likely deeper than similar relations which we can prove. Should we keep relations like this one hidden away on hard disks until the unknown and perhaps never to come day that they are proved? Of course not. The authors ask of the community that experimentally supported statements like this zeta identity be given considerable respect. In return, it is always acknowledged that these statements are not proved. Proofs are to be highly valued too, especially when they add insight.

Chapter 8 is a "Computational Conclusion."

**10. Communicating Mathematics in the Digital Era (2008), by Jonathan M Borwein, Eugenio M Rocha and Jose Francisco Rodrigues.**

**10.1. From the Publisher.**

The digital era has dramatically changed the ways that researchers search, produce, publish, and disseminate their scientific work. These processes are still rapidly evolving due to improvements in information science, new achievements in computer science technologies, and initiatives such as DML and open access journals, digitisation projects, scientific reference catalogues, and digital repositories. These changes have prompted many mathematicians to play an active part in the developments of the digital era, and have led mathematicians to promote and discuss new ideas with colleagues from other fields, such as technology developers and publishers. This book is a collection of contributions by key leaders in the field, offering the paradigms and mechanisms for producing, searching, and exploiting scientific and technical scholarship in mathematics in the digital era.

**10.2. Review by: Luiz Henrique de Figueiredo.**

https://www.maa.org/press/maa-reviews/communicating-mathematics-in-the-digital-era

What is the impact of the internet in mathematics? How does one preserve mathematical literature in digital form so that it will be available to present and future generations? Can it be made cost-free for readers? Do we still need hardcopies of papers? What is the future of journals? These are important questions that not only administrators and librarians but also mathematicians have to address in our digital era.

Although the articles collected in this book do not offer definite answers to those questions, they do raise several important and interesting points and describe some avenues that should be (and have been) explored. The articles in the first part of the book are especially relevant to those involved in making decisions about libraries and journal subscriptions.

**10.3. Review by: Kiril Bankov.**

*The Mathematical Gazette*

**94**(531) (2010), 557-559.

The digital era has dramatically changed the ways that researchers search, produce, publish, and disseminate their scientific work. These processes are still rapidly developing because of improvements in information science, new achievements in computer science technologies, and initiatives such as digital mathematics libraries and open access journals, digitisation projects, scientific reference catalogues, and digital repositories.

This book is a collection of essays and reports by key leaders in the field, offering examples and mechanisms for producing, searching, and exploiting scientific and technical scholarship in mathematics in the digital era. The contributions are organised in three parts. Part I consists of eight articles on electronic publishing and digital libraries. Part II contains six articles on technology of dissemination. Part III comprises five papers related to educational and cultural issues.

...

The book is written for a wide readership: mathematicians and scientists who are the main users of scientific publications; referees, editors, publishers, libraries, repositories, www-servers, readers, financial foundations, universities, and scientific societies. All users of these will be stimulated by the ideas, the problems, and the descriptions of the recent developments in the field. The authors of the papers represent a wide range of countries, which makes 'Communicating mathematics in the digital era' an internationally useful book.

**11. The Computer as Crucible: an Introduction to Experimental Mathematics (2008), by Jonathan M Borwein and Keith Devlin.**

**11.1. Note.**

Japanese edition 2009. German edition 2011.

**11.2. From the Publisher**

Keith Devlin and Jonathan Borwein, two well-known mathematicians with expertise in different mathematical specialties but with a common interest in experimentation in mathematics, have joined forces to create this introduction to experimental mathematics. They cover a variety of topics and examples to give the reader a good sense of the current state of play in the rapidly growing new field of experimental mathematics. The writing is clear and the explanations are enhanced by relevant historical facts and stories of mathematicians and their encounters with the field over time.

**11.3. From the Preface.**

Our aim in writing this book was to provide a short readable account of experimental mathematics. (Chapter 1 begins with an explanation of what the term "experimental mathematics" means.) It is not intended as a textbook to accompany a course (though good instructors could surely use it that way). In particular, we do not aim for comprehensive coverage of the field; rather, we pick and choose topics and examples to give the reader a good sense of the current state of play in the rapidly growing new field of experimental mathematics. Also, there are no large exercise sets. We do end each chapter with a brief section called "Explorations," in which we give some follow-up examples and suggest one or two things the reader might like to try. There is no need to work on any of those explorations to proceed through the book, but we feel that trying one or two of them is likely to increase your feeling for the subject. Answers to those explorations can be found in the "Answers and Reflections" chapter near the end of the book.

This book was the idea of our good friend and publisher (plus mathematics PhD) Klaus Peters of A K Peters, Ltd. It grew out of a series of three books that one of us (Borwein) co-authored on experimental mathematics, all published by A K Peters: Jonathan Borwein and David Bailey's

*Mathematics by Experiment*(2004); Jonathan Borwein, David Bailey, and Roland Girgensohn's

*Experimentation in Mathematics*(2004); and David Bailey, Jonathan Borweln, Neil J Calkin, Roland Girgensohn, D Russell Luke, and Victor H Moll's

*Experimental Mathematics in Action*(2007).

We both found this an intriguing collaboration. Borwein, with a background in analysis and optimisation, has been advocating and working in the new field of experimental mathematics for much of his career. This pursuit was considerably enhanced in 1993 when he was able to open the Centre for Experimental and Constructive Mathematics at Simon Fraser University, which he directed for a decade. (Many of the results presented here are due to Borwein, most often in collaboration with others, particularly Bailey.) Devlin, having focused on mathematical logic and set theory for the first half of his career, has spent much of the past twenty years looking at the emerging new field known as mathematical cognition, which tries to understand how the human brain does mathematics, how it acquires mathematical ability in the first place, and how mathematical thinking combines with other forms of reasoning, including machine computation. In working together on this book, written to explain to those not in the field what experimental mathematics is and how it is done, Borwein was on the inside looking out, and Devlin was on the outside looking in. We saw reassuringly similar scenes.

Experimental mathematics is fairly new. It is a way of doing mathematics that has been made possible by fast, powerful, and easy-to-use computers, by networks, and by databases.

The use of computers in mathematics for its own sake is a recent phenomenon - much more recent than the computer itself, in fact. (This surprises some outsiders, who assume, incorrectly, that mathematicians led the computer revolution. To be sure, mathematicians invented computers, but then they left it to others to develop them, with very few mathematicians actually using them until relatively recently)

In fact, in the late 1980s, the American Mathematical Society, noting that mathematicians seemed to be lagging behind the other sciences in seeing the potential offered by computers, made a deliberate effort to make the mathematical community more aware of the possibilities presented by the new technology. In 1998, their flagship newsletter, the Notices of the American Mathematical Society, introduced a "Computers and Mathematics" section, edited originally by the late Jon Barwise, then (from October 1992 through December 1994) by Devlin. Devlin's interest in how the use of computers can change mathematical practice was part of his growing fascination with mathematical cognition. Correspondingly, Borwein's experience led to a growing interest in mathematical visualisation and mathematical aesthetics.

A typical edition of the "Computers and Mathematics" section began with a commissioned feature article, followed by reviews of new mathematical software systems. Here is how Devlin opened his first "Computers and Mathematics" section: "Experimental mathematics is the theme of this month's feature article, written by the Canadian mathematical brothers Jonathan and Peter Borwein."

With this book, the circle is complete!

**11.4. Review by: Samuel S Wagstaff Jr.**

*Mathematical Reviews*MR2464847

**(2009j:00003)**.

Alchemists used to mix various substances together in a crucible and heat them to a high temperature to see what happened. Likewise, today's experimental mathematicians put numbers, formulas and algorithms into a computer and hope for interesting results. This book discusses many examples in which computer experiments have advanced mathematics.

Chapter 1 attempts to define experimental mathematics and discusses its role within mathematics. Chapters 2 and 7 present formulas for computing π and ways to discover efficient new formulas to do this. In Chapter 3, the authors tell how to identify a real number from a decimal approximation to it, often computed from an infinite series. This method is applied in Chapter 4 to study the irrationality of the Riemann zeta function at positive integers. Chapter 5 gives examples of integrals evaluated both numerically and formally by computer. In Chapter 6, the authors present some unexpected results that arose in computer experiments with infinite series or sequences. Chapter 8 gives examples of discoveries in which a computer algebra system, like Maple or Mathematica, evaluated something using methods more powerful than those known to the user. Chapter 9 studies ways to evaluate limits exactly. Chapter 10 gives examples in which a computer algebra system gave an answer that was misleading in some way. Chapter 11 presents a variety of examples from geometry and visualisation.

Each chapter ends with some computational exercises called "Explorations" related to the subject matter of that chapter. Answers to these exercises and comments about them are given in a final chapter.

**12. Convex Functions: Constructions, Characterizations and Counterexamples (2010), by Jonathan M Borwein and Jon D Vanderwerff**.

**12.1. From the Publisher.**

Like differentiability, convexity is a natural and powerful property of functions that plays a significant role in many areas of mathematics, both pure and applied. It ties together notions from topology, algebra, geometry and analysis, and is an important tool in optimisation, mathematical programming and game theory. This book, which is the product of a collaboration of over 15 years, is unique in that it focuses on convex functions themselves, rather than on convex analysis. The authors explore the various classes and their characteristics and applications, treating convex functions in both Euclidean and Banach spaces. The book can either be read sequentially for a graduate course, or dipped into by researchers and practitioners. Each chapter contains a variety of specific examples, and over 600 exercises are included, ranging in difficulty from early graduate to research level.

**12.2. From the Preface.**

This book on convex functions emerges out of 15 years of collaboration between the authors. It is far from being the first on the subject nor will it be the last. It is neither a book on convex analysis such as Rockafellar's foundational 1970 book nor a book on convex programming such as Boyd and Vandenberghe's excellent recent text. There are a number of fine books - both recent and less so - on both those subjects or on convexity and relatedly on variational analysis. [Various books] complement or overlap in various ways with our own focus which is to explore the interplay between the structure of a normed space and the properties of convex functions which can exist thereon. In some ways, among the most similar books to ours are those of Phelps and of Giles in that both also straddle the fields of geometric functional analysis and convex analysis - but without the convex function itself being the central character.

We have structured this book so as to accommodate a variety of readers. This leads to some intentional repetition. Chapter 1 makes the case for the ubiquity of convexity, largely by way of examples, many but not all of which are followed up in later chapters. Chapter 2 then provides a foundation for the study of convex functions in Euclidean (finite-dimensional) space, and Chapter 3 reprises important special structures such as polyhedrality, eigenvalue optimisation and semidefinite programming.

Chapters 4 and 5 play the same role in (infinite-dimensional) Banach space. Chapter 6 comprises a number of other basic topics such as Banach space selection theorems, set convergence, integral functionals, trace-class spectral functions and functions on normed lattices.

The remaining three chapters can be read independently of each other. Chapter 7 examines the structure of Legendre functions which comprises those barrier functions which are essentially smooth and essentially strictly convex and considers how the existence of such barrier functions is related to the geometry of the underlying Banach space; as always the nicer the space (e.g. is it reflexive, Hilbert or Euclidean?) the more that can be achieved. This coupling between the space and the convex functions which may survive on it is attacked more methodically in Chapter 8.

Chapter 9 investigates (maximal) monotone operators through the use of a specialised class of convex representative functions of which the Fitzpatrick function is the progenitor. We have written this chapter so as to make it more usable as a stand-alone source on convexity and its applications to monotone operators.

In each chapter we have included a variety of concrete examples and exercises - often guided, some with further notes given in Chapter 10. We both believe strongly that general understanding and intuition rely on having fully digested a good cross- section of particular cases. ...

We think this book can be used as a text, either primary or secondary, for a variety of introductory graduate courses. ... We hope also that this book will prove valuable to a larger group of practitioners in mathematical science; and in that spirit we have tried to keep notation so that the infinite-dimensional and finite-dimensional discussion are well comported and so that the book can be dipped into as well as read sequentially. This also requires occasional intentional redundancy. In addition, we finish with a 'bonus chapter' revisiting the boundary between Euclidean and Banach space and making comments on the earlier chapters.

**12.3. Review by: Heinz H Bauschke.**

*Mathematical Reviews*MR2596822

**(2011f:49001)**.

The systematic study of convex functions - and, more broadly, convex analysis and optimisation - can be traced back to a set of lecture notes by W Fenchel from 1951. The theory was greatly expanded upon in the 1960s in many works by J-J Moreau and by R T Rockafellar, and it continues to be a very active area of research today.

In the 521-page book under review, Borwein and Vanderwerff aim their focus on convex functions themselves. The book is organised into ten chapters, with extensive chapter notes and numerous exercises. A brief overview follows.

...

Borwein and Vanderwerff's book is particularly impressive due to its enormous breadth and depth. Necessarily, not all results are developed in complete detail - in fact, many proofs are left as exercises - which makes the text less ideal as a reference work; however, complete details would have easily at least doubled the number of pages. Some of the results are revisited, especially when simpler proofs are available in more restrictive settings. It is a beautiful experience to browse this inspiring book. The reviewer has not seen any source which is even close to presenting so many different and interesting convex functions and corresponding results. The book does contain typos but the authors provide a very detailed and complete list of errata, additional notes, and even solutions to many exercises on their website.

In summary, this delightful book is a most welcome addition to the library of any convex analyst or of any mathematician with an interest in convex functions.

**12.4. Review by: John R Cook**.

*Mathematical Association of America*(20 April 2010).

Jonathan Borwein and Jon Vanderwerff have written a valuable resource entitled

*Convex Functions: Constructions, Characterizations and Counterexamples*. As the authors note in the preface, the book is the result 15 years of collaboration; it is not a light read. However, neither is

*Convex Functions*a dry reference. It is a textbook written to guide the reader through the material.

*Convex Functions*tells a story from beginning to end. It starts with examples of convex functions in order to motivate the reader. It then progresses further and further into the theory, introducing special cases before proceeding to more general theory. The book closes with a retrospective, revisiting the differences between convex functions over finite and infinite dimensional spaces. The authors introduce a small amount of redundancy to make the book easier to read.

One way to read

*Convex Functions*is as a tour through a large amount of real and functional analysis, as seen from the perspective of convex functions. The book opens with simple geometric problems in Euclidean space and leads up to results on the classification of Banach spaces and the theory of monotone operators. As the authors suggest, the book could be used for a less advanced class by covering the early chapters and picking out the finite dimensional portions of the latter chapters.

**13. Selected Writings on Experimental and Computational Mathematics (2010), by Jonathan M Borwein and Peter B Borwein.**

**13.1. From the Publisher.**

A quiet revolution in mathematical computing and scientific visualisation took place in the latter half of the 20th century. These developments have dramatically enhanced modes of mathematical insight and opportunities for "exploratory" computational experimentation. This volume collects the experimental and computational contributions of Jonathan and Peter Borwein over the past quarter century.

**13.2. From the Preface.**

This is a representative collection of most of our writings about the nature of mathematics over the past twenty-five years. Many are jointly authored, others are by one or other of us with various co-authors - often with our long-time collaborator and friend David Bailey who has generously written a foreword to the volume. We have included some technical papers only when they form the basis for discussion in the more general papers collected. We would like to thank Joshua Borwein for writing the index for our book.

For each article in this selection, we have written a few paragraphs of introduction situating the given paper in the collection and where appropriate bringing it up to date. For the most part, we let the articles speak for themselves and suggest the reader consult the index which we have added to improve navigation between selections.

**13.3. Foreword by David Bailey.**

Great things often begin in humble ways.

My 25-year-long collaboration with Jonathan and Peter Borwein began in 1985, when I read their paper on fast methods for computing elementary constants and functions, which article is included as the first chapter of this volume. After reading this engaging and intriguing article, I was inspired to try to implement some of these algorithms on the computer, using a high-precision arithmetic software package that I wrote specifically for this task. Indeed, the Borwein algorithms worked as advertised, yielding many thousands of correct digits in just a few seconds on a state-of-the-art computer at the time. I was particularly intrigued by the Borwein algorithm for

*π*. I then contacted the Borweins to tell of my interest in their work, and, as they say, the rest is history.

As can be readily seen in the articles here, the Borweins are inarguably the world's leading exponents of utilising state-of-the-art computer technology to discover and prove new and fundamental mathematical results. They employ raw numerical computation, symbolic processing and advanced visualisation facilities in their work. The Borweins have spread this gospel in countless fascinating and cogent lectures, as well as in numerous books and over 200 published papers. As a direct result of the Borweins' influence, hundreds of researchers worldwide are now engaged in "experimental" and computationally-assisted mathematics, and the pace of mathematical discovery has measurably quickened.

In reading through these papers, I am struck that the Borweins have also made an important contribution to a more fundamental problem in the field: How can researchers, labouring at the state of the art in the very difficult and demanding arena of modern mathematics, communicate the excitement of their work to young minds who potentially will form the next generation of mathematicians?

The Borweins have found the answer: (a) Bring computers into all aspects of the mathematical research arena, thereby attracting thousands of young, computer-savvy students into the fold, and letting them experience first-hand the excitement of discovering heretofore unknown facts of mathematics; and (b) Highlight the numerous intriguing connections of this work to other fields of mathematics, computer science and modern scientific philosophy.

These articles exemplify the exploratory spirit of truly pioneering work. They are destined to be read over and over again for decades to come. What's more, in most cases these papers can be read and comprehended even by persons of modest mathematical training. Enjoy!

**14. Modern Mathematical Computation with Maple (2011), by Jonathan M Borwein and Matthew Skerritt.**

**14.1. From the Preface.**

Thirty years ago mathematical, as opposed to applied numerical, computation was difficult to perform and so relatively little used. Three threads changed that: the emergence of the personal computer; the discovery of fibre-optics and the consequent development of the modern internet; and the building of the Three "M's" Maple, Mathematica and Matlab.

We intend to persuade that Mathematica and other similar tools are worth knowing, assuming only that one wishes to be a mathematician, a mathematics educator, a computer scientist, an engineer or scientist, or anyone else who wishes/needs to use mathematics better. We also hope to explain how to become an "experimental mathematician" while learning to be better at proving things. To accomplish this our material is divided into three main chapters followed by a postscript. These cover elementary number theory, calculus of one and several variables, introductory linear algebra, and visualisation and interactive geometric computation.

**14.2. Review by: David S Mazel.**

*Mathematical Association of America*(30 June 2012).

In

*An Introduction to Modern Mathematical Computing with Maple*, Borwein and Skerritt show that computers are an excellent companion for learning mathematics. They do not do this with an essay on the advantages of computers, such as, fewer sign errors or quicker algebraic manipulations (both of which are true). Rather, they show readers that a particular computer algebra program, in their case Maple, is so flexible and powerful that it can work alongside students to show them insights that may be otherwise difficult to see.

To that end, the authors go through many aspects of Maple such as: algebraic manipulations, graphics, matrix manipulations, integration, differentiation, sums, limits, and number theory. Their treatment of the program is thorough, well explained, and instructive. This book is a good companion to the user manual and, maybe even better than the manual on some topics because the authors' examples are succinct and clearly illustrate many facets of the program. (It is not a replacement for the user's guide, however.)

The theme of the book is that Maple can supplement mathematics learning and, what is more, can do much of the mathematics for the students. That is certainly true. What is missing in this book is just how will Maple actually help students understand mathematics when the students are still learning the maths.

**15. An Introduction to Modern Mathematical Computing with Mathematica (2012), by Jonathan M Borwein and Matthew Skerritt.**

**15.1. From the Preface.**

Identical to 14.1 above.

**16. Exploratory experimentation in mathematics: Selected works (2012), by David H Bailey and Jonathan M Borwein.**

**16.1. From the Publisher.**

The authors - pioneers in their field - have collaborated on the subject of experimental mathematics for a quarter of a century. The book collects sixteen articles written together or separately and with co-authors. These works reflect their work on and their views about the changing face of computer-assisted "high-performance" mathematics.

**16.2.Review by: Richard E Crandall.**

In the field of experimental mathematics, one discovers truths in much the same way that new particles can b e discovered in a physics laboratory; or in the manner by which DNA was discovered... to be of helical structure.... Bailey and Borwein show how to use numerical tools and intuition to bear mathematical fruit.

**17. Lattice Sums Then and Now (2013), by Jonathan M Borwein, M L Glasser, R C McPhedran, J G Wan and I J Zucker.**

**17.1. From the Publisher.**

The study of lattice sums began when early investigators wanted to go from mechanical properties of crystals to the properties of the atoms and ions from which they were built (the literature of Madelung's constant). A parallel literature was built around the optical properties of regular lattices of atoms (initiated by Lord Rayleigh, Lorentz and Lorenz). For over a century many famous scientists and mathematicians have delved into the properties of lattices, sometimes unwittingly duplicating the work of their predecessors. Here, at last, is a comprehensive overview of the substantial body of knowledge that exists on lattice sums and their applications. The authors also provide commentaries on open questions, and explain modern techniques which simplify the task of finding new results in this fascinating and ongoing field. Lattice sums in one, two, three, four and higher dimensions are covered.

**17.2. From the Preface.**

The study of lattice sums is an important topic in mathematics, physics, and other areas of science. It is not a new field, dating back at least to the work of Appell in 1884, and has attracted contributions from some of the most eminent practitioners of science (M Born and A Landé). Despite this, it has not been widely recognised as a field with its own important tradition, results, and techniques. This has led to independent discoveries and rediscoveries of important formulae and methods and has impeded progress in some topics owing to the lack of knowledge of key results.

In order to solve this problem, Larry Glaser and John Zucker published in 1980 a seminal paper, the first comprehensive review of what was then known about the analytic aspects of lattice sums. This work was immensely valuable to many researchers but now is out of date and lacks the immediate electronic accessibility expected by today's researchers.

Hence, we have the genesis of the present project, the composition of this monograph. It contains a slightly corrected version of the paper of Glasser and Zucker as well as additions reflecting the progress of the subject since 1980. The emphasis of the results collected here is on analytic techniques for evaluating lattice sums and results obtained using them. We will nevertheless touch upon numerical methods for evaluating sums and how these may be used in the spirit of experimental mathematics to discover new formulae for sums. Several chapters in this monograph are based on published material and, as such, we have tried to retain their original styles. Each chapter has its own reference list while a complete bibliography is also provided at the end of the book.

**17.3. Review by: S L Kalla.**

*Mathematical Reviews*MR3135109

This book is well written and will encourage researchers to continue work in this field. Lattice sums will continue to be a topic of interest to coming generations of researchers.

**18. Neverending Fractions. An Introduction to Continued Fractions (2014), by Jonathan Borwein, Alf van der Poorten, Jeffrey Shallit and Wadim Zudilin.**

**18.1. From the Publisher.**

Despite their classical nature, continued fractions are a never-ending research area, with a body of results accessible enough to suit a wide audience, from researchers to students and even amateur enthusiasts. Neverending Fractions brings these results together, offering fresh perspectives on a mature subject. Beginning with a standard introduction to continued fractions, the book covers a diverse range of topics, from elementary and metric properties, to quadratic irrationals, to more exotic topics such as folded continued fractions and Somos sequences. Along the way, the authors reveal some amazing applications of the theory to seemingly unrelated problems in number theory. Previously scattered throughout the literature, these applications are brought together in this volume for the first time. A wide variety of exercises guide readers through the material, which will be especially helpful to readers using the book for self-study, and the authors also provide many pointers to the literature.

**18.2. From the Preface.**

This book arose from many lectures the authors delivered independently at different locations to students of different levels.

'Theory' is a scientific name for 'story'. So, if the reader somehow feels uncomfortable about following a theory of continued fractions, he or she might be more content to read the story of never-ending fractions.

The queen of mathematics - number theory - remains one of the most accessible parts of significant mathematical knowledge. Continued fractions form a classical area within number theory, and there are many textbooks and monographs devoted to them. Despite their classical nature, continued fractions remain a never-ending research field, many of whose results are elementary enough to be explained to a wide audience of graduates, postgraduates and researchers, as well as teachers and even amateurs in mathematics. These are the people to whom this book is addressed.

After a standard introduction to continued fractions in the first three chapters, including generalisations such as continued fractions in function fields and irregular continued fractions, there are six 'topics' chapters. In these we give various amazing applications of the theory (irrationality proofs, generating series, combinatorics on words, Somos sequences, Diophantine equations and many other applications) to seemingly unrelated problems in number theory. The main feature that we would like to make apparent through this book is the naturalness of continued fractions and of their expected appearance in mathematics. The book is a combination of formal and informal styles. The aforementioned applications of continued fractions are, for the most part, not to be found in earlier books but only in scattered scientific articles and lectures.

We have included various remarks and exercises but have been sparing with the latter. In the topics chapters we do not always give full details. Needless to say, all topics can be followed up in the end notes for each chapter and through the references.

...

Finally, Alf van der Poorten (1942-2010) died before this book could be brought to fruition. He was both a good friend and a fine colleague. We offer this book both in his memory and as a way of bringing to a more general audience some of his wonderful contributions to the area. Chapters 4, 5 and 6 originate in lectures Alf gave in the last few years of his life and, for matters of both taste and necessity, they are largely left as presentations in his unique and erudite style.

**18.3. Review by: Poj Lertchoosakul.**

*Mathematical Reviews*MR3468515

Continued fractions form a classical area within number theory, and their origin can be traced back to the Euclidean algorithm. Ever since John Wallis and many great mathematicians like Euler, Lambert, Lagrange and Gauss discovered various fundamental properties and important applications of continued fractions, these fascinating objects have been a very active field of research and have made their appearances in many other fields. Despite their classical nature, continued fractions remain a never-ending research area. The book under review arose from many lectures the authors delivered independently on different occasions to students of different levels. Its main aim is to provide an introduction to continued fractions for a wide audience of graduates, postgraduates and researchers, as well as teachers and even amateurs in mathematics. The book covers a diverse range of topics, including elementary and metric properties of continued fractions, quadratic irrationals and more exotic topics such as folded continued fractions and Somos sequences. In addition, it provides various applications of the theory to seemingly unrelated problems in number theory, such as irrationality proofs, generating series, combinatorics on words and Diophantine equations. This book is a combination of formal and informal styles of expository writing and a mixture of introductory textbook and topical survey. Many of the special topics and the aforementioned applications of continued fractions are not to be found in other books but only in scattered articles and lectures. As for full details with regard to these topics, the reader is referred to the original research papers listed in the rich bibliography. At the end of each chapter, the authors put a set of notes providing additional remarks and hints for further reading. Some figures of calculation and experiments are also presented to illustrate special topics throughout the book. Moreover, a wide variety of exercises are included to guide the reader to acquire complementary knowledge through independent work.

**19. Tools and mathematics: Instruments for learning (2016), by John Monaghan, Luc Troche and Jonathan Borwein.**

**19.1. From the Publisher.**

This book is an exploration of tools and mathematics and issues in mathematics education related to tool use. The book has five parts. The first part reflects on doing a mathematical task with different tools, followed by a mathematician's account of tool use in his work. The second considers prehistory and history: tools in the development from ape to human; tools and mathematics in the ancient world; tools for calculating; and tools in mathematics instruction. The third part opens with a broad review of technology and intellectual trends, circa 1970, and continues with three case studies of approaches in mathematics education and the place of tools in these approaches. The fourth part considers issues related to mathematics instructions: curriculum, assessment and policy; the calculator debate; mathematics in the real world; and teachers' use of technology. The final part looks to the future: task and tool design and new forms of activity via connectivity and computer games.

**19.2. Review by: Woong Lim.**

*Mathematical Association of America*(1 October 2017).

Classroom teachers may find this book too theoretical and philosophical; this may be especially true when looking for practical ideas to teach through the use of technology. I believe, however, that this book is a must-have reference for researchers and faculty in mathematics education. First, each chapter is rich with rich data, illustrations, history, mathematics, and critical perspectives, taking the reader into the minds of leading experts of tool use in mathematics. Second, and related, each part of the book has a discussion chapter that helps the reader reflect on preceding chapters and connect to emerging or related critical issues. In particular, I enjoyed reading the dialogues among the authors in these discussion chapters, as they provide intimate access to the thinking and humours of several brilliant minds. Third, the extensive references cited in the book could be an important source of future readings for novice academics, especially those within the U.S. context. Relying too much on recent studies for a literature review sometimes leads us to lose perspective on history and origin of the questions.

The authors exemplify how to capture the grand idea of tool use in mathematics education. Thanks to the breadth and depth of the book, this book will be successful as a course text or reference on your bookshelf. If you are a graduate student with interest in technology in mathematics education, you would appreciate the first six chapters to build your theoretical basis. I also recommend reading chapter 19 for new research ideas. If you are teaching mathematics education seminars, no matter what theories or practices your course may cover, I am confident you will find most chapters quite useful for a reading assignment.

**20. Pi: The Next Generation. A Sourcebook on the Recent History of Pi and its Computation (2016), by David H Bailey and Jonathan M Borwein.**

**20.1. About this book.**

This book contains a compendium of 25 papers published since the 1970s dealing with pi and associated topics of mathematics and computer science. The collection begins with a Foreword by Bruce Berndt. Each contribution is preceded by a brief summary of its content as well as a short key word list indicating how the content relates to others in the collection. The volume includes articles on actual computations of pi, articles on mathematical questions related to pi (e.g., "Is pi normal?"), articles presenting new and often amazing techniques for computing digits of pi (e.g., the "BBP" algorithm for

*π*, which permits one to compute an arbitrary binary digit of

*π*without needing to compute any of the digits that came before), papers presenting important fundamental mathematical results relating to

*π*, and papers presenting new, high-tech techniques for analysing

*π*(i.e., new graphical techniques that permit one to visually see if

*π*and other numbers are "normal").

This volume is a companion to

*Pi: A Source Book*whose third edition released in 2004. The present collection begins with 2 papers from 1976, published by Eugene Salamin and Richard Brent, which describe "quadratically convergent" algorithms for

*π*and other basic mathematical functions, derived from some mathematical work of Gauss. Bailey and Borwein hold that these two papers constitute the beginning of the modern era of computational mathematics. This time period (1970s) also corresponds with the introduction of high-performance computer systems (supercomputers), which since that time have increased relentlessly in power, by approximately a factor of 100,000,000, advancing roughly at the same rate as Moore's Law of semiconductor technology. This book may be of interest to a wide range of mathematical readers; some articles cover more advanced research questions suitable for active researchers in the field, but several are highly accessible to undergraduate mathematics students.

**20.2. From the Preface.**

This volume is a companion to

*Pi: A Source Book*(by Lennart Berggren, Jonathan Borwein, and Peter Borwein, Springer-Verlag), which was first published in 1997, with a third edition released in 2004.

Rather than produce an even heftier fourth edition, the current authors have prepared a collection of papers written between 1975 and the present. Since a number of the collected papers contain substantial historical material, the reader can glean an accurate picture of the life of π from the current volume. That said, the focus in this book is on "

*π*in the digital age." The reader will note that many of the papers have substantial algorithmic material and it is recommended that where possible he or she explore such material at the computer.

Each of the 25 papers comprising this volume is preceded by a brief summary of its contents, and this is accompanied by a very brief, key word, indication of some of the ways the content of the given paper relates to that of others in the collection.

For the most part, however, we are happy to let the papers speak for themselves. The present authors have been fascinated by

*π*throughout their academic lives and hope that this volume will help readers share this fascination and potentially even contribute to the ever-growing literature on the subject.

We are also delighted that Bruce Berndt agreed to add a Foreword to the volume.

**20.3. Review by: Thomas Sonar.**

*London Mathematical Society Newsletter*(November 2017).

*Pi: The Next Generation*is compiled as a sourcebook on the recent history of π from 1975 on, and on computational issues. ... Reading the papers in this book I found many aspects on the mathematics and history of

*π*which I did not know before and I enjoyed reading it very much. As the older book on

*π*this one will also soon become a standard reference tool for working mathematicians and historians of mathematics alike.

**20.4. Review by: Jeffrey O Shallit**.

*Mathematical Reviews*MR3559747.

This is a follow-up volume to a previous book [J L Berggren, J M Borwein and P B Borwein, Pi: a source book], containing reprints of 25 papers (or excerpts from books) concerning the celebrated real number

*π*= 3.14159... published since 1976. Each reprinted paper is accompanied by a brief introduction explaining its significance. The papers range from historical surveys to popular expositions to research articles.

Although I knew most of the papers already, I still found it delightful to browse at random. It would make a good selection for a high school or college library.

Unfortunately the quality of the (re)printing of some of the earlier papers is not very high, making them hard to read.

**20.5. Review by: Adhemar Bultheel.**

*European Mathematical Society Newsletter*(23 August 2016).

In

*Pi. A source book*the editors L Berggren, J Borwein and P Borwein, assembled a number of reprints that sketch the history of pi, its mathematical importance and the broad interest that it has received through the centuries from the Rhind papyrus till modern times. The last edition (3rd edition, 2004, to which I will refer as SB3) added several papers that related to the computation of the digits of

*π*by digital computers. Rather than extending this with more recent developments (SB3 was already some 800 pages), it was decided to collect this computational aspect in a new volume. This "The next generation" volume got the rightful subtitle "a source book on the recent history of

*π*and its computation". Because it extends the papers on digital computation that were added in SB3, the trailing papers of SB3 are reprised here. The papers are ordered chronologically, so of the first 14 papers in this book, 12 were already at the end of SB3.

...

Many of the papers have authors that are the main players in the field: David Bailey, Bruce Berndt, and Jonathan and Peter Borwein. As this book was being printed one of its editors, Jonathan (Jon) Borwein, passed away on 2 August 2016. So it was probably too late to add a dedication or a note in this book. This collection he helped to compile and containing several papers that he co-authored, can be considered one of his last gifts to the scientific community.

**20.6. Review by: Allen Stenger.**

*Mathematical Association of America*(11 October 2016).

... the book consists of reprints of the most important papers in the field. (Some of you may be surprised to hear that π is a field, but with several entire scholarly and popular books on it, I think it qualifies.) Most of the papers deal with calculating

*π*to many digits of precision, dealing both with the computer implementation and with new algorithms. In the 1970s very fast parallel and vector computers were becoming available, and there are several papers with detailed studies of how these new architectures can be used to calculate

*π*. There are also a few papers on the question of whether

*π*is a normal number (in other words, whether the digits are asymptotically equi-distributed), some surveys of

*π*and its lore, and a few miscellaneous papers on the irrationality of

*π*and on spigot algorithms (algorithms that produce the exact digits one at a time, rather than producing successively-better approximations and reading the digits from there).

The new book follows the same layout and selection criteria as its predecessor, so if you liked that one, you'll like this one too. Very Good Feature: an extensive index (this is unusual for collections of papers).

Last Updated September 2023