# Peter Benjamin Borwein

### Quick Info

St Andrews, Fife, Scotland

Burnaby, British Columbia, Canada

**Peter Borwein**was a Scottish born mathematician who spent most of his career in Canada. He is perhaps best known for his research and publications on π.

### Biography

**Peter Borwein**was the son of David Borwein and Bessie Flax. David Borwein had been born in Lithuania but was brought up and educated to B.Sc. level in South Africa. He married Bessie, who was studying Botany and Zoology at the University of Witwatersrand, on 30 June 1946 in Yeoville, Johannesburg. They emigrated to Britain in 1948 and David was awarded a Ph.D. in mathematics from the University of London in 1950. In the same year he was appointed as a lecturer in mathematics at the University of St Andrews in Scotland. In St Andrews the Borwein family lived at Alvie, Buchanan Gardens (1950-1953), West View Cottage, Lade Braes Lane (1954-1958), and then 4 Abbotsford Place (1959). They had a house built at 10 Strathkinness High Road (West Acres) and moved into the house in 1960. David and Bessie's three children were all born in St Andrews: Jonathan Michael Borwein (known as Jon), an exceptionally talented mathematician, was born on 20 May 1951; Peter Benjamin Borwein, the subject of this biography, was born on 10 May 1953; and Sarah Tanya Borwein, who became a medical doctor, was born in 1961.

Jonathan and Peter had a happy childhood. Peter Borwein said [10]:-

I have fond memories of growing up with Jon, as young boys in St Andrews, Scotland. An early memory ... is of us fishing off the pier in St Andrews with our dad David. We would catch rock cod, and make my mother Bessie clean them and make fish cakes. Jon and I also spent much of our childhood with our guinea pigs, Gilbert and Gulliver. My dad built a cage for them with no bottom, so that the guinea pigs would move around the lawn, acting as lawnmowers. As young boys we also shared a bedroom in St Andrews. I used to keep Jon awake at night, by reminding him that the universe was infinite. This bothered him to no end. It's no surprise that he ended up as a mathematician.In fact not only did the boys share a bedroom but, from 1954 to 1957, they shared a bed (end-to-end). Because of their mother's zoology interests, the boys often had unusual pets, for example a Xenopus Toad. St Andrews is famed for its beaches with golden sands. During the summer months, children had donkey rides on the West Sands. This proved a problem for Peter, however, since he loved donkeys but hated sand! Another thing that he loved to do was to camp in the garden [8]:-

As kids a big treat was to be allowed to camp and freeze in the garden - in a one-man tent that had been to the top of K2 in the early 1950s.David Borwein attended the International Congress of Mathematicians in Edinburgh in August 1958 and delivered the short communication

*Multiplication of $(C, -\kappa)$-summable series*. Five year old Peter went to Edinburgh with his father and was registered as a participant and given a name badge. He began his studies in the Madras Primary (called the 'Kinder') immediately after the Congress and was proud to wear his Congress name badge, pinned on his Madras blazer, on his first day at school. In 1959 he caught scarlet fever. At that time it was a notifiable disease which had to be reported to the public-health authorities. Peter had to be quarantined in his home for a month.

In September 1963 the Borwein family moved to London, Ontario, Canada when David Borwein joined the Mathematics Department at the University of Western Ontario. Peter continued his education at London Central Secondary School at the corner of Dufferin Avenue and Waterloo Street. He graduated in 1971 and began his studies at the University of Western Ontario. His brother Jonathan writes [8]:-

Peter lived with a future CBC Morningside producer, an ex porn publisher, and a large python.In the summer of 1972 Peter Borwein and his brother Jonathan travelled in Europe following the "ten dollars a day" book. Back at the University of Western Ontario, in 1973 he met Jennifer Moore who was studying psychology. In the following year he graduated with a B.Sc. from the University of Western Ontario and, together with Jennifer, he went to the University of British Columbia in Vancouver to study for his Master's Degree. His advisor was John James Francis Fournier who had been awarded a Ph.D. by the University of Wisconsin-Madison in 1967 for his thesis

*Extensions of a Fourier Multiplier Theorem of Paley*. In 1976 Borwein was awarded an M.Sc. from the University of British Columbia and continued to study there for a Ph.D. advised by David W Boyd. Boyd had been awarded a B.Sc. from Carlton University in 1963 and a Ph.D. from the University of Toronto in 1966 for his thesis

*The Hilbert Transformation on Rearrangement Invariant Banach Spaces*. After appointments at the University of Alberta and the California Institute of Technology, he had been appointed to the University of British Columbia in 1971.

In the summer of 1978 Peter and Jonathan again travelled in Europe. Peter's research was going well and he published his first paper in 1978, namely

*Arbitrarily Slow Rational Approximations on the Positive Real Line*. The paper begins:-

This paper will exhibit positive, non-decreasing, infinitely differentiable functions f with the property that the best rational approximations of degree n in the supremum norm to 1/f on [0, ∞) tend to zero arbitrarily slowly. Furthermore, such f can be chosen to have very general growth characteristics at infinity. In particular, this demonstrates that two conjectures of Erdős and Reddy are false.It contains the acknowledgement:-

The author wishes to thank Dr D W Boyd for many useful discussions.In 1979 Peter Borwein was awarded a Ph.D. for his thesis

*Restricted Uniform Rational Approximations*. He published the paper

*Approximations by Rational Functions with Positive Coefficients*(1980) which considers the problem of approximating analytic functions with positive coefficients by rational functions with positive coefficients. He gives the following acknowledgement:-

Some of the results of this paper occur in the author's Ph.D. dissertation at the University of British Columbia under the supervision of Dr D Boyd, to whom the author wishes to express his gratitude.This was one of three papers which he published in 1980 containing results from his thesis. Supported by the National Science and Engineering Research Council of Canada, he then spent the year 1979-80 in England as a postdoctoral student at the University of Oxford. During that year he submitted for publication the paper

*Approximation of $x^{n}$ by reciprocals of polynomials*(1981).

Back in Canada after the year in England, Peter Borwein was appointed to Dalhousie University in 1980 joining his brother Jonathan who was already on the faculty there. Later that year, on Thursday, 27 November 1980, Peter married Jennifer: they had three daughters, Alexandra Borwein (born 1987, known as Alex), Sophie Borwein (born 1989), and Tessa Borwein (born 1993).

The brothers were now producing excellent research papers and both were keen on collaborations, yet they had not worked together on any mathematics projects up to this time. In fact, surprisingly, they had not even discussed mathematics at any time in their lives up to this point. Now both were on the faculty at Dalhousie University in Halifax, Nova Scotia, this changed with Jonathan and Peter making up for lost time and undertaking many joint research collaborations. Before we look at this aspect of Peter Borwein's career, perhaps we should give a few details about Peter's sister Sarah Borwein. Sarah was ten years younger than Jonathan and eight years younger than Peter so by the start of the 1980s she was an undergraduate student. With a father and two brothers all enthusiastic mathematics researchers, it was natural for Sarah to move towards that topic. She completed a mathematics and economics degree from Queen's University in Kingston, Ontario, Canada and then went to Oxford, England on a Commonwealth Scholarship where she completed an M.Sc. in economics. She saw the passion for mathematics that her father and two brothers had and realised that, although she had the ability, she did not have that passion. She studied medicine, completed an MD at the University of Toronto, married the computer scientist Andrew Nevin, and became a medical doctor in Hong Kong.

Peter and Jonathan Borwein's first joint paper was

*A very rapidly convergent product expansion for*π (1983). The paper begins:-

There has recently been considerable interest in an essentially quadratic method for computing π. The algorithm, first suggested by Salamin, is based upon an identity known to Gauss. This iteration has been used by two Japanese researchers, Y Tamura and Y Kanada, to compute $2^{23}$ decimal digits of π in under 7 hours. They have now successfully computed $2^{24}$ digits (more than 16.7 million places). This is reported in the January 1983 'Scientific American' and the February 1983 'Discover Magazine'. In the process of surveying this and related fast methods of computing elementary functions, the authors discovered a new quadratically convergent product expansion for π. Our algorithm, like Salamin's, is intimately related to the Gaussian arithmetic-geometric mean iteration. However, it requires considerably less elliptic function theory to establish.This was the first of many joint papers on computing π, the next few being:

*The arithmetic-geometric mean and fast computation of elementary functions*(1984);

*Cubic and higher order algorithms for π*(1984);

*Explicit algebraic nth order approximations to π*(1984);

*More quadratically converging algorithms for π*(1986); and

*An explicit cubic iteration for π*(1986). In 1987 they published the book

*Pi and the AGM*which demonstrated not only their mathematical skills but also their ability to explain and entertain. Peter Cass writes [11]:-

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.Jet Wimp writes [26]:-

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 ...For more information about this book, and other books by Peter Borwein, see THIS LINK.

Now we must not give the impression that Peter Borwein's publications were all with his brother from 1983 onwards. MathSciNet lists 35 joint publications by Jonathan and Peter Borwein, while a total of 183 publications by Peter Borwein are listed. David Bailey writes in [4]:-

Peter Borwein's published research spanned classical analysis, computational number theory, Diophantine number theory, symbolic computing, experimental mathematics and applied mathematics. This research includes several rather important results. As one example, Peter Borwein, together with Tamás Erdélyi, Ronald Ferguson, and Richard Lockhart, settled Littlewood's Problem 22. ... In another notable example, Peter, in collaboration with Edward Dobrowolski and Michael J Mossinghoff, solved a problem of Lehmer on polynomials with odd coefficients.Peter Borwein's most remarkable result on π was, in fact, not done in collaboration with his brother. This is in the paper

*On the rapid computation of various polylogarithmic constants*(1997) written in collaboration with David Bailey and Simon Plouffe. The paper [4]:-

... arose when Peter posed the question to some students and postdocs of whether there was any economical way to calculate digits in some base of a mathematical constant such as π, beginning at a given digit position, without needing to calculate the preceding digits. Peter himself subsequently found [a] surprisingly simple scheme for binary digits of log 2 ...Both brothers lectured at the Ramanujan Centennial International Conference held 15-18 December 1987 in Annamalai Nagar in the Cuddalore district of the state of Tamil Nadu, India.

Peter Borwein was one of the leading figures in a strike at Dalhousie University in 1988. The faculty voted to strike in September of that year and the strike took place from 4 November to 22 November. The majority of faculty, including both Borwein brothers, did not teach during the strike. Peter Borwein was in charge of public relations and created the slogan for the strike, "Building a better university - sorry for the inconvenience!"

Both Peter Borwein and his brother Jonathan joined the Department of Mathematics of Simon Fraser University in 1993 and they founded the Centre for Experimental and Constructive Mathematics. This university is in Burnaby, British Columbia, Canada. It was in this year that Borwein was awarded the first of many prestigious prizes. The Chauvenet Prize 1993 was awarded jointly to Peter Borwein, Jonathan Borwein and David Bailey for their paper

*Ramanujan, Modular Equations, and Approximations to Pi, or, How to Compute One Billion Digits of Pi*which had been published in the

*American Mathematical Monthly*. The Chauvenet Prize is awarded by the Mathematical Association of America and that Association also awarded the same three authors their Merten M Hasse Prize 1993 for the same paper. Peter Borwein received further awards such as 'Academic of the Year for 1996' from the Confederation of University Faculty Associations of British Columbia, the University of Western Ontario National Alumni Merit Award 1999, and the Lester R Ford Award 2002, jointly with Loki Jorgenson, for their paper

*Visible Structures in Number Theory*in the

*American Mathematical Monthly*.

Peter Borwein also gave much to the mathematical community in his role as a journal editor. He served on the Editorial Boards of:

*Mathematics of Computation*;

*Canadian Journal of Mathematics*;

*Journal of Approximation Theory*;

*Computational Complexity*;

*Ramanujan Quarterly*; and

*Electronic Transactions in Numerical Analysis*.

On 6 July 1999 Borwein gave a lecture in Budapest. This report of the lecture by Peter Cameron illustrates his humour:-

Peter Borwein gave a lovely talk interspersed with cartoons. My favourite was the research institute with two signs: "Unanswered questions" and "Unquestioned answers." He asked, "If Erdős had got the job at the IAS in Princeton, would he have touched our lives in the way he did?"In 2004 Peter Borwein founded the Centre for Interdisciplinary Research in the Mathematical and Computational Sciences (IRMACS) at Simon Fraser University. He became the director of IRMACS. Carol Thorbes writes [25]:-

Move over creators of Max Head-room, Matrix and Metropolis. What researchers can accomplish at Simon Fraser University's IRMACS centre rivals the high tech feats of the most memorable futuristic films.Already by the time IRMACS was up and running, Peter Borwein had made known that he was suffering from multiple sclerosis. In fact he had been diagnosed with multiple sclerosis in 1999 but it only became widely known in 2004. David Bailey writes that multiple sclerosis [3]:-

The $14 million centre's acronym stands for Interdisciplinary Research in the Mathematical and Computational Sciences. The centre's expansive view of the Lower Mainland from atop Burnaby Mountain echoes its limitless potential as a facility dedicated to fostering interdisciplinary research among scientists whose primary laboratory tool is the computer.

A newly constructed 2,500 square metre space atop the applied sciences building, the centre has eight labs, five meeting rooms and a presentation theatre, seating up to 100 people. They are equipped with easily upgradable computational, multimedia, internet and remote conferencing (including satellite) technology. High performance distributed computing and clustering technology, designed at SFU, and access to WestGrid, an ultra high-speed, interprovincial network, with shared computing and multimedia resources, make IRMACS unique in Western Canada.

... left him confined to a wheelchair, increasingly unable to pursue his research, and increasingly dependent on family and caregivers. I recall visiting Peter in January 2019 at his home in Burnaby, British Columbia. In spite of his paralysis and infirmity, I was astonished at his pleasant demeanour and ever-present humour. Would that we could all bear our misfortunes with as much strength and courage!Tributes to Peter Borwein at his 60th birthday meeting in May 2013 were by Kevin Hare, Karl Dilcher and Mike Mossinghoff. They appear in [8] and, in shortened form, in [4]. Kevin Hare said:-

I guess some of the things that come to mind are how my first paper with Peter came about. We were at the International Symposium on Symbolic and Algebraic Computation conference in Vancouver. Peter came up to me over lunch and said something to the effect of "This afternoon looks really boring, did you want to go hiking instead?" Next thing I knew, he had organised a few people to go up the grouse grind. On the way up he decided to start asking me math questions. (And I guess more importantly, I was able to ask him math questions without him getting distracted by phone calls.)Karl Dilcher said:-

I found his approach to supervising was basically, toss lots of questions at me and hope that one of them sticks. If anybody has ever sat beside him during a conference talk, they know exactly what I mean by that. Or, I guess anybody that has been sitting with him in a pub/coffee shop while conference talks are going on also knows what I am talk about.

The only other thing I can think of really is his addiction to coffee. I remember going to a conference in Portland while he was driving us all in a mini-van. I suspect during what should have been a 6 hour drive, we stopped for coffee about 8 times. It was during this drive that he decided on the naming conventions for all of the Mitacs machines, (all after some type of coffee). Since then, he got his espresso machine in his office, and he is probably even more addicted.

Peter was my "official" Post Doctoral Fellow supervisor, and I believe I was his first Post Doctoral Fellow. Without his support and encouragement I would probably not be where I am now. What I admired (and still admire) most about Peter is the fact that he always has a problem, or problems, on his mind; he will ask you, prod you, share insights with you and be very persistent. When he has found a truly exciting and worthwhile problem, he will not let go until it is solved. He will hack away at it from all directions, will try to get others interested and involved (and often succeed in this), and more often than not he will eventually make substantial progress, either alone, or in collaboration with others.Mike Mossinghoff said:-

An example for this is the spectacular Bailey-Borwein-Plouffe formula. Already during his time at Dalhousie he often said that he wanted to find the 10 billionth (I believe) digit of pi without having to know all the previous digits. I had no doubt that he would eventually succeed, and I consider Bailey-Borwein-Plouffe as a result of this (though hexadecimal instead of decimal).

Peter likes funny anecdotes, often related to mathematics or mathematicians. [One] I remember: A visit with daughter Alex as a toddler to Point Pleasant Park, when Alex pointed to the Sailors Memorial and said, 'look, Daddy, a big plus!' Peter was and is an excellent writer and expositor. But there wasn't a name, usually foreign, that he wouldn't find a way to misspell. One time in the mid-90s I got a (positive) referee report in which every name was incorrect; I KNEW that Peter had to be the referee. When I gave him an off-print later on, I remarked that I thought he might already be familiar with the paper. Peter, very surprised, "How do you know I refereed it?" I had to tell him the truth.

It's always a pleasure to visit Peter and his family and friends in "Mahler measure heaven", as Jeff Vaaler has called Vancouver. But one of my most memorable meetings with Peter was in Nashville, at an approximation theory meeting in the late 90's. I was just passing through, driving from North Carolina to Texas, but my wife and I briefly crashed the conference to join Peter for lunch at a local Indian buffet. I'd visited Simon Fraser University the prior summer, and we'd had some correspondence on some problems. But it was at lunch in Nashville that we hammered out plans for our first joint paper.A great sadness to Peter Borwein was the sudden death of his brother Jonathan in 2016. Peter died of pneumonia on Sunday 23 August 2020 in Ward 3D of Burnaby Hospital with his wife Jennie by his side. The pneumonia was a secondary condition caused by the multiple sclerosis which he [2]:-

Working with Peter has always been just as comfortable. Since that lunch meeting, I have visited Peter some six or seven times at Simon Fraser and the Centre for Experimental and Constructive Mathematics and IRMACS, and we now have some nine papers together.

Coming to Simon Fraser University was always an exciting time. Peter seems to have an incredible knack for suggesting just the right sort of problem to his collaborators - problems that are not only irresistible, but very well-suited to the listener's interests. He's a lot like Erdős in this respect. Each time I arrived in Vancouver, Peter always had a fascinating new project that was just irresistible to join.

... bore with remarkable grace and courage. He was until the end smart, funny, mischievous, kind, interesting and interested.

### References (show)

- G E Andrews, Review: Pi and the AGM - A Study of Analytic Number Theory and Computational Complexity, by Jonathan M Borwein and Peter B Borwein,
*Bull. Amer. Math. Soc.***22**(1) (1990), 198-201. - R Askey, Review: Pi and the AGM - A Study of Analytic Number Theory and Computational Complexity, by Jonathan M Borwein and Peter B Borwein,
*The American Mathematical Monthly***95**(9) (1988), 895-897. - D H Bailey, Peter Borwein dies at 67,
*Math Scholar*(29 August 2020).

https://mathscholar.org/2020/08/peter-borwein-dies-at-67/ - D H Bailey, Peter Borwein: A Visionary Mathematician,
*Notices Amer. Math. Soc.***70**(4) (2023), 610-613. - D H Bailey, J M Borwein, P B Borwein and S Plouffe, The Quest for Pi,
*NAS Technical Report NAS-96-015, NASA Ames Research Center*(June 1996). - D H Bailey, J M Borwein, P B Borwein and S Plouffe, The Quest for Pi,
*Mathematical Intelligencer***19**(1) (1997), 50-57. - B C Berndt, Review: Pi and the AGM - A Study of Analytic Number Theory and Computational Complexity, by Jonathan M Borwein and Peter B Borwein,
*Mathematics of Computation***50**(181) (1988), 352-354. - J M Borwein, Peter Borwein revisited,
*Computer Assisted Research Mathematics and Its Applications, University of Newcastle, Australia*(2008).

https://carmamaths.org/resources/jon/pbb.pdf - J M Borwein, P B Borwein and D H Bailey, Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi,
*The American Mathematical Monthly***96**(3) (1989), 201-219. - P Borwein, My brother Jon,
*Jonathan Borwein Memorial Website*(27 August 2016).

https://jonborwein.org/2016/08/my-brother-jon-by-peter-borwein/ - P Cass, Review: Pi and the AGM - A Study of Analytic Number Theory and Computational Complexity, by Jonathan M Borwein and Peter B Borwein,
*The Mathematical Gazette***83**(497) (1999), 334-335. - V Jungic, Peter Borwein 1953-2020,
*Canadian Mathematical Society Notes*.

https://notes.math.ca/en/article/peter-borwein-1953-2020/ - S V Konyagin, Review: Computational excursions in analysis and number theory, by Peter Borwein,
*Mathematical Reviews*MR1912495**(2003m:11045)**. - N Lord, Review: A dictionary of real numbers, by Jonathan Borwein and Peter Borwein,
*The Mathematical Gazette***74**(470) (1990), 395. - D S Lubinsky, Review: Polynomials and Polynomial Inequalities, by Peter Borwein and Tamás Erdelyi,
*Mathematical Reviews*MR1367960**(97e:41001)**. - R E O'Malley Jr, Review: Mathematicians on Creativity,
*SIAM Review***57**(1) (2015), 164. - Peter Benjamin Borwein (May 10, 1953-August 23, 2020),
*Constructive Approximation***52**(2020), 181-182. - Peter Borwein, Centre for Experimental and Constructive Mathematics,
*Simon Fraser University*.

http://www.cecm.sfu.ca/people/pm/pborwein.shtml - Publications by Peter Borwein, Centre for Experimental and Constructive Mathematics,
*Simon Fraser University*.

http://www.cecm.sfu.ca/~pborwein/PAPERS/papers.html - Remembering Peter Benjamin Borwein,
*Vancouver Sun*(29 August 2020)

https://vancouversunandprovince.remembering.ca/obituary/peter-borwein-1079935869 - A C Robin, Review: Pi: A source book (3rd edn.), by Lennart Berggren, Jonathan Borwein and Peter Borwein,
*The Mathematical Gazette***90**(518) (2006), 375-376. - SF News, A piece of pi: SFU mathematicians set two world records for calculating pi,
*SFU Mathematics and Statistics Newsletter*(June 1995). - S Strauss, Math professor figures kid can make pi history,
*Globe and Mail*(18 October 1995). - Talks by Peter Borwein, Centre for Experimental and Constructive Mathematics,
*Simon Fraser University*.

http://www.cecm.sfu.ca/~pborwein/SLIDES/slides.html - C Thorbes, Centre seen as 'serious nirvana',
*SFU News***32**(7) (7 April 2005), 31. - J Wimp, Review: Pi and the AGM - A Study of Analytic Number Theory and Computational Complexity, by Jonathan M Borwein and Peter B Borwein,
*SIAM Review***30**(3) (1988), 530-533. - W Van Assche, Review: Polynomials and Polynomial Inequalities, by Peter Borwein and Tamás Erdelyi,
*SIAM Review***38**(4) (1996), 705-706. - D Zeilberger, Review: Mathematics by Experiments, by J Borwein and D Bailey; and Experimentation in Mathematicsm by J Borwein, D Bailey and R Girgensohn,
*Rutgers University*(12 November 2004).

https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/mathexp.pdf

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Written by J J O'Connor and E F Robertson

Last Update September 2023

Last Update September 2023