Jump to ratings and reviews
Rate this book

The Calculus of Friendship: What a Teacher and a Student Learned about Life While Corresponding about Math

Rate this book
The Calculus of Friendship is the story of an extraordinary connection between a teacher and a student, as chronicled through more than thirty years of letters between them. What makes their relationship unique is that it is based almost entirely on a shared love of calculus. For them, calculus is more than a branch of mathematics; it is a game they love playing together, a constant when all else is in flux. The teacher goes from the prime of his career to retirement, competes in whitewater kayaking at the international level, and loses a son. The student matures from high school math whiz to Ivy League professor, suffers the sudden death of a parent, and blunders into a marriage destined to fail. Yet through it all they take refuge in the haven of calculus--until a day comes when calculus is no longer enough.



Like calculus itself, The Calculus of Friendship is an exploration of change. It's about the transformation that takes place in a student's heart, as he and his teacher reverse roles, as they age, as they are buffeted by life itself. Written by a renowned teacher and communicator of mathematics, The Calculus of Friendship is warm, intimate, and deeply moving. The most inspiring ideas of calculus, differential equations, and chaos theory are explained through metaphors, images, and anecdotes in a way that all readers will find beautiful, and even poignant. Math enthusiasts, from high school students to professionals, will delight in the offbeat problems and lucid explanations in the letters.

For anyone whose life has been changed by a mentor, The Calculus of Friendship will be an unforgettable journey.

184 pages, Hardcover

First published July 5, 2009

Loading interface...
Loading interface...

About the author

Steven H. Strogatz

14 books819 followers
Steven Strogatz is the Schurman Professor of applied mathematics at Cornell University. A renowned teacher and one of the world’s most highly cited mathematicians, he has been a frequent guest on National Public Radio’s Radiolab. Among his honors are MIT's highest teaching prize, membership in the American Academy of Arts and Sciences, and a lifetime achievement award for communication of math to the general public, awarded by the four major American mathematical societies. He also wrote a popular New York Times online column, “The Elements of Math,” which formed the basis for his new book, The Joy of x. He lives in Ithaca, New York with his wife and two daughters.

Ratings & Reviews

What do you think?
Rate this book

Friends & Following

Create a free account to discover what your friends think of this book!

Community Reviews

5 stars
86 (25%)
4 stars
136 (39%)
3 stars
101 (29%)
2 stars
15 (4%)
1 star
6 (1%)
Displaying 1 - 30 of 51 reviews
251 reviews10 followers
October 2, 2009
[As always, I rarely post my reviews of math books here because they are written for a mathy audience. That said, I loved this book so much I want to rave about it to anyone I can. It was wonderful:]

Anyone who has spent a significant amount of time as an educator knows about the special relationship that can develop between a teacher and a student, which can be especially striking when the relationship evolves and the student becomes the teacher. I know that I have former teachers -- as well as former students -- who I have kept in touch with long after our professional relationship was over, and these relationships are some of the most important in my life even to this day. Steven Strogatz, who is probably most well known for his book Sync as well as his contributions to the fields of chaos and dynamical systems, has written about one such relationship in a marvelous new book.

In The Calculus of Friendship, Strogatz writes about his relationship with Don Joffray and how it evolved over thirty years of letter writing from the time that Joffray taught Strogatz calculus at Loomis Chaffee, a prep school in Windsor Connecticut. The friendship between these two men was based largely on their love of mathematics in general, and calculus in particular. Over time they slowly revealed more and more of their personal lives with one another, but the bulk of their letters are devoted to discussing mathematics and the bulk of the book is devoted to reprinting the letters. For example, Strogatz barely mentions his first marriage or divorce to Joffray and Joffray only writes of his son's death in passing. On the other hand, they spend page after page writing about Hero's formula for the area of a triangle or Wallis's formula for 2⁄π.

Most of the chapters begin with Strogatz setting up a batch of letters by catching the readers up with the next phase of his life, or of Joffray's, to fill us in on the autobiographical background of the two men that they tend not to share with each other. Often he uses various mathematical topics -- ranging from chaos theory to the 'Monks on a Mountain' problem from a calculus course -- as metaphors for life and for their relationship. OK, I know what you're thinking, but somehow this is not as painful as it sounds, and Strogatz is able to get away with this type of writing in a way that feels both natural and heartwarming:

"Yet in another way, calculus is fundamentally naive, almost childish in its optimism. Experience teaches us that change can be sudden, discontinuous, and wrenching. Calculus draws its power by refusing to see that. It insists on a world without accidents, where one thing leads logically to another. Give me the initial conditions and the law of motion, and with calculus I can predict the future -- or better yet, reconstruct the past. I wish I could do that now."

As time continues to pass, Strogatz and Joffray slowly reverse their roles as teacher and student, and the evolution of their relationship is beautiful to watch. The affection that they have for each other shines through in their letters, whether they are talking about mathematics or teaching or kayaking. Most importantly, the 'characters' (if that is the right term for two people in a nonfiction book) are as well developed as one would find in any novel, and both Strogatz and Joffray come across as people I would genuinely want to know -- and to learn mathematics from.

While this is a book that contains quite a few equations and diagrams, most mathematically inclined readers probably won't learn a lot of new mathematics from The Calculus of Friendship, as the problems that Joffray and Strogatz write are for the most part standard gems of the field. More specifically, the title of the book is not to be taken literally, as Strogatz does not model friendship mathematically -- although readers interested in that topic might want to check out Strogatz' column from the NY Times this past summer on mathematical models of love, or otherwise look to the (fictional) book by Charlie Eppes on Numb3rs entitled Friendship: As Easy as Pi. However, while the mathematics itself may not be new it is still a joy to read, and I really enjoyed reading mathematics done as an exploratory dialogue in the way that many of us actually work.

If I haven't been clear already, let me say that I loved this book, and I devoured it in a single evening. I plan to show portions of it to my students the next time I teach calculus. However, I am not sure how this book would be received by people who are not interested in reading about mathematics. Yes, it got good blurbs from Alan Alda and John Cleese and a glowing review on the book review website Bookslut, but with equations appearing on two out of every three pages it is hard for me to imagine recommending the book to someone who is not at least passingly familiar with -- and interested in -- mathematics. However, it is exactly the kind of book I would like non-mathematicians to read, as the leisurely way that Strogatz and Joffray write about problem solving in general as well as solve particular problems is more illustrative about how certain kinds of mathematics is done than anything I have ever read. And ultimately, this is not a book about mathematics but rather a book about friendship and mathematics and the way they are intertwined in the lives of these two men. This might sound strange to some people, but I think Strogatz puts it best when argues that a love of calculus is no stranger a thing to build a friendship off of than a love of opera or of baseball: "For them [calculus:] is more than a science. It is a game they love playing together -- so often the basis of friendship between men -- a constant while all around them is in flux."
Profile Image for Cory.
104 reviews1 follower
August 30, 2020
It's a short and wholesome book, but it could have been just as effective as a 5-10 YouTube video. It's about Steven's letters to his former math teacher, Mr. Joffray. They exchange math problems and short blips of news, usually preferring math over more devastating news (death of a family member or divorce). Corresponding about math helped them through the hard bits of life, but Steven wanted to know more about Joffray as a person. I'm afraid that the math was a little too prevailing and a whole picture wasn't quite drawn out.

The math in this book is written at a pretty high level. I have a degree in math and I couldn't follow it... I would have to sit down with a pen and paper and check each step line-by-line (in a few cases, I did). So in a way, I didn't really read the book. I skipped the math. But I have faith that if I really wanted to know a problem, I could sit down and follow the proof. I sadly didn't have the interest to do that this time around. I'm not sure if the math is there so people can follow it or so that there's a reference. I do enjoy all of the life philosophies connected to mathematical concepts. Every chapter has a theme that connects to where Steven and Mr. Joffray were in life and the math.

Writing "pop math" is extremely hard and Steven Strogatz is considered one of the best. Reading pop math books in high school was the reason I learned to love math, so I'm indebted to them. But... there are very few good pop math books. Most of them contain the same 14 or so math concepts. (Once, I tried to write an essay about why pop math books are usually bad, so I've thought about this a lot.) This book is unique in that the math is at a much higher level, mostly calculus, and has a greater theme about correspondence between a student and a teacher. I think the idea for the book was fantastic, but it wasn't executed as well as I had hoped. I think what pushes this into three-star territory over two-star territory is how short it is and how easy it would be to pull out a chapter and think about a problem later on. It's now a good resource and place of inspiration.

So, as the summer comes to a close, let's hope I can keep up reading throughout the school year. And also, please help me find a good pop math book.
25 reviews
August 18, 2017
Wonderful book. The author, an applied mathematician, writes about his correspondence with his High School teacher. The correspondence is mostly about Calculus problems. It includes some really wonderful solutions, such as a relativity-inspired solution to a problem involving dogs chasing each other. But the book is also about life and how time passes. It's beautifully written and for those of us that enjoy math, it has many great ideas.

I was impressed with the organization. The topics for each chapter come from Calculus (or advanced Calculus), but somehow connect with the life story of the author or of the High School teacher (or both). I was very impressed.

I highly recommend the book to any math enthusiasts.
Profile Image for Shane Orr.
217 reviews3 followers
August 4, 2019
This tells the fascinating story of a student and teacher who wrote occasional letters back and forth to each other about calculus for over 30 years. What’s interesting is that Strogatz only had Joffray for his junior year and didn’t really have a close relationship with him even then. But, for some reason, sent him that first letter about a calculus problem during his freshman year of college. I really wanted to like this book, but the letters were way too deep into the actual calculus and pretty light on any personal connection. The two corresponded about their personal lives only in passing, despite some tragedy.
Profile Image for Michelle.
151 reviews
July 6, 2016
The math was beyond me even though I teach intro calculus.
The message of friendship, caring, and looking beyond yourself is life changing.
Profile Image for Jessada Karnjana.
457 reviews4 followers
April 22, 2022
เรื่องราวที่บอกเล่าผ่านจดหมายโต้ตอบระหว่างลูกศิษย์ (Steven Strogatz) กับครูแคลคูลัสสมัยมัธยม (Joff หรือ Mr. Don Joffray) จากโรงเรียน Loomis Chaffee ในสิ่งที่พวกเขาต่างมีความสนใจร่วมกัน นั่นคือ แคลคูลัส ปัญหาที่ทั้งคู่เขียนถึงกันนั้นมีทั้งปัญหาที่นักเรียนของ Joff สงสัยแล้วขอคำแนะนำจาก Steven และปัญหาที่ Joff ตั้งข้อสงสัย รวมทั้งปัญหาที่ทั้งคู่หาวิธีพิสูจน์ในแบบของตัวเองที่คิดว่าสวย ที่มาพร้อมกับแคลคูลัสคือเรื่องราวความผูกพันธ์ ความใส่ใจกัน สำหรับผู้อ่านที่ไม่มีพื้นฐานแคลคูลัสอาจจะติดตามปัญหาของพวกเขาไม่ทัน หรือแม้แต่ผู้อ่านที่ผ่านแคลคูลัสพื้นฐานมาบ้างแล้วก็อาจต้องใช้เวลาในการทำความเข้าใจพอสมควรกับบางปัญหา แต่ผมคิดว่า จุดเน้นจริง ๆ ของ Strogatz ไม่ได้ต้องการให้หนังสือกึ่งบันทึกกึ่งคำสารภาพที่แสนจะจริงใจ และบางตอนก็ชวนซาบซึ้งใจ กลายเป็นตำราเฉลยโจทย์แคลคูลัสไป นั่นคือเราสามารถอ่านมันได้เหมือนอ่านไดอารีพร้อมจดหมายของใครบางคน

เรื่องที่เขาคุยกันมีหลากหลาย ตั้งแต่ Geometry (เช่น พิสูจน์ว่ารากที่สองของสองเป็นจำนวนอตรรกยะด้วยวิธีการเรขาคณิต) Trigonometry (เช่น พิสูจน์สูตรพื้นที่ของ Hero) Probability กับ discrete math (เช่น ปัญหา Monty Hall เป็นจดหมายอยู่ในช่วงที่ Marilyn กำลังโต้แย้งกับนักคณิตศาสตร์บางคนอยู่พอดี) Calculus (เช่น ฟังก์ชั่นแกมม่า, การ differentiation ภายใต้ integral - วิธีที่ฟายน์แมนพูดถึงใน surely, you're joking - dimensional analysis) Differential Equation (เช่น ปัญหา chase) Fourier series, Complex variable, Asymptotic methods ไปจนถึง Calculus of variations (เช่น เส้นโค้ง brachistochrone กับเส้นโค้ง tautochrone)
94 reviews2 followers
April 30, 2019
Some of the reviews for this book indicated that understanding the math was not a requirement for reading and enjoying this book.
That may be so, but if you remove all of the math problems from the text, the remainder is mostly the author kicking himself for poor social skills and lacking a spine as he rarely makes a decision without confiding in his mother or a brother. .

My undergraduate calculus skills were never profound in the first place and are now pretty much extinct almost forty years later. So, while I can appreciate that a true mathematician would enjoy that part of the book, all other readers will probably feel left out of the conversation.

The author struggles to tie each chapter to a standard mathematics or physics term but the correlation between his relationship with his teacher and the chosen term is a stretch in my opinion.

The fundamental take-away is probably the transition of the relationship from student -teacher to teacher-student which is kind of a coming of age story.
This is valid but is lost in the math and never really gets completed.
I don’t have the feeling the author really reached an emotional Tate of maturity.

Oh well, maybe others will get more out of it than I.

Profile Image for Swan.
54 reviews12 followers
May 2, 2020
Interesting math, some that is beyond me and I will be excited in the future once I can comprehend it. (I'm taking three math courses online this summer!) I liked reading about the relationship between a student who becomes a professor with his high school calculus teacher after they surprisingly reignite contact with lots of letters of calculus problems many years after Strogatz graduates. I definitely want to spend more time talking to math professors beyond just listening to them in lectures, but distance learning is giving me an actual decent excuse to stay relatively anonymous. It is so incredible the number of things mathematics can be used to gain a better understanding of, and I also liked early on the problem the author spent so much time trying to solve as a teenagers about a dog chasing a duck in a circle that even now he claims is a problem that cannot be graphed with any formula. That is disappointing, but it is a neat contrast to all the extremely nice problems given in textbooks that nearly always give beautiful results.
Profile Image for Ehsan Misaghi.
16 reviews1 follower
August 20, 2019
The Calculus of Friendship is an interesting book. I love calculus and math, so the problems discussed through the letters in the book were very interesting to me. However, the more interesting aspect of the book was the development of a friendship between a student-turned-teacher and a teacher-turned-student. Some of the struggles the author went through and the awkwardness when the relationship shifted towards more of a friendship is something a lot of us struggle with. It's a very easy to read book (unless you want to check every step of some of the given proofs for the math concepts) and I would recommend it to anyone who has an admiration for math and building friendships.
Profile Image for Josh.
793 reviews
October 10, 2020
I really appreciated how this book did not sanitize the math involved. I couldn't follow most of it. (ok, maybe I could have followed it, but I'm lazy and that would have taken me about 100 times longer to read this book). But the important lessons of the book really shine through. There isn't enough communication in this world about those ordinary people who had a positive impact on our lives. Strogatz was able to tell us about one of these people in this little book and also to show the reader some interesting mathematics while he was at it. Recommended if you like math even a little bit.
83 reviews
July 14, 2017
The relationship that was developed between teacher and student is wonderful to see through the letters and commentary that Strogatz provides. I wish that more of the letters were included, but understand how the personal nature of the letters would have led from excluding them. The math included is wonderful for someone like myself (a fellow high school math teacher) who has not seen these ideas before or it has been awhile since I last saw them.
Profile Image for Derek.
8 reviews
April 19, 2019
Male expression and all the ways of expressing intimacy and demonstrating fellowship but without actually saying it. Despite studying calculus some years ago, I jumped through the calculus example pages, it grew tiring after awhile.
1,157 reviews7 followers
September 3, 2019
I liked this little book, even though I really didn't understand all the math in it. I would like to have had more touchy-feely writing about these two guys, but I guess mathematicians aren't that emotional.
Profile Image for Ronah.
33 reviews1 follower
August 19, 2020
This is how a teacher should be, not just able to give students facts but who is also able to inspire and let students learn outside the classroom. Their correspondence was both intellectual and personal, a rare kind of friendship born from a teacher-student relationship.
33 reviews1 follower
March 23, 2021
This is how a teacher should be, not just able to give students facts but who is also able to inspire and let students learn outside the classroom. Their correspondence was both intellectual and personal, a rare kind of friendship born from a teacher-student relationship.
Profile Image for Yasemin Esen.
41 reviews
July 19, 2017
steve'e ifrit olsam da sıef Joffray Hoca'nın samimiyeti için okunur bu kitap.
Profile Image for meryem.
80 reviews1 follower
September 23, 2019
“...matematiğin o yanını severdim. Yapısında adalet vardır. Doğru yerden başlar, çok çalışır ve her şeyi doğru yaparsan zahmetli olsa da sonunda kazanacağın kesindir. Çözüm, ödülün olur.”
Profile Image for Esther.
472 reviews5 followers
June 11, 2020
Interesting reading even though I often got lost in the proofs.
4 reviews
June 16, 2020
Interesting book about Mathematicians and their thinking. Looking forward to reading another book by the same author: Infinite Powers.
Profile Image for Bonny.
712 reviews26 followers
August 6, 2015
I'm familiar with Steven Strogatz from being an avid Radiolab listener, so I have anxiously awaited The Calculus of Friendship. It's a beautiful, poignant story of the intense, special, and evolving relationship between student & teacher. It is also an amazing writing accomplishment that successfully combines math and memoir. While I can't say that I completely understand all the math, the mathematics only adds to this story rather than detracting from it. It shows the true elegance & beauty of calculus, along with the elegance & beauty of working together as student and teacher to solve the calculus. Mr. Joffray is truly an extraordinary teacher, one that allows his students to teach him. I honestly wish that I could learn math from both of these men, and they are portrayed so well through their letters that I would also like to share an interesting dinner & discussion with them. Maybe they could (attempt to) further explain "differentiating under the integral sign" to me! I'm going to insist that my son read this book as he has already had some exemplary math (and chemistry!) teachers that have greatly influenced his life, and is embarking on his college career where I fervently hope he meets more of the same.
77 reviews
May 28, 2011
This is the story of the author's friendship with his high school calculus teacher, Don Joffray. Over the course of 30 years, they maintain a correspondence based mostly on their love of math, sharing interesting puzzles and solutions with one another. Before long, the pupil has surpassed the teacher, and their roles reverse.

My husband has math friendships like the one portrayed here, and I chose this book for that aspect. I was hoping for a lot more relationship and a little less math. I could not coax my lazy brain into doing any but the simplest problems in the book, and even then, I only followed along the proofs and didn't work them out myself.

The picture that emerged of "Joff," the teacher, is a beautiful one, and I fell in love with him almost immediately. His side of the correspondence is lively and colorful, and he interjects glimpses of his life outside of math. My impression of the author is a lot less flattering and soured further as the relationship evolved. I suspect that he really does care deeply for Joff in his own way, but as Strogatz himself puts it, he spends a lot of his life in his own head...and it shows.
25 reviews
October 6, 2013
The book goes down the mathematical life of Prof. Steven Strogatz, a renowned applied mathematician currently at Cornell University, and his correspondence with Don Joffray, his math teacher from high school. In a collection of 10 or so chapters covering various phases of Strogatz's career, the author reproduces his letters with Joff, as he is affectionatedly called, which mainly comprise of little gems of mathematical results.
Some of them which I enjoyed are:

1. The evaluation of \sum_{k=1}^\infty (sin k)/k, using both Fourier series in pg 51 (Kindle edition), and via complex analysis in Pg 59.

2. The "differentiation under integration" trick in pg 57 which is used to derive the integral formula for n! (and thereby generalizing to the Gamma function).

3. The resolution of the Monty Hall puzzle in Pg 90.

4. The integration \int_0^{2pi} cos^{2m} 2t dt

5. The evaluation of \pi using only 2's given in page 113.

A quote from the preface which caught my attention: "...the best student a teacher can have [is] someone with perfect preparation and an evident sense of delight and gratitude".
Profile Image for Jobert.
219 reviews
April 11, 2012
"But now I also see that I did learn something from him, something profoundly mathematical, about how to live. From his hobbies to the way he faces ups and downs in his life... He rolls with it and tries to make peace with it. And where he can, he even plays with it. Jazz piano, windsurfing, whitewater kayaking - all of these balance the inevitable against the unforeseeable, the two sides if change in this world. The orderly and the chaotic. The changes that calculus can tame, and the ones it cannot."


Great book! Syempre, tungkol sa math eh, haha! Haaay, nakakamiss. Nakakatuwa na yung iba eh alam ko na, tas yung iba hindi ko pa alam pero interesanteng aralin. Fibonacci, chase problems, logarithmic spiral, irrationality, shift operators, convergence tests, Fourier series, differentiating under the integral sign, Scientific American riddles, complete elliptic integral of the 1st kind, Ask Marilyn, infinity and limits, chaos, inverted cycloid, Hero's formula, at kung anu-ano pa. Saya noh? ^^
Profile Image for Melody.
2,622 reviews251 followers
July 20, 2010
The math was far, far over my head, but that hardly ever detracts from my enjoyment of a book. However this book, which appeared from the blurbs to be a collection of letters, was actually more of a collection of math problems. The author was pretty honest about what a clueless jerk he was for much of the time period the book covers, but his honesty didn't make me like him any better. I wonder if there's a lot left out of this book, or if he really is a guy so mathy that he doesn't see anything but numbers.

In summation, I wanted to like this book, but I didn't. The writing wasn't bad, mind you- just so emotionless and flat that I couldn't connect with the non-math parts, and the math parts were so unintelligible (to me, I know, I know the rest of you got 'em) that I was overwhelmed with ennui.
Profile Image for jersey9000.
Author 3 books19 followers
July 4, 2012
A great book for math people (like me) and teachers (also me) and anyone who has always felt that nagging feeling that they have let a relationship fade away for no reason other than a general sense of being busy (me again). Told via a series of letters, the book traces the relationship between a calculus teacher and a former student. They mostly communicate through calculus problems, but as they both get older and their lives change (and diminish towards an end point, which is ironically how calculus works, kind of), their letters change. A great exploration of what teaching means, and how life can change/challenge us in unexpected ways. Only 3 stars because the math can be super complicated if you don't have the background for it, but if you have a basic calculus background it isn't too bad. And you could always skip the math bits.
Profile Image for Laurent.
51 reviews15 followers
May 21, 2012
In The Calculus of Friendship, the author, Steven Strogatz narrates the epistolary relationship he maintained with his maths high school teacher for 30 years. Their letters contained a lot of math which what kept them close to each other but there's also this very touching side of their inadequacy to get to a more personal level. I really enjoy reading it.

Dans le Calcul de l'amitie, l'auteur, Steven Strogatz narre la relation epistolaire qu'il a maintenue avec son prof de maths au lycee pendant 30 ans. Leurs lettres contiennent beaucoup de maths ce qui les a rapproches mais il y aussi ce cote tres touchant de leur insuffisance a atteindre un niveau plus personnel. J'ai vraiment aime lire ce livre.
Profile Image for Will Schwalbe.
Author 7 books1,210 followers
September 20, 2009
I was privileged to edit this author's first trade book, SYNC. And I'm also in the acknowledgments for this one. So I'm biased. But I still think this is an amazing book about the lifelong friendship between a math teacher and his student. There is a ton of formulas in this book -- they were really corresponding about math. But you can also read this book and just ignore the math and get a huge amount out of it. It's a love story -- a teacher for his prize student, a student for a beloved teacher and mentor, and two men for a subject that held for them inestimable beauty and challenge: math.
Profile Image for David.
259 reviews30 followers
May 24, 2010
Only rarely do I find a book that mentions mathematics in the title that Jessica actually wants to read. After hearing Steve talk about the book on an NPR program, and then hearing him read from it at one of the downtown bookstores, she was sold. The reason, of course, is that for her, the book was mostly about the evolution of a friendship of mathematical conversations between a teacher and a student. Ultimately, that turned out to be the main attraction for me as well -- I have a couple friendships built almost entirely on mathematical conversations, too.
Displaying 1 - 30 of 51 reviews

Can't find what you're looking for?

Get help and learn more about the design.