## What is Calculus?: Limits

This is the second in a series of short articles exploring calculus. The first article explored the origins of calculus. The next few articles explore the mathematics of calculus. This article focuses on the foundation of calculus: limits.

The first article in this series asked the question: What is calculus? I promised then that the second article in the series would explore the substance of calculus, the mathematics of calculus. So let’s dive right in.

Here’s my two-part answer to the “what is calculus?” question:

Calculus is a mindset—a dynamics mindset.
Contentwise, calculus is the
mathematics of infinitesimal change.

The second sentence describes the mathematics of calculus. But I don’t expect you to understand that sentence just yet. That’s where the first sentence comes in. If you ask me, all of calculus flows from a more fundamental—and intuitive—principle, articulated in the first sentence: the notion that calculus is a dynamics mindset. Let me explain.

Calculus: A New Way of Thinking

The mathematics that precedes calculus—often called “pre-calculus,” which includes algebra and geometry—largely focuses on static problems: problems devoid of change. By contrast, change is central to calculus. Calculus is all about dynamics. Example:

• What’s the perimeter of a square of side length 2 feet? ← Pre-calculus problem.
• How fast is the square’s perimeter changing if its side length is increasing at the constant rate of 2 feet per second? ← Calculus problem.

Now that I’ve sensitized you to thinking “calculus!” whenever you read about or infer the presence of change, take a quick look at the second sentence in my two-part answer above. What’s the last word? Change. But it’s a new type of change—infinitesimal change—and this requires some explaining. That’s our next stop.

A Philosopher Walks Into a Starbucks

First, a rough definition of “infinitesimal change”:

“Infinitesimal change” means: as close to zero change as you can imagine, but not zero change.

“What?!” I hear you saying. So let me illustrate this definition via my friend, Zeno of Elea (c. 490-430 BC). This ancient Greek philosopher thought up a set of paradoxes arguing that motion is not possible. One such paradox—the Dichotomy Paradox—can be stated as follows:

To travel a certain distance you must first traverse half of it.

Makes perfect sense. Two is one plus one. And one is one-half plus one-half. But don’t be fooled by this seemingly innocent reasoning; it’s a trap! (Admiral Ackbar!)

To appreciate what’s going on—and connect Zeno’s paradox back to calculus—let’s pretend Zeno is in line at Starbucks, two feet away from the cash register. He’s almost done scanning the menu when the barista yells out “next!” And that’s when poor Zeno panics. He must now walk two feet, but because of his mindset, he walks only half that distance with his first step. He then walks half of the remaining distance with his second step. (Can you imagine how annoying those in line behind him are getting?) The figure below keeps track of the total distance d Zeno has walked and the change in distance Δd after each of his steps.

Fig. 1.2: Zeno trying to walk a distance of 2 feet by traversing half the remaining distance with each step. (Reprinted, with permission, from Calculus Simplified.)

Here’s a tabular representation of the action:

Table 1.1: The distance d and change in distance Δd after each of Zeno’s steps. (Reprinted, with permission, from Calculus Simplified.)

Each change d in Zeno’s distance is half the previous one. So as Zeno continues his walk, d gets closer to zero but never becomes zero.[1] If we checked back in with Zeno after he’s taken an infinite

amount of steps—what a patient barista!—the change d resulting from his next step would be . . . drum roll please . . . an infinitesimal change—as close to zero as you can imagine but not equal to zero.

This example, in addition to illustrating what an infinitesimal change is, also does two more things. First, it illustrates the dynamics mindset of calculus. We discussed Zeno walking; we thought about the change in the distance he traveled; we visualized the situation with a figure and a table that each conveyed movement. (Calculus is full of action verbs!) Second, the example challenges us. Clearly, one can walk 2 feet. But as Table 1.1 suggests, that doesn’t happen during Zeno’s walk—he approaches the 2-foot mark with each step yet never arrives. How do we describe this fact with an equation? (That’s the challenge.) No pre-calculus equation will do. We need a new concept that quantifies our very dynamic conclusion. That new concept is the mathematical foundation of calculus: limits.

Limits: The Foundation of Calculus

In modern calculus speak we paraphrase the main takeaway of Table 1.1 this way: the distance d traveled by Zeno approaches 2 as Δd approaches zero. It’s important to note that d never equals 2 and Δd never equals 0. Today we express these conclusions more compactly by writing

read “the limit of d as Δd approaches zero (but is never equal to zero) is 2.” This new equation—and what we take it to mean—remind us that d is always approaching 2 yet never arrives at 2. (Oh, the dynamics!) The same idea holds for Δd: it is always approaching 0 yet never arrives at 0. Said more succinctly:

Limits approach indefinitely (and thus never arrive).

You’ve now met the foundational concept of calculus—limit. You’ve also gotten a glimpse of what infinitesimal change means and how a limit encodes that notion. Finally, you’ve seen many times how a dynamics mindset is at the core of calculus’ new way of thinking about mathematics. In the next article in this series we’ll employ a dynamics mindset and limits to solve the three Big Problems that drove the development of calculus—instantaneous speed, the tangent line problem, and the area problem (discussed in the first post in this series). See you then!

Footnote: [1] Because each d is always half of a positive number.

Calculus Simplified
By Oscar E. Fernandez

Calculus is a beautiful subject that most of us learn from professors, textbooks, or supplementary texts. Each of these resources has strengths but also weaknesses. In Calculus Simplified, Oscar Fernandez combines the strengths and omits the weaknesses, resulting in a “Goldilocks approach” to learning calculus: just the right level of detail, the right depth of insights, and the flexibility to customize your calculus adventure.

Fernandez begins by offering an intuitive introduction to the three key ideas in calculus—limits, derivatives, and integrals. The mathematical details of each of these pillars of calculus are then covered in subsequent chapters, which are organized into mini-lessons on topics found in a college-level calculus course. Each mini-lesson focuses first on developing the intuition behind calculus and then on conceptual and computational mastery. Nearly 200 solved examples and more than 300 exercises allow for ample opportunities to practice calculus. And additional resources—including video tutorials and interactive graphs—are available on the book’s website.

Calculus Simplified also gives you the option of personalizing your calculus journey. For example, you can learn all of calculus with zero knowledge of exponential, logarithmic, and trigonometric functions—these are discussed at the end of each mini-lesson. You can also opt for a more in-depth understanding of topics—chapter appendices provide additional insights and detail. Finally, an additional appendix explores more in-depth real-world applications of calculus.

Learning calculus should be an exciting voyage, not a daunting task. Calculus Simplified gives you the freedom to choose your calculus experience, and the right support to help you conquer the subject with confidence.

• An accessible, intuitive introduction to first-semester calculus
• Nearly 200 solved problems and more than 300 exercises (all with answers)
• No prior knowledge of exponential, logarithmic, or trigonometric functions required
• Additional online resources—video tutorials and supplementary exercises—provided

## Dana Johnson on Will This Be on the Test?

Getting into college takes plenty of hard work, but knowing what your professors expect of you once you get there can be even more challenging. Will This Be on the Test? is the essential survival guide for high-school students making the transition to college academics. In this entertaining and informative book, Dana Johnson shares wisdom and wit gleaned from her decades of experience as an award-winning teacher in the freshman classroom—lessons that will continue to serve you long after college graduation.

What inspired you to write this book?

I’ve taught college freshmen for decades and have seen the trouble they have because they don’t realize how high school and college are different. Some don’t figure out how to be successful in their coursework and end up doing poorly or even dropping out. This book is my best advice to students based on my experiences and other professors I’ve known and worked with. I’ve wanted to write the book for many years, but finally made time to do it because I realized how much it could help.

How have students changed since you started teaching?

Students are less self-reliant and have more expectations of reminders, extensions, re-do’s on assignments, and extra credit. They want to be told information, rather than take charge of their own learning. They are more likely to blame someone or something else than take responsibility. With the advent of email, students prefer to send electronic messages rather than call or come to the offices of their professors, which means they have less of an academic relationship. Students seem less likely to meet many of their classmates as they are primarily connected via their phones and social media to friends they know through other contexts.

When should students (and parents) read this book?

Students should read it before going to college and again at the end of the first semester or two. The re-reading will help them pick up some tips that are more meaningful after they have experienced some college courses. Parents should read the book before their students are in high school so they understand what high school and the family should be preparing them for.

High school teachers and counselors could benefit from reading it too, so they’re aware of habits, skills, and a mindset that will help students make the transition successfully from high school to college.

What is the biggest mistake students make in college academics?

I’ll give you two:

1. Skipping class. Since no one is calling their parents when they don’t go to class, it seems easy to sleep in or give preference to other activities.
2. Procrastinating. There are fewer intermediate deadlines, reminders, reviews, prompts, and safety nets in college than in high school. At first, the assignment deadlines and exams seem so far away, and students wait too long before starting the work or studying.

An example of the comics found in Will This Be on the Test?. Art by Jeremy Tamburello.

Are the cartoons featured throughout the book based on real events?

The ideas all originated in something I experienced or was told to me. Every professor tells stories about bizarre, rude, amusing, or naïve behavior on the part of students, and students have told me their stories also. Some of them seem a little unbelievable – but they are all based on true stories!

What should students know about professors that they generally don’t?

Professors are experts in a special slice of their fields. They love their content, and they love their work. This is not just a job for them, it is their intellectual life. You can learn a lot by talking with them outside of class. Professors enjoy having their students visit office hours, and they want to pass on what they know. Students can think of this as a form of networking, which is a skill that will pay off after college too.

Dana Johnson taught for many years at the College of William and Mary, where she twice won the Simon Prize for Excellence in the Teaching of Mathematics, and has three decades of experience teaching college freshmen. She lives in Williamsburg, Virginia.

## In Dialogue: Christopher Phillips and Tim Chartier on Sports & Statistics

Question: How would you describe the intersection between statistics and sports? How does one inform the other?

Christopher Phillips, author of Scouting and Scoring: Sports have undoubtedly become one of the most visible and important sites for the rise of data analytics and statistics. In some respects, sports seem to be an easy, even inevitable place to apply new statistical tools: most sports produce a lot of data across teams and seasons; games have fixed rules and clear measures of success (e.g., wins or points); players and teams have incentives to adjust in order to gain a competitive edge.

But as I discuss in my new book Scouting and Scoring: How We Know What We Know About Baseball, it is also easy to fall prey to myths about the use of statistics in sports. Though these myths apply across many sports, it is easiest to hone in on baseball, as that has been one of the most consequential areas for statistics.

Perhaps the most persistent and pernicious myth is that data emerge naturally from sporting events. There is no doubt that new video-, Doppler-, and radar-based technologies, especially when combined with increasingly cheap computing power and storage capability, have dramatically expanded the amount of data that can be collected. But it takes a huge about of labor to create, collect, clean, and curate data, even before anyone tries to analyze them. Moreover, some data, like errors in baseball, are inescapably the product of individual judgment which has to be standardized and monitored.

The second myth is that sport statistics emerged only recently, particularly after the rise of the electronic computer. In fact, statistical analysis in sports goes back decades: in baseball, playing statistics were being used to evaluate players for year-end awards and negotiate contracts for as long as professional baseball has existed. (And statistics were collected and published for cricket decades before baseball’s rules were formalized.) As new methods of statistical analysis emerged in the early twentieth century in fields like psychology and physiology, some observers immediately tried to apply them to sports. In the 1910 book Touching Second, the authors promoted the use of data for shifting around fielders and for scouting prospects, two of the most important uses of statistical data in the modern era as well. There’s certainly been a flurry of new statistics over the last twenty years, but the general idea isn’t new—consider that Allen Guttmann’s half-century-old book From Ritual to Record, highlights the “numeration of achievement” and the “quantification of the aesthetic” as defining features of modern sport.

Finally, it’s a myth that there is a fundamental divide between those who look at performance statistics (i.e., scorers) and those who evaluate bodies (i.e., scouts). The usual gloss is that scouts are holistic, subjective judges of quality whereas scorers are precise, objective measurers. In reality, baseball scouts have long used methods of quantification, whether for the pricing of amateur prospects, or for the grading of skills, or the creation of single metrics like the Overall Future Potential that reduce a player to a single number. There’s a fairly good case to be made that scouts and other evaluators of talent are even more audacious quantifiers than scorers in that the latter mainly analyze things that can be easily counted.

Tim Chartier, author of Math Bytes: Data surrounds us. The rate at which data is produced can make us seem like specks in the cavernous expanse of digital information.  Each day 3 billion photos and videos are shared on Snapchat.  In the last minute, 300 hours of video were uploaded to YouTube.  Data is offering new possibilities for insight. Sports is an area where data has a traditional role and newfound possibilities, in part, due to the enlarging datasets.

For years, there are a number of constants in baseball that include the ball, bat, bases, and statistics like balls, strikes, hits and outs.  Statistics are and have simply been a part of the game.  You can find from the 1920 box score that Babe Ruth got 2 hits in 4 at-bats in his first game as a Yankee. While new metrics have emerged with analytical advances, the game has been well studied for some time. As Ford C. Frick stated in Games, Asterisks and People,

“Baseball is probably the world’s best documented sport.”

While this is true, the prevalence of data does not necessarily result in trusting the recommendations of those who study it.  For example, Manager Bobby Bragen stated, “Say you were standing with one foot in the oven and one foot in an ice bucket. According to the percentage people, you should be perfectly comfortable.”  This underscores an important aspect of data and analytics.  Data, inherently, can lead to insight but it becomes actionable when one trusts in how accurately it reflects our world.

Other sports, while not as statistically robust as baseball also have an influx of data.  In basketball, cameras positioned in the rafters report the (x,y) position of every player on the court and the (x,y,z) position of the ball throughout the entire game every fraction of a second.  As such, we can replay aspects of games via this data for years to come.  With such information comes new information.  For example, we know that Steph Curry, while averaging just over 34 minutes a game, runs, on average, just over 2.6 miles per game. He also runs almost a quarter of a mile more on offense than defense.

While such data can be stunning with its size and detail, it also comes with challenges. How do you recognize a pick and roll versus an isolation play simply from essentially dots moving in a plane?  Further, basketball, like football but unlike baseball, generally involves multiple players at a time.  How much credit do players get for a basket on offense?  A player’s position may open up possibilities for scoring, even if that player didn’t touch the ball.  As such, metrics have been and continued to be created in order to better understand the game.

Sports are played with a combination of analytics, gut and experience.  What combination depends on the sport, player, coach and context.  Nonetheless, data is here and will continue to give insight on the game.

## Pi: A Window into the World of Mathematics

Mathematicians have always been fascinated by Pi, the famous never-ending never-repeating decimal that rounds to 3.14. But why? What makes Pi such an interesting number? Every mathematician has their own answer to that question. For me, Pi’s allure is that it illustrates perfectly the arc of mathematics. Let me explain what I mean by taking you on a short mathematical adventure.

Picture yourself in a kitchen, rummaging the pantry for two cans of food. Let’s say you’ve found two that have circular bases of different diameters d1 and d2. Associated with each circle is a circumference value, the distance you’d measure if you walked all the way around the circle.

Were you to perfectly measure each circle’s circumference and diameter you would discover an intriguing relationship:

In other words, the ratio of each circle’s circumference to its diameter doesn’t change, even though one circle is bigger than the other. (This circumference-to-diameter number is  (“Pi”), the familiar 3.14-ish number.) This is the first stop along the arc of mathematics: the discovery of a relationship between two quantities.

Where this story gets very interesting is when, after grabbing even more cans and measuring the ratio of their circumferences to their diameters—you seem to have lots of free time on your hands—you keep finding the same ratio. Every. Time. This is the second stop along the arc of mathematics: the discovery of a pattern. Shortly after that, you begin to wonder: does every circle, no matter its size, have the same circumference-to-diameter ratio? You have reached the third stop along the arc of mathematics: conjecture. (Let’s call our circumference-to-diameter conjecture The Circle Conjecture.)

At first you consider proving The Circle Conjecture by measuring the ratio C/d for every circle. But you soon realize that this is impossible. And that’s the moment when you start truly thinking like a mathematician and begin to wonder: Can I prove The Circle Conjecture true using mathematics? You have now reached the most important stop along the arc of mathematics: the search for universal truth.

One of the first thinkers to make progress on The Circle Conjecture was the Greek mathematician Euclid of Alexandria. Euclid published a mammoth 13-book treatise text called Elements circa 300 BC in which he, among other accomplishments, derived all the geometry you learned in high school from just five postulates. One of Euclid’s results was that the ratio of a circle’s area A to the square of its diameter d2 is the same for all circles:

This is close to what we are trying to prove in The Circle Conjecture, but not the same. It would take another giant of mathematics—the Greek mathematician Archimedes of Syracuse—to move us onto what is often the last stop on the arc of mathematics: thinking outside the box.

Archimedes went back to Euclid’s five postulates, all but one of which dealt with lines, and extended some of Euclid’s postulates to handle curves. With these new postulates Archimedes was able to prove in his treatise Measurement of a Circle (circa 250 BC) that the area, circumference, and radius r of a circle are related by the equation:

(You may recognize this as the area of a triangle with base C and height r. Indeed, Archimedes’ proof of the formula effectively “unrolls” a circle to produce a triangle and then calculates its area.) Combining Archimedes’ formula with Euclid’s result, and using the fact that r = d/2, yields:

Et Voilà! The Circle Conjecture is proved! (To read more about the mathematical details involved in proving The Circle Conjecture, I recommend this excellent article.)

This little Pi adventure illustrated the core arc of mathematics: discovery of a relationship between to quantities; discovery of a more general pattern; statement of a conjecture; search for a proof of that conjecture; and thinking outside the box to help generate a proof. Let me end our mathematical adventure by encouraging you to embark on your own. Find things you experience in your life that are quantifiable and seem to be related (e.g., how much sleep you get and how awake you feel) and follow the stops along the arc of mathematics. You may soon afterward discover another universal truth: anyone can do mathematics! All it takes is curiosity, persistence, and creative thinking. Happy Pi Day!

Oscar E. Fernandez is associate professor of mathematics at Wellesley College. He is the author of Calculus Simplified, Everyday Calculus, and The Calculus of Happiness (all Princeton).

## Browse our 2019 Mathematics Catalog

Our new Mathematics catalog includes an exploration of mathematical style through 99 different proofs of the same theorem; an outrageous graphic novel that investigates key concepts in mathematics; and a remarkable journey through hundreds of years to tell the story of how our understanding of calculus has evolved, how this has shaped the way it is taught in the classroom, and why calculus pedagogy needs to change.

If you’re attending the Joint Mathematics Meetings in Baltimore this week, you can stop by Booth 500 to check out our mathematics titles!

Integers and permutations—two of the most basic mathematical objects—are born of different fields and analyzed with different techniques. Yet when the Mathematical Sciences Investigation team of crack forensic mathematicians, led by Professor Gauss, begins its autopsies of the victims of two seemingly unrelated homicides, Arnie Integer and Daisy Permutation, they discover the most extraordinary similarities between the structures of each body. Prime Suspects is a graphic novel that takes you on a voyage of forensic discovery, exploring some of the most fundamental ideas in mathematics. Beautifully drawn and wittily and exquisitely detailed, it is a once-in-a-lifetime opportunity to experience mathematics like never before.

99 Variations on a Proof offers a multifaceted perspective on mathematics by demonstrating 99 different proofs of the same theorem. Each chapter solves an otherwise unremarkable equation in distinct historical, formal, and imaginative styles that range from Medieval, Topological, and Doggerel to Chromatic, Electrostatic, and Psychedelic. With a rare blend of humor and scholarly aplomb, Philip Ording weaves these variations into an accessible and wide-ranging narrative on the nature and practice of mathematics. Readers, no matter their level of expertise, will discover in these proofs and accompanying commentary surprising new aspects of the mathematical landscape.

Exploring the motivations behind calculus’s discovery, Calculus Reordered highlights how this essential tool of mathematics came to be. David Bressoud explains why calculus is credited to Isaac Newton and Gottfried Leibniz in the seventeenth century, and how its current structure is based on developments that arose in the nineteenth century. Bressoud argues that a pedagogy informed by the historical development of calculus presents a sounder way for students to learn this fascinating area of mathematics.

## Browse our 2019 Computer Science Catalog

Our new Computer Science catalog includes an introduction to computational complexity theory and its connections and interactions with mathematics; a book about the genesis of the digital idea and why it transformed civilization; and an intuitive approach to the mathematical foundation of computer science.

If you’re attending the Information Theory and Applications workshop in San Diego this week, you can stop by the PUP table to check out our computer science titles!

Mathematics and Computation provides a broad, conceptual overview of computational complexity theory—the mathematical study of efficient computation. Avi Wigderson illustrates the immense breadth of the field, its beauty and richness, and its diverse and growing interactions with other areas of mathematics. With important practical applications to computer science and industry, computational complexity theory has evolved into a highly interdisciplinary field that has shaped and will further shape science, technology, and society.

A few short decades ago, we were informed by the smooth signals of analog television and radio; we communicated using our analog telephones; and we even computed with analog computers. Today our world is digital, built with zeros and ones. Why did this revolution occur? The Discrete Charm of the Machine explains, in an engaging and accessible manner, the varied physical and logical reasons behind this radical transformation, and challenges us to think about where its future trajectory may lead.

Discrete mathematics is the basis of much of computer science, from algorithms and automata theory to combinatorics and graph theory. This textbook covers the discrete mathematics that every computer science student needs to learn. Guiding students quickly through thirty-one short chapters that discuss one major topic each, Essential Discrete Mathematics for Computer Science can be tailored to fit the syllabi for a variety of courses. Fully illustrated in color, it aims to teach mathematical reasoning as well as concepts and skills by stressing the art of proof.

## Calling Girls Who Love Math: Register for Girls’ Angle’s SUMIT 2019!

Get ready for a new mathematical adventure! SUMIT 2019 is coming April 6 and 7 with an all-new plot and math problems galore.

If you’re a 6th-11th grade girl who loves math, you’ll love SUMIT! There will be challenges for all levels and key leadership roles to fulfill. You’ll emerge with an even greater love of math, new friends, and lasting memories.

Princeton University Press has been a major sponsor of SUMIT since its inception in 2012, and is always proud to promote this magical escape-the-room-esque event where girls join forces to overcome challenges and become the heroines of an elaborate mathematical saga. The event offers one of the most memorable opportunities to do math while forming lasting friendships with like-minded peers. Together, girls build mathematical momentum and frequently surprise themselves with what they’re able to solve. All previous SUMITs have garnered overall ratings of 10 out of 10 by participants.

Created by Girls’ Angle, a nonprofit math club for girls, together with a team of college students, graduate students, and mathematicians, SUMIT 2019 takes place in Cambridge, MA.

Registration opens at 2 pm ET on Sunday, February 10 on a first-come-first-served basis and there are limited slots, so register quickly!

## SUMIT 2018: A math collaboration

by C. Kenneth Fan
President and Founder of Girls’ Angle, an organization that connects mentors with girls who love math

For decades, math extracurricular activity in the United States has been dominated by the math competition. I, myself, participated in and enjoyed math competitions when I was growing up. Many school math clubs are centered on math contest prep. Today, there are dozens upon dozens of math competitions. While many students gain much from math competitions, many others, for a variety of good reasons, do not find inspiration in math competitions to do more math, and the best way to learn math is to do math.

When I founded Girls’ Angle over ten years ago, a main task was to create new, non-competitive, mathematically compelling avenues into math that appeal to those who, for whatever reason, may not be so inspired by math competitions. To celebrate the end of our first year, we baked a brownie for the girls, but it wasn’t a rectangular brownie—it was a trapezoid, and nobody could have any brownie until members figured out how to split the brownie into equal pieces for all. We were counting on them to succeed because we wanted brownie!

It became a Girls’ Angle tradition to celebrate the conclusion of every semester with a collaborative math puzzle, and every semester the puzzle has grown more elaborate. It finally dawned on me that these collaborative end-of-session math puzzles could well serve as robust, mathematically-intense, but fully collaborative alternatives to the math competition. To directly contrast the concept with that of the math competition, we called these events “math collaborations.” On January 21, 2012, after 4 years of in-house development, we took the concept out of Girls’ Angle with SUMIT 2012, which took place at MIT in conjunction with MIT’s Undergraduate Society of Women in Mathematics. Then, on March 7, 2012, the Buckingham, Browne, and Nichols Middle School became the first school to host a math collaboration. The success of these events led to annual math collaborations at Buckingham, Browne, and Nichols, and, to date, over 100 other math collaborations at schools, libraries, and other venues, such as Girl Scout troops.

The upcoming SUMIT 2018 is going to be our biggest and best math collaboration ever. For girls in grades 6-10, participants will be put in a predicament from which they must extricate themselves using the currency of the world they’ll find themselves immersed in: mathematics! They must self-organize and communicate well as there will be no one to help them but themselves. It’ll be an epic journey where participants must become the heroines of their own saga.

Should they succeed, they’ll be rewarded with the knowledge of genuine accomplishment—and gifts, such as Marc Chamberland’s captivating book, Single Digits: In Praise of Small Numbers courtesy of long-time SUMIT sponsor Princeton University Press.

The best way to learn math is to do math, and what better way to do math than to do it while laughing out loud and making new friends?

There are a limited number of spots still available for 9th and 10th graders. Register today!

## Keith Devlin: Fibonacci introduced modern arithmetic —then disappeared

More than a decade ago, Keith Devlin, a math expositor, set out to research the life and legacy of the medieval mathematician Leonardo of Pisa, popularly known as Fibonacci, whose book Liber abbaci has quite literally affected the lives of everyone alive today. Although he is most famous for the Fibonacci numbers—which, it so happens, he didn’t invent—Fibonacci’s greatest contribution was as an expositor of mathematical ideas at a level ordinary people could understand. In 1202, Liber abbaci—the “Book of Calculation”—introduced modern arithmetic to the Western world. Yet Fibonacci was long forgotten after his death. Finding Fibonacci is a compelling firsthand account of his ten-year quest to tell Fibonacci’s story. Devlin recently answered some questions about his new book for the PUP blog:

KD: This is my third book about the history of mathematics, which already makes it different from most of my books where the focus was on abstract concepts and ideas, not how they were discovered. What makes it truly unique is that it’s the first book I have written that I have been in! It is a first-person account, based on a diary I kept during a research project spread over a decade.

If you had to convey the book’s flavor in a few sentences, what would you say?

KD: Finding Fibonacci is a first-person account of a ten-year quest to uncover and tell the story of one of the most influential figures in human history. It started out as a diary, a simple record of events. It turned into a story when it became clear that it was far more than a record of dates, sources consulted, places visited, and facts checked. Like any good story, it has false starts and disappointments, tragedies and unexpected turns, more than a few hilarious episodes, and several lucky breaks. Along the way, I encountered some amazing individuals who, each for their own reasons, became fascinated by Fibonacci: a Yale professor who traced modern finance back to Fibonacci, an Italian historian who made the crucial archival discovery that brought together all the threads of Fibonacci’s astonishing story, an American math professor who fought against cancer to complete the world’s first (and only) modern language translation of Liber abbaci, and the widow who took over and brought his efforts to fruition after he lost that battle. And behind it all, the man who was the focus of my quest. Fibonacci played a major role in creating the modern commercial world. Yet he vanished from the pages of history for five hundred years, made “obsolete,” and in consequence all but forgotten forever, by a new technology.

What made you decide to write this book?

KD: There were really two key decisions that led to this book. One was deciding, back in the year 2000, to keep a diary of my experiences writing The Man of Numbers. My first history book was The Unfinished Game. For that, all I had to do was consult a number of reference works. It was not intended to be original research. Basic Books asked me to write a short, readable account of a single mathematical document that changed the course of human history, to form part of a series they were bringing out. I chose the letter Pierre De Fermat wrote to his colleague Blaise Pascal in 1654, which most experts agree established modern probability theory, in particular how it can be used to predict the future.

In The Man of Numbers, in contrast, I set out to tell a story that no one had told before; indeed, the consensus among the historians was that it could not be told—there simply was not enough information available. So writing that book would require engaging in a lot of original historical research. I had never done that. I would be stepping well outside my comfort zone. That was in part why I decided to keep a diary. The other reason for keeping a record was to ensure I had enough anecdotes to use when the time came to promote the book—assuming I was able to complete it, that is. (I had written enough popular mathematics books to appreciate the need for author promotional activities!)

The second decision, to turn my diary into a book (which only at the end found the title, Finding Fibonacci), came after The Man of Numbers was published in 2011. The ten-year process of researching and writing that book had turned out to be so rich, and so full of unexpected twists and turns, including several strokes of immense luck, that it was clear there was a good story to be told. What was not clear was whether I would be able to write such a book. All my other books are third-person accounts, where I am simply the messenger. In Finding Fibonacci, I would of necessity be a central character. Once again, I would be stepping outside my comfort zone. In particular, I would be laying out on the printed page, part of my inner self. It took five years and a lot of help from my agent Ted Weinstein and then my Princeton University Press editor Vickie Kearn to find the right voice and make it work.

Who do you expect will enjoy reading this book?

KD: I have a solid readership around the world. I am sure they will all read it. In particular, everyone who read The Man of Numbers will likely end up taking a look. Not least because, in addition to providing a window into the process of writing that earlier book, I also put in some details of that story that I did not fully appreciate until after the book had been published. But I hope, and in fact expect, that Finding Fibonacci will appeal to a whole new group of readers. Whereas the star of all my previous books was a discipline, mathematics, this is a book about people, for the most part people alive today. It’s a human story. It has a number of stars, all people, connected by having embarked on a quest to try to tell parts of the story of one of the most influential figures in human history: Leonardo of Pisa, popularly known as Fibonacci.

Now that the book is out, in one sentence if you can, how would you summarize writing it?

KD: Leaving my author’s comfort zone. Without a doubt. I’ve never been less certain how a book would be received.

Keith Devlin is a mathematician at Stanford University and cofounder and president of BrainQuake, an educational technology company that creates mathematics learning video games. His many books include The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter That Made the World Modern and The Man of Numbers: Fibonacci’s Arithmetic Revolution. He is the author of Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World.

## Marc Chamberland: Why π is important

On March 14, groups across the country will gather for Pi Day, a nerdy celebration of the number Pi, replete with fun facts about this mathematical constant, copious amounts pie, and of course, recitations of the digits of Pi. But why do we care about so many digits of Pi? How big is the room you want to wallpaper anyway? In 1706, 100 digits of Pi were known, and by 2013 over 12 trillion digits had been computed. I’ll give you five reasons why someone may claim that many digits of Pi is important, but they’re not all good.

Reason 1
It provides accuracy for scientific measurements

This argument had merit when only a few digits were known, but today this reason is as empty as space. The radius of the universe is 93 billion light years, and the radius of a hydrogen atom is about 0.1 nanometers. So knowing Pi to 38 places is enough to tell you precisely how many hydrogen atoms you need to encircle the universe. For any mechanical calculations, probably 3.1415 is more than enough precision.

Reason 2
It’s neat to see how far we can go

It’s true that great feats and discoveries have been done in the name of exploration. Ingenious techniques have been designed to crank out many digits of Pi and some of these ideas have led to remarkable discoveries in computing. But while this “because it is there” approach is beguiling, just because we can explore some phenomenon doesn’t mean we’ll find something valuable. Curiosity is great, but harnessing that energy with insight will take you farther.

Reason 3
Computer Integrity

The digits of Pi help with testing and developing new algorithms. The Japanese mathematician Yasumasa Kanada used two different formulas to generate and check over one trillion digits of Pi. To get agreement after all those arithmetic operations and data transfers is strong evidence that the computers are functioning error-free. A spin-off of the expansive Pi calculations has been the development of the Fast Fourier Transform, a ground-breaking tool used in digital signal processing.

Reason 4
It provides evidence that Pi is normal

A number is “normal” if any string of digits appears with the expected frequency. For example, you expect the number 4 to appear 1/10 of the time, or the string 28 to appear 1/100 of the time. It is suspected that Pi is normal, and this was evidenced from the first trillion digits when it was seen that each digit appears about 100 billion times. But proving that Pi is normal has been elusive. Why is the normality of numbers important? A normal number could be used to simulate a random number generator. Computer simulations are a vital tool in modeling any dynamic phenomena that involves randomness. Applications abound, including to climate science, physiological drug testing, computational fluid dynamics, and financial forecasting. If easily calculated numbers such as Pi can be proven to be normal, these precisely defined numbers could be used, paradoxically, in the service of generating randomness.

Reason 5
It helps us understand the prime numbers

Pi is intimately connected to the prime numbers. There are formulas involving the product of infinitely numbers that connect the primes and Pi. The knowledge flows both ways: knowing many primes helps one calculate Pi and knowing many digits of Pi allows one to generate many primes. The Riemann Hypothesis—an unsolved 150-year-old mathematical problem whose solution would earn the solver one million dollars—is intimately connected to both the primes and the number Pi.

And you thought that Pi was only good for circles.

Marc Chamberland is the Myra Steele Professor of Mathematics and Natural Science at Grinnell College. His research in several areas of mathematics, including studying Pi, has led to many publications and speaking engagements in various countries. His interest in popularizing mathematics resulted in the recent book Single Digits: In Praise of Small Numbers with Princeton University Press. He also maintains his YouTube channel Tipping Point Math that tries to make mathematics accessible to a general audience. He is currently working on a book about the number Pi.

## Mircea Pitici on the best mathematics writing of 2016

The Best Writing on Mathematics 2016 brings together the year’s finest mathematics writing from around the world. In the 2016 edition, Burkard Polster shows how to invent your own variants of the Spot It! card game, Steven Strogatz presents young Albert Einstein’s proof of the Pythagorean Theorem, Joseph Dauben and Marjorie Senechal find a treasure trove of math in New York’s Metropolitan Museum of Art, and Andrew Gelman explains why much scientific research based on statistical testing is spurious. And there’s much, much more. Read on to learn about how the essays are chosen, what is meant by the ‘best’ mathematics writing, and why Mircea Pitici, the volume editor, enjoys putting this collection together year after year:

What is new in the new volume of The Best Writing on Mathematics series?

The content is entirely new, as you expect! The format is the same as in the previous volumes—with some novelties. Notably, this volume has figures in full color, in line with the text (not just an insert section of color figures). Also, the reference section at the end of the book is considerably more copious than ever before; besides a long list of notable writings and a list of special journal issues on mathematical topics, I offer two other resources: references for outstanding book reviews on mathematics and references for interviews with mathematical people. I included these additional lists to compensate for the rule we adopted from the start of the series, namely that we will not include in the selection pieces from these categories. Yet book reviews and interviews are important to the mathematical community. I hope that the additional bibliographic research required to do these lists is worth the effort; these references can guide the interested readers if they want to find materials of this sort on their own. The volume is not only an anthology to read and enjoy but also a research tool for the more sophisticated readers.

What do you mean by “the best” writings—and are the pieces you include in this volume really the best?

The superlative “best” in the title caused some controversy at the beginning. By now, perhaps most readers understand (and accept, I hope) that “best” denotes the result of a comparative, selective, and subjective procedure involving several people, including pre-selection reviewers who remain anonymous to me. Every year we leave out exceptional writings on mathematics, due to the multiple constraints we face when preparing these anthologies. With this caveat disclosed, I am confident that the content satisfies the most exigent of readers.

Where do you find the texts you select for these anthologies?

I survey an immense body of literature on mathematics published mainly in academic journals, specialized magazines, and mass media. I have done such searches for many years, even before I found a publisher for the series. I like to read what people write about mathematics. A comprehensive survey is not possible but I aspire to it; I do both systematic and random searches of publications and databases. A small proportion of the pieces we consider are suggested to me either by their authors or by other people. I always consider such pieces; some of them made it into the books, most did not.

Who are the readers you have in mind, for the volumes in this series?

The books are addressed to the public, in the sense that a curious reader with interest toward mathematics can understand most of the content even if their mathematical training is not sophisticated. And yet, at the same time, the series may also interest mathematical people who want to place mathematics in broad social, cultural, and historical contexts. I am glad that we struck a good balance, making this series accessible to these very different audiences.

Mathematics is to a high degree self-contained and self-explanatory, in no need for outside validation. One can do mathematics over a lifetime and not care about “the context.” From a broader intellectual perspective though, interpreting mathematics in social-historical contexts opens up the mind to grasping the rich contribution made by mathematics and mathematicians to ubiquitous aspects of our daily lives, to events, trends, and developments, and to imagining future possibilities. Writing about mathematics achieves such a contextual placement, unattainable by doing mathematics.

What drives you to edit the volumes in this series?

Curiosity, interest in ideas, joy in discovering talented people who show me different perspectives on mathematics; foremost, fear of dogmatism. This last point might sound strange; I readily admit that it is rooted in my life experience, growing up in Romania and emigrating to the U.S. (now I am a naturalized citizen here). Editing this series comes down to a simple recipe: I edit books I will enjoy reading; that sets a high bar by default, since I am a demanding reader. Editing this series allows me to have a personal rapport with mathematics, different from the rapport everyone else has with it. It’s my thing, my placement in relationship with this complicated human phenomenon we call ‘mathematics.’ Or, rather, it is one facet of my rapport to mathematics, one that transpired to the public and gained acceptance. I relate to mathematics in other ways, also important to me—but those facets remain unacknowledged yet, despite my (past) efforts to explain them. Most dramatically, once I went to a business school full of ideas about mathematics and how it relates to the world. At that well-known business school, a handful of faculty dressed down my enthusiasm so efficiently that I learned to be guarded in what I say. After that misadventure of ideas in a place that supposedly encouraged creative thinking, I lost confidence in my persuasive abilities and, disappointed, I gave up on expressing my views on mathematics. Instead, I now rejoice in accomplishing the next best thing: finding and promoting other people’s originality, not mine!

Are you working on the next volume in the series?

The content of the next volume is already selected. We are close to approaching the production stage.

Mircea Pitici holds a PhD in mathematics education from Cornell University and is working on a master’s degree in library and information science at Syracuse University. He has edited The Best Writing on Mathematics since 2010.

## Cipher challenge #1 from Joshua Holden: Merkle’s Puzzles

The Mathematics of Secrets by Joshua Holden takes readers on a tour of the mathematics behind cryptography. Most books about cryptography are organized historically, or around how codes and ciphers have been used in government and military intelligence or bank transactions. Holden instead focuses on how mathematical principles underpin the ways that different codes and ciphers operate. Discussing the majority of ancient and modern ciphers currently known, The Mathematics of Secrets sheds light on both code making and code breaking. Over the next few weeks, we’ll be running a series of cipher challenges from Joshua Holden. Presenting the first, on Merkle’s puzzles.

For over two thousand years, everyone assumed that before Alice and Bob start sending secret messages, they’d need to get together somewhere where an eavesdropper couldn’t overhear them in order to agree on the secret key they would use. In the fall of 1974, Ralph Merkle was an undergraduate at the University of California, Berkeley, and taking a class in computer security. He began wondering if there was a way around the assumption that everyone had always made. Was it possible for Alice to send Bob a message without having them agree on a key beforehand? Systems that do this are now called public-key cryptography, and they are a key ingredient in Internet commerce. Maybe Alice and Bob could agree on a key through some process that the eavesdropper couldn’t understand, even if she could overhear it.

Merkle’s idea, which is now commonly known as Merkle’s puzzles, was slow to be accepted and went through several revisions. Here is the version that was finally published. Alice starts by creating a large number of encrypted messages (the puzzles) and sends them to Bob.

The beginning of Merkle’s puzzles.

Merkle suggested that the encryption should be chosen so that breaking each puzzle by brute force is “tedious, but quite possible.” For our very small example, we will just use a cipher which shifts each letter in the message by a specified number of letters. Here are ten puzzles:

```VGPVY QUGXG PVYGP VAQPG UKZVG GPUGX GPVGG PBTPU XSNHT JZFEB
GJBAV ARSVI RFRIR AGRRA GJRYI RFRIR AGRRA VTDHC BMABD QMPUP
AFSPO JOFUF FOUFO TFWFO UXFOU ZGJWF TFWFO UFFOI RCXJQ EHHZF
JIZJI ZNDSO RZIOT ADAOZ ZINZQ ZIOZZ IWOPL KDWJH SEXRJ IKAVV
YBJSY DSNSJ YJJSY BJSYD KNAJX JAJSK TZWXJ AJSYJ JSFNY UZAKM
QCTCL RFPCC RUCLR WDMSP RCCLD GDRCC LQCTC LRCCL JLXUW HAYDT
ADLUA FMVBY ALUVU LVULZ LCLUZ LCLUA LLUGE AMPWB PSEQG IKDSV
JXHUU VYLUJ XHUUJ UDDYD UIULU DJUUD AUTRC SGBOD ALQUS ERDWN
RDUDM SDDMS VDMSX RDUDM SDDMM HMDSD DMRHW SDDMR DUDMS DDMAW
BEMTD MBEMV BGBPZ MMMQO PBMMV AMDMV NQDMA MDMVB MMVUR YCEZC
```

Alice explains to Bob that each puzzle consists of three sets of numbers. The first number is an ID number to identify the puzzle. The second set of numbers is a secret key from a more secure cipher which Alice and Bob could actually use to communicate. The last number is the same for all puzzles and is a check so that Bob can make sure he has solved the puzzle correctly. Finally, the puzzles are padded with random letters so that they are all the same length, and each puzzle is encrypted by shifting a different number of letters.

Bob picks one of the puzzles at random and solves it by a brute force search. He then sends Alice the ID number encrypted in the puzzle.

Bob solves the puzzle.

For example, if he picked the puzzle on the fifth line above, he might try shifting the letters:

```YBJSY DSNSJ YJJSY BJSYD KNAJX JAJSK TZWXJ AJSYJ JSFNY UZAKM
zcktz etotk zkktz cktze lobky kbktl uaxyk bktzk ktgoz vabln
adlua fupul allua dluaf mpclz lclum vbyzl clual luhpa wbcmo
bemvb gvqvm bmmvb emvbg nqdma mdmvn wczam dmvbm mviqb xcdnp```

```qtbkq vkfkb qbbkq tbkqv cfsbp bsbkc lropb sbkqb bkxfq mrsce
ruclr wlglc rcclr uclrw dgtcq ctcld mspqc tclrc clygr nstdf
svdms xmhmd sddms vdmsx ehudr dudme ntqrd udmsd dmzhs otueg
twent ynine teent wenty fives evenf ourse vente enait puvfh```

Now he knows the ID number is “twenty” and the secret key is 19, 25, 7, 4. He sends Alice “twenty”.

Alice has a list of the decrypted puzzles, sorted by ID number:

 ID secret key check zero nineteen ten seven twentyfive seventeen one one six twenty fifteen seventeen two nine five seventeen twelve seventeen three five three ten nine seventeen ⋮ ⋮ ⋮ seventeen twenty seventeen nineteen sixteen seventeen twenty nineteen twentyfive seven four seventeen twentyfour ten one one seven seventeen

So she can also look up the secret key and find that it is 19, 25, 7, 4. Now Alice and Bob both know a secret key to a secure cipher, and they can start sending encrypted messages. (For examples of ciphers they might use, see Sections 1.6, 4.4, and 4.5 of The Mathematics of Secrets.)

Alice and Bob both have the secret key.

Can Eve the eavesdropper figure out the secret key? Let’s see what she has overheard. She has the encryptions of all of the puzzles, and the check number. She doesn’t know which puzzle Bob picked, but she does know that the ID number was “twenty”. And she doesn’t have Alice’s list of decrypted puzzles. It looks like she has to solve all of the puzzles before she can figure out which one Bob picked and get the secret key. This of course is possible, but will take her a lot longer than the procedure took Alice or Bob.

Eve can’t keep up.

Merkle’s puzzles were always a proof of concept — even Merkle knew that they wouldn’t work in practice. Alice and Bob’s advantage over Eve just isn’t large enough. Nevertheless, they had a direct impact on the development of public-key systems that are still very much in use on the Internet, such as the ones in Chapters 7 and 8 of The Mathematics of Secrets.

Actually, the version of Merkle’s puzzles that I’ve given here has a hole in it. The shift cipher has a weakness that lets Eve use Bob’s ID number to figure out which puzzle he solved without solving them herself. Can you use it to find the secret key which goes with ID number “ten”?