Raffi Grinberg: Survival Techniques for Proof-Based Math

GrinbergReal analysis is difficult. In addition to learning new material about real numbers, topology, and sequences, most students are also learning to read and write rigorous proofs for the first time. The Real Analysis Lifesaver by Raffi Grinberg is an innovative guide that helps students through their first real analysis course while giving them a solid foundation for further study. Below, Grinberg offers an introduction to proof-based math:




Raffi Grinberg is an entrepreneur and former management consultant. He graduated with honors from Princeton University with a degree in mathematics in 2012. He is the author of The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs.

Emperor Nero and the “unteachability of mankind”

the emperor nero barrett jacketAncient Rome has long been a source of fascination and enjoys a significant presence in popular culture, though in film and fiction, the life depicted is often highly romanticized. In their new book The Emperor Nero: A Guide to the Ancient Sources, Anthony Barrett, Elaine Fantham, and John Yardley use source material to examine the life of one of Rome’s more notorious and extravagant figures: murderer, tyrant, and likely madman Emperor Nero. The book offers a comprehensive history of Nero’s personal life in the context of historical events that happened during his rule, such as the great fire of Rome. The three authors recently answered some questions on the enduring allure of Rome and in Nero.

There seems to be considerable popular interest in ancient Rome at the moment. Can you explain this?

AB & EF & JY: This is not a uniquely modern phenomenon. Interest in antiquity does seem to have waned during the middle ages, but it enjoyed a vigorous revival with the renaissance, beginning on the fifteenth century, and that interest has never died away. That said, we do seem to be particularly fascinated by the ancient world at the moment. It may be that modern life, so utterly dependent on machines and technology, where so many of our daily transactions are conducted through the computer, without human contact, has created a void, and a general attachment to the past, when life seemed so much more interesting and romantic, is one of the things that we use to fill it. Within the general area of antiquity, the Romans have particular appeal, for the West at least, perhaps because their empire represents the first manifestation of a global superpower, governed by people who are in many respects so different from us, yet, in their ambitions and their motivations, are strikingly similar to us. There are in addition two fortuitous factors. One is that the traditional birth of Christ occurred at the time of the birth of the Roman empire. The over-towering place of the Christ story in the thought of the West has by association kept the Roman empire in our consciousness. Also, at a mundane level, Rome just lends itself well to film and television, with its rich use of imagery and symbols to convey the phenomenon of power. As a consequence, some of the most popular cinematic spectacles, from Quo Vadis, to Spartacus, to Ben Hur, to Gladiator have Rome as their setting and inspiration.

There is a general fascination with Roman Emperors, but Nero seems to attract more attention than most. Why do you think that is?

AB & EF & JY: That is a undoubtedly true. Basically, people find villains engrossing. We may admire Mother Teresa or Saint Francis of Assisi, but for most of us the Jack the Rippers and Vlad the Impalers of this world are far more compelling and entertaining characters. Nero has a special reputation for villainy, and a number of factors have come together to foster that reputation. Perhaps first and foremost, he appears in the Christian tradition as the Antichrist, the first emperor to persecute the Christians after their supposed role in the Great Fire. And he is traditionally blamed for the martyrdoms of the two earliest great champions of the Christian cause, Saints Peter and Paul. We also have reasonably detailed accounts of Nero from three ancient authors, Suetonius, Tacitus and Dio, which ensures a rich store of anecdotes. These anecdotes may be of highly dubious validity, but that does not prevent them from being vastly entertaining. Additionally, Nero was not only cruel, other emperors were no less so, but he disgraced himself in Roman eyes by his public performances on the stage and on the racecourse, providing yet another store of irresistible anecdotes. Finally, there is the simple chance fact that a combination of larger-than-life historical events, which have provided themes for generations of writers and artists over the centuries, occurred during his reign: the murder of his mother, the Great Fire of Rome, the rebellion of Boudica, his melodramatic yet tawdry suicide. Caligula, with a similarly villainous reputation, for similar sorts of reasons, comes a close second, but Nero gets the top billing.

There is a popular belief that Nero was mad. Is that your conclusion too?

AB & EF & JY: Defining and identifying madness is a difficult process, and the tag of ‘mad’ is used loosely to cover anything from wacky eccentricity to severe mental illness. Assessing someone’s psychological state is a great challenge; it is striking that experts who testify in court proceedings after lengthy one-on-one interviews and full access to the patient’s clinical history are often met with scepticism, even ridicule. Thus one has to be even more wary about trying to assess the mental state of someone who lived two thousand years ago and whose conduct is known from incomplete records, produced well after the fact by writers who are hostile to their subject and as often as not relish the prospect of telling the lewdest and most outrageous anecdotes they can amass. It would be dangerous to make broad statements about Nero’s mental health. His conduct does seem to be outrageous at times, but we have to remember that he was only sixteen when he was thrust overnight into a position of enormous power, surrounded by fawning toadies willing to applaud his each and every act. It is probably little wonder that he behaved at times like a spoilt teenager.

There is perhaps one troublesome pattern detectable throughout his adult life that does seem to point to something disturbing. Nero seems to have had a tendency to fall under the spell of powerful women, and his ultimate response to their dominance was invariably a violent one, thus he murdered his mother Agrippina and he reputedly kicked to death his wife Poppaea. He also had a supposed proclivity for look-alikes of these powerful women, using courtesans and actresses (essentially powerless females) to impersonate them. The stories might be fabrications, of course, but the fact that they form a repeated pattern gives them an aura of authenticity. Perhaps something of interest there for the psychologists.

Academics who write about ancient history seem to be more interested in the sources than in the actual events. Can you explain this?

AB & EF & JY: No history, of any period, can be a perfectly accurate record of events. The instant we report on the past our reports are contaminated by the social and intellectual baggage that as historians we carry into the discussion. But for the history of much of the world in recent centuries we do have a considerable body of archives and material records that makes possible a fairly reliable reconstruction of past events. As a broad principle the further back we go, the more tenuous the records. This does not always hold true: we have, for instance, fairly detailed accounts of the Rome in the last century BC, but have very sketchy information about the early middle ages from the fifth century AD on. But as a general broad principle, the historian of a period of history separated from our own is going to face enormous difficulties. The records are in most cases lost or fragmentary, and the contemporary accounts make no pretence of the principled search for the truth that we expect of modern historians. Consequently it is often difficult for the students of ancient history to reconstruct even a reliable outline of events, let alone identify broad historical development. It is admittedly the case that for the Julio-Claudian period we are relatively fortunate, and can draw on the accounts of a number of ancient writers. Yet the material that they have preserved is often inconsistent and even contradictory, and at times reaches levels of absurdity that beggar belief. For much of Nero’s reign we would be hard put to say in which city he was present at any given time, in whose company he passed his time, how he spent the large part of his day. As a consequence as historians we have to spend much time and effort in an attempt simply to work out what was happening. The narrative of past events can be a stimulating and exciting one, but first of all we must work out how to put a reliable version of that narrative together.

What is the main thing that Nero can teach us today?

AB & EF & JY: It is always risky to draw close analogies between events that happen in widely separated periods that have different social and political contexts. That said, it is possible to discern some identifiable common themes that seem to run throughout human history. Perhaps the main thing that Nero teaches us, ironically, is what Churchill called the confirmed unteachability of mankind. In AD 37 the Roman world, including its governing classes, embraced with gusto a youthful and almost totally unknown emperor who had no proven talent for government and virtually no experience of it. His qualifications seem to have consisted exclusively of a general affability and good family connections. This youthful emperor was Caligula, and his subsequent reign was a disaster. Less that 20 years later, in AD 54, Rome went through almost the same scenario, when an even younger and even less experienced Nero was enthusiastically greeted as the new emperor, because of those very same personal qualities, family connections and an amiable manner. It is unsurprising that the net result was quite similar, but it does seem surprising, even astonishing, that Romans had not learned at all from their previous experience. But, as Churchill seemed to have perceived, that reason may derive from a basic flaw in our make-up. The concept of ‘never again’ has a brief shelf-life in the store of human experience.

Anthony A. Barrett is professor emeritus of classics at the University of British Columbia. His books include Livia: First Lady of Imperial Rome. Elaine Fantham is the Giger Professor of Latin, emerita, at Princeton University. Her books includeRoman Literary Culture: From Plautus to Macrobius. John C. Yardley is professor emeritus of classics at the University of Ottawa. His books include Alexander the Great: Historical Sources in Translation. All three recently collaborated on The Emperor Nero: A Guide to the Ancient Sources.

An interview with John Stillwell on Elements of Mathematics

elements of mathematics jacketNot all topics that are part of today’s elementary mathematics were always considered as such, and great mathematical advances and discoveries had to occur in order for certain subjects to become “elementary.” Elements of Mathematics: From Euclid to Gödel, by John Stillwell gives readers, from high school students to professional mathematicians, the highlights of elementary mathematics and glimpses of the parts of math beyond its boundaries.

You’ve been writing math books for a long time now. What do you think is special about this one?

JS: In some ways it is a synthesis of ideas that occur fleetingly in some of my previous books: the interplay between numbers, geometry, algebra, infinity, and logic. In all my books I try to show the interaction between different fields of mathematics, but this is one more unified than any of the others. It covers some fields I have not covered before, such as probability, but also makes many connections I have not made before. I would say that it is also more reflective and philosophical—it really sums up all my experience in mathematics.

Who do you expect will enjoy reading this book?

JS: Well I hope my previous readers will still be interested! But for anyone who has not read my previous work, this might be the best place to start. It should suit anyone who is broadly interested in math, from high school to professional level. For the high school students, the book is a guide to the math they will meet in the future—they may understand only parts of it, but I think it will plant seeds for their future mathematical development. For the professors—I believe there will be many parts that are new and enlightening, judging from the number of times I have often heard “I never knew that!” when speaking on parts of the book to academic audiences.

Does the “Elements” in the title indicate that this book is elementary?

JS: I have tried to make it as simple as possible but, as Einstein is supposed to have said, “not simpler”. So, even though it is mainly about elementary mathematics it is not entirely elementary. It can’t be, because I also want to describe the limits of elementary mathematics—where and why mathematics becomes difficult. To get a realistic appreciation of math, it helps to know that some difficulties are unavoidable. Of course, for mathematicians, the difficulty of math is a big attraction.

What is novel about your approach?

JS: It tries to say something precise and rigorous about the boundaries of elementary math. There is now a field called “reverse mathematics” which aims to find exactly the right axioms to prove important theorems. For example, it has been known for a long time—possibly since Euclid—that the parallel axiom is the “right” axiom to prove the Pythagorean theorem. Much more recently, reverse mathematics has found that certain assumptions about infinity are the right axioms to prove basic theorems of analysis. This research, which has only appeared in specialist publications until now, helps explain why infinity appears so often at the boundaries of elementary math.

Does your book have real world applications?

JS: Someone always asks that question. I would say that if even one person understands mathematics better because of my book, then that is a net benefit to the world. The modern world runs on mathematics, so understanding math is necessary for anyone who wants to understand the world.

John Stillwell is professor of mathematics at the University of San Francisco. His many books include Mathematics and Its History and Roads to Infinity. His most recent book is Elements of Mathematics: From Euclid to Gödel.

Even celebrities misquote Albert Einstein

Calaprice_QuotableEinstein_pb_cvrAlice Calaprice is the editor of The Ultimate Quotable Einstein, a tome mentioned time and again in the media because famous folks continue to attribute words to Einstein that, realistically, he never actually said. Presidential candidates, reality stars, and more have used social media make erroneous references to Einstein’s words, perhaps hoping to give their own a bit more credibility. From the Grapevine recently compiled the most recent misquotes of Albert Einstein by public figures and demonstrated how easy it is to use The Ultimate Quotable Einstein to refute those citations:

Albert Einstein was a wise man, even outside the science laboratory. He has inspired painters, young students and comic book creators. Even budding romantics take advice from him.

So it should come as no surprise, then, that so many people today quote Einstein. Or, to be more precise, misquote Einstein.

“I believe they quote Einstein because of his iconic image as a genius,” Alice Calaprice, an Einstein expert, tells From The Grapevine. “Who would know better and be a better authority than the alleged smartest person in the world?”

Read more here.


Nicholas J. Higham: The Top 10 Algorithms in Applied Mathematics


From “Computational Science” by David E. Keyes in Princeton Companion to Applied Mathematics

In the January/February 2000 issue of Computing in Science and Engineering, Jack Dongarra and Francis Sullivan chose the “10
algorithms with the greatest influence on the development and practice of science and engineering in the 20th century” and presented a group of articles on them that they had commissioned and edited. (A SIAM News article by Barry Cipra gives a summary for anyone who does not have access to the original articles). This top ten list has attracted a lot of interest.

Sixteen years later, I though it would be interesting to produce such a list in a different way and see how it compares with the original top ten. My unscientific—but well defined— way of doing so is to determine which algorithms have the most page locators in the index of The Princeton Companion to Applied Mathematics (PCAM). This is a flawed measure for several reasons. First, the book focuses on applied mathematics, so some algorithms included in the original list may be outside its scope, though the book takes a broad view of the subject and includes many articles about applications and about topics on the interface with other areas. Second, the content is selective and the book does not attempt to cover all of applied mathematics. Third, the number of page locators is not necessarily a good measure of importance. However, the index was prepared by a professional indexer, so it should reflect the content of the book fairly objectively.

A problem facing anyone who compiles such a list is to define what is meant by “algorithm”. Where does one draw the line between an algorithm and a technique? For a simple example, is putting a rational function in partial fraction form an algorithm? In compiling the following list I have erred on the side of inclusion. This top ten list is in decreasing order of the number of page locators.

  1. Newton and quasi-Newton methods
  2. Matrix factorizations (LU, Cholesky, QR)
  3. Singular value decomposition, QR and QZ algorithms
  4. Monte-Carlo methods
  5. Fast Fourier transform
  6. Krylov subspace methods (conjugate gradients, Lanczos, GMRES,
  7. JPEG
  8. PageRank
  9. Simplex algorithm
  10. Kalman filter

Note that JPEG (1992) and PageRank (1998) were youngsters in 2000, but all the other algorithms date back at least to the 1960s.

By comparison, the 2000 list is, in chronological order (no other ordering was given)

  • Metropolis algorithm for Monte Carlo
  • Simplex method for linear programming
  • Krylov subspace iteration methods
  • The decompositional approach to matrix computations
  • The Fortran optimizing compiler
  • QR algorithm for computing eigenvalues
  • Quicksort algorithm for sorting
  • Fast Fourier transform
  • Integer relation detection
  • Fast multipole method

The two lists agree in 7 of their entries. The differences are:

PCAM list 2000 list
Newton and quasi-Newton methods The Fortran Optimizing Compiler
Jpeg Quicksort algorithm for sorting
PageRank Integer relation detection
Kalman filter Fast multipole method

Of those in the right-hand column, Fortran is in the index of PCAM and would have made the list, but so would C, MATLAB, etc., and I draw the line at including languages and compilers; the fast multipole method nearly made the PCAM table; and quicksort and integer relation detection both have one page locator in the PCAM index.

There is a remarkable agreement between the two lists! Dongarra and Sullivan say they knew that “whatever we came up with in the end, it would be controversial”. Their top ten has certainly stimulated some debate, but I don’t think it has been too controversial. This comparison suggests that Dongarra and Sullivan did a pretty good job, and one that has stood the test of time well.

Finally, I point readers to a talk Who invented the great numerical algorithms? by Nick Trefethen for a historical perspective on algorithms, including most of those mentioned above.

This post originally appeared on Higham’s popular website.

Higham jacketNicholas J. Higham is the Richardson Professor of Applied Mathematics at The University of Manchester. He most recently edited The Princeton Companion to Applied Mathematics.

Happy Birthday, Albert Einstein!

What a year. Einstein may have famously called his own birthday a natural disaster, but between the discovery of gravitational waves in February and the 100th anniversary of the general theory of relativity this past November, it’s been a big year for the renowned physicist and former Princeton resident. Throughout the day, PUP’s design blog will be celebrating with featured posts on our Einstein books and the stories behind them.

HappyBirthdayEinstein Graphic 3

Here are some of our favorite Einstein blog posts from the past year:

Was Einstein the First to Discover General Relativity? by Daniel Kennefick

Under the Spell of Relativity by Katherine Freese

Einstein: A Missionary of Science by Jürgen Renn

Me, Myself and Einstein by Jimena Canales

The Revelation of Relativity by Hanoch Gutfreund

A Mere Philosopher by Eoghan Barry

The Final Days of Albert Einstein by Debra Liese


Praeteritio and the quiet importance of Pi

pidayby James D. Stein

Somewhere along my somewhat convoluted educational journey I encountered Latin rhetorical devices. At least one has become part of common usage–oxymoron, the apparent paradox created by juxtaposed words which seem to contradict each other; a classic example being ‘awfully good’. For some reason, one of the devices that has stuck with me over the years is praeteritio, in which emphasis is placed on a topic by saying that one is omitting it. For instance, you could say that when one forgets about 9/11, the Iraq War, Hurricane Katrina, and the Meltdown, George W. Bush’s presidency was smooth sailing.

I’ve always wanted to invent a word, like John Allen Paulos did with ‘innumeracy’, and πraeteritio is my leading candidate–it’s the fact that we call attention to the overwhelming importance of the number π by deliberately excluding it from the conversation. We do that in one of the most important formulas encountered by intermediate algebra and trigonometry students; s = rθ, the formula for the arc length s subtended by a central angle θ in a circle of radius r.

You don’t see π in this formula because π is so important, so natural, that mathematicians use radians as a measure of angle, and π is naturally incorporated into radian measure. Most angle measurement that we see in the real world is described in terms of degrees. A full circle is 360 degrees, a straight angle 180 degrees, a right angle 90 degrees, and so on. But the circumference of a circle of radius 1 is 2π, and so it occurred to Roger Cotes (who is he? I’d never heard of him) that using an angular measure in which there were 2π angle units in a full circle would eliminate the need for a ‘fudge factor’ in the formula for the arc length of a circle subtended by a central angle. For instance, if one measured the angle D in degrees, the formula for the arc length of a circle of radius r subtended by a central angle would be s = (π/180)rD, and who wants to memorize that? The word ‘radian’ first appeared in an examination at Queen’s College in Belfast, Ireland, given by James Thomson, whose better-known brother William would later be known as Lord Kelvin.

The wisdom of this choice can be seen in its far-reaching consequences in the calculus of the trigonometric functions, and undoubtedly elsewhere. First semester calculus students learn that as long as one uses radian measure for angles, the derivative of sin x is cos x, and the derivative of cos x is – sin x. A standard problem in first-semester calculus, here left to the reader, is to compute what the derivative of sin x would be if the angle were measured in degrees rather than radians. Of course, the fudge factor π/180 would raise its ugly head, its square would appear in the formula for the second derivative of sin x, and instead of the elegant repeating pattern of the derivatives of sin x and cos x that are a highlight of the calculus of trigonometric functions, the ensuing formulas would be beyond ugly.

One of the simplest known formulas for the computation of π is via the infinite series 𝜋4=1−13+15−17+⋯

This deliciously elegant formula arises from integrating the geometric series with ratio -x^2 in the equation 1/(1+𝑥^2)=1−𝑥2+𝑥4−𝑥6+⋯

The integral of the left side is the inverse tangent function tan-1 x, but only because we have been fortunate enough to emphasize the importance of π by utilizing an angle measurement system which is the essence of πraeteritio; the recognition of the importance of π by excluding it from the discussion.

So on π Day, let us take a moment to recognize not only the beauty of π when it makes all the memorable appearances which we know and love, but to acknowledge its supreme importance and value in those critical situations where, like a great character in a play, it exerts a profound dramatic influence even when offstage.

LA MathJames D. Stein is emeritus professor in the Department of Mathematics at California State University, Long Beach. His books include Cosmic Numbers (Basic) and How Math Explains the World (Smithsonian). His most recent book is L.A. Math: Romance, Crime, and Mathematics in the City of Angels.

Where would we be without Pi?

Pi Day, the annual celebration of the mathematical constant π (pi), is always an excuse for mathematical and culinary revelry in Princeton. Since 3, 1, and 4 are the first three significant digits of π, the day is typically celebrated on 3/14, which in a stroke of serendipity, also happens to be Albert Einstein’s birthday. Pi Day falls on Monday this year, but Princeton has been celebrating all weekend with many more festivities still to come, from a Nerd Herd smart phone pub crawl, to an Einstein inspired running event sponsored by the Princeton Running Company, to a cocktail making class inside Einstein’s first residence. We imagine the former Princeton resident would be duly impressed.

Einstein enjoying a birthday/ Pi Day cupcake

Einstein enjoying a birthday/ Pi Day cupcake

Pi Day in Princeton always includes plenty of activities for children, and tends to be heavy on, you guessed it, actual pie (throwing it, eating it, and everything in between). To author Paul Nahin, this is fitting. At age 10, his first “scientific” revelation was,  If pi wasn’t around, there would be no round pies! Which it turns out, is all too true. Nahin explains:

Everybody “knows’’ that pi is a number a bit larger than 3 (pretty close to 22/7, as Archimedes showed more than 2,000 years ago) and, more accurately, is 3.14159265… But how do we know the value of pi? It’s the ratio of the circumference of a circle to a diameter, yes, but how does that explain how we know pi to hundreds of millions, even trillions, of decimal digits? We can’t measure lengths with that precision. Well then, just how do we calculate the value of pi? The symbol π (for pi) occurs in countless formulas used by physicists and other scientists and engineers, and so this is an important question. The short answer is, through the use of an infinite series expansion.

NahinIn his book In Praise of Simple Physics, Nahin shows you how to derive such a series that converges very quickly; the sum of just the first 10 terms correctly gives the first five digits. The English astronomer Abraham Sharp (1651–1699) used the first 150 terms of the series (in 1699) to calculate the first 72 digits of pi. That’s more than enough for physicists (and for anybody making round pies)!

While celebrating Pi Day has become popular—some would even say fashionable in nerdy circles— PUP author Marc Chamberland points out that it’s good to remember Pi, the number. With a basic scientific calculator, Chamberland’s recent video “The Easiest Way to Calculate Pi” details a straightforward approach to getting accurate approximations for Pi without tables or a prodigious digital memory. Want even more Pi? Marc’s book Single Digits has more than enough Pi to gorge on.

Now that’s a sweet dessert.

If you’re looking for more information on the origin of Pi, this post gives an explanation extracted from Joseph Mazur’s fascinating history of mathematical notation, Enlightening Symbols.

You can find a complete list of Pi Day activities from the Princeton Tour Company here.

James D. Stein: Putting Excitement Back in High-School Education

High school has been failing its students, according to James D. Stein, mostly by presenting to disinterested students an overwhelming mass of information that they aren’t likely to find interesting or useful. As the author of L. A. Math: Romance, Crime, and Mathematics in the City of Angels, Stein is an expert at keeping subjects interesting for the most reluctant math students.

by James D. Stein

Let me start by repeating something I said in the last post. Where we’ve shortchanged students is at the secondary level. This is where I think we’ve lost sight of the purpose of education, which is to give students a broad general background in subjects deemed necessary but which they probably won’t use, and to prepare them for life as a productive citizen. So here’s what I’d recommend: revamp high school education to give students an enjoyable way to absorb a basic general background in subjects that they probably won’t use later on, and find out what they find interesting and give them a full dose of that.

In 1961, Richard Feynman delivered an introductory lecture at Caltech in which he made the following oft-quoted statement. “If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis that all things are made of atoms — little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied.”

Let’s tweak what Feynman said a little.

If, in some cataclysm, all of the knowledge of humanity were to be destroyed, and only one book passed on to the next generation of creatures, what book would contain the most information about humanity in the fewest words? It would be a book summarizing the Top Ten most important achievements in the most important areas of natural science, social science, the humanities and history, ranked in order of importance by a panel of experts who have devoted their lives to the study of these subjects.

All of a sudden, acquiring a broad general background becomes both achievable and enjoyable – and in a reasonably short period of time. A basic education should tell you what’s important in the important subjects —AND NOBODY KNOWS WHAT THEY ARE!!! Oh, sure, in the sciences you could probably come up with a fairly good list (although the ORDER of the items would not be known, and that’s a key part of this idea) —but other than World Wars I and II, what are the important events in world history? How can we teach the important material in the important subjects, when we don’t even have a consensus as to what they are?

And let’s do it using the Top Ten format, because not only can we find out what are the most important achievements—which should form the basis for a broad general background—but because the Top Ten format is almost universally engaging. Publish a Top Ten list backed by experts, and you’ll know you’ve got a reasonable approximation of the biggies. Moreover, Top Ten lists invite further study and critical thinking.

Just think of the following assignment in a high-school history course: using the Top Ten list in American history as a guide, construct your own Top Ten list of the ten most important events in American history, and justify your choices. I’m guessing that you’d see raging debates in the classroom, with teachers serving as enlightened moderators rather than just ‘sages on the stage’. Maybe I’m overly optimistic, but instead of arguing about Top Ten football teams or Top Ten TV shows, you just might find students suddenly arguing about the relative importance of the Civil War and the American Revolution in American history. You might find students actually doing research to support their points of view. You’d find students thinking about important ideas, rather than memorizing stuff to regurgitate on standardized exams.

Two decades ago, Carl Sagan wrote The Demon-Haunted World: Science As a Candle in the Dark, in which he decried the deplorable lack of scientific knowledge in the general public. I’ll bet if you simply had a list of the Top Ten achievements in physics, chemistry, biology, and mathematics, and if you taught that in a one-semester course, you’d have taken a giant step toward rectifying the problem that so concerned Sagan.

Almost every teacher in every subject feels the same way: students just don’t know what’s important. Let’s find out what is the important stuff in the important subjects, and give every high school student an opportunity to acquire that knowledge—relatively quickly and enjoyably. And then let’s get on with the business of enabling students to become productive members of society by enabling them to take courses at the high school level in what really interests them. It hurts me—a little—to say this, but if a student wants to become a video-game designer, I’d rather have them become a really good video-game designer than a barely passing algebra student. School should be a place where you go to help you fulfill your dreams. And I’m willing to bet you’d find a lot more students getting interested in science and history once they know what experts think is important—and once they’ve had an opportunity to think critically about it for themselves.

LA MathJames D. Stein is emeritus professor in the Department of Mathematics at California State University, Long Beach. His books include Cosmic Numbers (Basic) and How Math Explains the World (Smithsonian). His most recent work is L. A. Math: Romance, Crime, and Mathematics in the City of Angels.

James D. Stein on teaching math in the liberal arts

Time and technology have changed the education system, but James D. Stein insists that we still have room for improvement, particularly in how the mathematics curriculum is handled in high school. In his latest book, L.A. Math: Romance, Crime, and Mathematics in the City of Angels, Stein offers a unique approach that teaches mathematical techniques through liberal arts, making the subject more accessible to those who might otherwise avoid it. Today Stein discusses the challenge of providing students with a broad general background in subjects deemed necessary but which they probably won’t pursue professionally.

Abraham Lincoln and American High Schools

by James D. Stein

February 12th was Lincoln’s birthday.  Like almost everyone in my generation, I was given the official story of Abraham Lincoln and the value of education. You probably know it, how Honest Abe, realizing at an early age the value of education, would trudge miles through snow-covered forest from his log cabin in order to attend school.

I have no doubt that he did indeed so trudge, but over the years I’ve become skeptical of this ‘realizing at an early age the value of education’ explanation. I think Abe, like the vast majority of children (and adults), was basically a pleasure-seeker. Put yourself in his shoes – no TV, no video games, no Facebook. Which is better – a lonely log cabin in the middle of the woods, or a small school, with other children and the opportunity to hear stories far more interesting than anything he could find at home? I’m guessing he went to school in large part because it was a lot more interesting than what he found at home.

Today, however, schools face a problem – its students DO have TV, video games, and Facebook – and they’re stiff competition. Let’s be honest with ourselves; although there are a few students who will find factoring polynomials as interesting as Facebook, most won’t. And let’s continue to be honest with ourselves; although students who plan on entering a career in a STEM subject – science, technology, engineering, mathematics – need to be familiar with algebra, the only time anyone else will encounter an algebra problem during the rest of their life is when one of their children asks them for help with algebra.

And what do we want then? We don’t want both parents to tell their children that they had a really bad experience with math and don’t remember anything, This is not likely to encourage the next generation to pursue the STEM subjects on which our future well-being as a society depends.

So, having cursed the darkness, let me try to light a candle. Our education system does a reasonable job at the primary school level. It’s not perfect, but we do a pretty good job of teaching the three Rs in a highly diverse society. We also do a great job of education at the level of college and graduate school; after all, students come from all over the world to study at our institutions of higher learning, and generally the chief reason our college students go elsewhere is to participate in an exchange program.

Where we truly shortchange students is at the secondary level, where I think we’ve lost sight of the purpose of education – to give students a broad general background in subjects deemed necessary but which they probably won’t use, and to prepare them for life as a productive citizen.

My only expertise is in mathematics, but as I look at the California Framework for Mathematics, insofar as it deals with the high school level, I’m thinking – will anyone other than STEM students use algebra, geometry, or trigonometry in later life? Or even statistics? Probably not. It would be helpful if they understood how statistics functions and what it is used for, rather than knowing how to compute a standard deviation or a confidence interval – which they’ll almost certainly have forgotten within a year.

So here’s what I’d recommend – revamp high school education to give students an enjoyable way to absorb a basic general background in subjects that they probably won’t use later on, and find out what they find interesting and concentrate on doing a solid job of giving them a full dose of that. After all, that’s what we do in college – except for the enjoyable part.

Stay tuned for Jim Stein’s next post on how to give students an enjoyable way to absorb a general background.

LA MathJames D. Stein is emeritus professor in the Department of Mathematics at California State University, Long Beach. His books include Cosmic Numbers (Basic) and How Math Explains the World (Smithsonian). His most recent book is L.A. Math: Romance, Crime, and Mathematics in the City of Angels.

Nick Higham on beautiful equations

The Most Beautiful Equations in Applied Mathematics

By Nick Higham


From p. 171 of PCAM,
typeset in all its splendour in the Lucida Bright font.

The BBC Earth website has just published a selection of short articles on beautiful mathematical equations and is asking readers to vote for their favourite.

I wondered if we had included these equations in The Princeton Companion to
Applied Mathematics
(PCAM), specifically in Part III: Equations, Laws, and Functions of Applied Mathematics. We had indeed included the ones most
relevant to applied mathematics. Here are those equations, with links to the
BBC articles.

  • The wave equation (which quotes PCAM author Ian Stewart). PCAM has a short
    article by Paul Martin of the same title (III.31), and the wave equation
    appears throughout the book.
  • Einstein’s field equation. PCAM has a 2-page article Einstein’s Field
    (note the plural), by Malcolm MacCallum (article III.10).
  • The Euler-Lagrange equation. PCAM article III.12 by Paul Glendinning is about
    these equations, and more appears in other articles, especially The
    Calculus of Variations
    (IV.6), by Irene Fonseca and Giovanni Leoni.
  • The Dirac equation. A 3-page PCAM article by Mark Dennis (III.9) describes
    this equation and its quantum mechanics roots.
  • The logistic map. PCAM article The logistic equation (III.19), by Paul
    Glendinning treats this equation, in both differential and difference forms.
    It occurs in several places in the book.
  • Bayes’ theorem. This theorem appears in the PCAM article Bayesian Inference in Applied Mathematics (V.11), by Des Higham, and in other articles employing
    Bayesian methods.

A natural equation is: Are there other worthy equations that are the
subject of articles in Part III of PCAM that have not been included in the BBC
list? Yes! Here are some examples (assuming that only single equations are
allowed, which rules out the Cauchy-Riemann equations, for example).

  • The Black-Scholes equation.
  • The diffusion (or heat) equation.
  • Laplace’s equation.
  • The Riccati equation.
  • Schrödinger’s equation.

Higham jacketThis article is cross posted to Nick Higham’s blog.

Check out the Princeton Companion to Applied Math here.

Solving last week’s L.A. Math challenge

LA MathWe’re back with the conclusion to last week’s LA Math challenge, The Case of the Vanishing Greenbacks, (taken from chapter 2 of the book). After the conclusion of the story, we’ll talk a little more with the author, Jim Stein. Don’t forget to check out the fantastic trailer for LA Math here.

Forty‑eight hours later I was bleary‑eyed from lack of sleep. I had made no discernible progress. As far as I could tell, both Stevens and Blaisdell were completely on the up‑and‑up.   Either I was losing my touch, or one (or both) of them were wasting their talents, doctoring books for penny‑ante amounts.   Then I remembered the envelope Pete had sealed. Maybe he’d actually seen something that I hadn’t.

I went over to the main house, to find Pete hunkered down happily watching a baseball game. I waited for a commercial break, and then managed to get his attention.

“I’m ready to take a look in the envelope, Pete.”

“Have you figured out who the guilty party is?”

“Frankly, no. To be honest, it’s got me stumped.” I moved to the mantel and unsealed the envelope. The writing was on the other side of the piece of paper. I turned it over. The name Pete had written on it was “Garrett Ryan and the City Council”!

I nearly dropped the piece of paper. Whatever I had been expecting, it certainly wasn’t this. “What in heaven’s name makes you think Ryan and the City Council embezzled the money, Pete?”

“I didn’t say I thought they did. I just think they’re responsible for the missing funds.”

I shook my head. “I don’t get it. How can they be responsible for the missing funds if they didn’t embezzle them?”

“They’re probably just guilty of innumeracy. It’s pretty common.”

“I give up. What’s innumeracy?”

“Innumeracy is the arithmetical equivalent of illiteracy. In this instance, it consists of failing to realize how percentages behave.” A pitching change was taking place, so Pete turned back to me. “An increase in 20% of the tax base will not compensate for a reduction of 20% in each individual’s taxes.   Percentages involve multiplication and division, not addition and subtraction. A gain of 20 dollars will compensate for a loss of 20 dollars, but that’s because you’re dealing with adding and subtracting. It’s not the same with percentages, because the base upon which you figure the percentages varies from calculation to calculation.”

“You may be right, Pete, but how can we tell?”

Pete grabbed a calculator. “Didn’t you say that each faction was out $198,000?”

I checked my figures. “Yeah, that’s the amount.”

Pete punched a few numbers into the calculator. “Call Ryan and see if there were 99,000 taxpayers in the last census. If there were, I’ll show you where the money went.”

I got on the phone to Ryan the next morning. He confirmed that the tax base in the previous census was indeed 99,000. I told Pete that it looked like he had been right, but I wanted to see the numbers to prove it.

Pete got out a piece of paper. “I think you can see where the money went if you simply do a little multiplication. The taxes collected in the previous census were $100 for each of 99,000 individuals, or $9,900,000. An increase of 20% in the population results in 118,800 individuals. If each pays $80 (that’s the 20% reduction from $100), the total taxes collected will be $9,504,000, or $396,000 less than was collected after the previous census. Half of $396,000 is $198,000.”

I was convinced. “There are going to be some awfully red faces down in Linda Vista. I’d like to see the press conference when they finally announce it.” I went back to the guesthouse, called Allen, and filled him in. He was delighted, and said that the check would be in the mail.   As I’ve said before, when Allen says it, he means it. Another advantage of having Allen make the arrangements is that I didn’t have to worry about collecting the fee, which is something I’ve never been very good at.

I wondered exactly how they were going to break the news to the citizens of Linda Vista that they had to pony up another $396,000, but as it was only about $3.34 per taxpayer I didn’t think they’d have too much trouble. Thanks to a combination of Ryan’s frugality and population increase, the tax assessment would still be lower than it was after the previous census, and how many government agencies do you know that actually reduce taxes? I quickly calculated that if they assessed everyone $3.42 they could not only cover the shortage, but Allen’s fee as well. I considered suggesting it to Ryan, but then I thought that Ryan probably wasn’t real interested in hearing from someone who had made him look like a bungler.

My conscience was bothering me, and I don’t like that. I thought about it, and finally came up with a compromise I found acceptable. I went back to the main house.

Pete was watching another baseball game. The Dodgers fouled up an attempted squeeze into an inning‑ending double play. Pete groaned. “It could be a long season,” he sighed.

“It’s early in the year.” I handed him a piece of paper. “Maybe this will console you.”

“What’s this?” He was examining my check for $1,750. “Your rent’s paid up.”

“It’s not for the rent, Pete. It’s your share of my fee.”

“Fee? What fee?”

“That embezzling case in Orange County. It was worth $3,500 to me to come up with the correct answer. I feel you’re entitled to half of it. You crunched the numbers, but I had the contacts and did the legwork.”

Pete looked at the check. “It seems like a lot of money for very little work. Tell you what. I’ll take $250, and credit the rest towards your rent.”

A landlord with a conscience! Maybe I should notify the Guinness Book of Records. “Seems more than fair to me.”

Pete tucked the check in the pocket of his shirt. “Tell me, Freddy, is it always this easy, doing investigations?”

I summoned up a wry laugh. “You’ve got to be kidding. So far, I’ve asked you two questions that just turned out to be right down your alley. I’ve sometimes spent months on a case, and come up dry. That can make the bottom line look pretty sick. What’s it like in your line of work?”

“I don’t really have a line of work. I have this house and some money in the bank. I can rent out the guesthouse and make enough to live on. People know I’m pretty good at certain problems, and sometimes they hire me. If it looks like it might be interesting, I’ll work on it.” He paused. “Of course, if they offer me a ridiculous amount of money, I’ll work on it even if it’s not interesting. Hey, we’re in a recession.”

“I’ll keep that in mind.”   I turned to leave the room. Pete’s voice stopped me.

“Haven’t you forgotten something?”

I turned around. “I give up. What?”

“We had a bet. You owe me five bucks.”

I fished a five out of my wallet and handed it over. He nodded with satisfaction as he stuffed it in the same pocket as the check, and then turned his attention back to the game.

What made you include this particular idea in the book?

JS: The story features one of the most common misunderstandings about percentages.  There are innumerable mistakes made because people assume that percentages work the same way as regular quantities.  But they don’t — if a store lowers the cost of an item by 30% and then by another 20%, the cost of the item hasn’t been lowered by 50% — although many people make the mistake of assuming that it has.  I’m hoping that the story is sufficiently memorable that if a reader is confronted by a 30% discount followed by a 20% discount, they’ll think “Wasn’t there something like that in The Case of the Vanishing Greenbacks?

There are 14 stories in the book, and each features a mathematical point, injected into the story in a similar fashion as the one above.  I think the stories are fun to read, and if someone reads the book and remembers just a few of the points, well, I’ve done a whole lot better than when I was teaching liberal arts math the way it is usually done.

James D. Stein is emeritus professor in the Department of Mathematics at California State University, Long Beach. His books include LA Math, Cosmic Numbers (Basic) and How Math Explains the World (Smithsonian).