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

pcam-p346-newton.jpg

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,
    minres)
  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

pcam-p171-wave.jpg

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
    Equations
    (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).

Try your hand at solving an L.A. Math mystery

If you caught the rather incredible trailer for L.A. Math, you know it’s not your typical scholarly math book. Romance, crime, and mathematics don’t often go hand in hand, but emeritus professor in the Department of Mathematics at California State University Jim Stein cooked up the idea for an unconventional literary math book that would speak to students in his liberal arts math class. The end result is an entertaining, backdoor approach to practical mathematics knowledge, ranging from percentages and probability to set theory, statistics, and the mathematics of elections. Recently, Stein spoke to us about writing L.A. Math. Not only that, he left us with a mathematical mystery to solve.

L.A. Math is definitely an unusual book.  Brian Clegg described it by saying “It’s as if Ellery Queen, with the help of P. G. Wodehouse, spiced up a collection of detective tales with a generous handful of practical mathematics.”  How did you happen to write it?

JS: I absolutely loved it when he described it that way, because I was brought up on Ellery Queen.  For younger readers, Ellery Queen was one of the greatest literary detectives of the first half of the twentieth century, specializing in classic Sherlock Holmes type cases.  The Ellery Queen stories were written by the team of Manfred Dannay and Frederick Lee — and my mother actually dated one of them!

LA MathThe two other mystery writers who influenced me were Agatha Christie and Rex Stout.  Rex Stout wrote a series featuring Nero Wolfe and Archie Goodwin; the books are presumably written by Archie Goodwin describing their cases, so I used that as the model for Freddy Carmichael.  The relationship between Archie and Nero also served, somewhat, as a parallel for the relationship between Freddy and Pete.  Nero and Pete both have addictions — Nero wants to spend his time eating elaborate cuisine and raising orchids, and Pete wants to spend his time watching and betting on sports.  It’s up to Archie and Freddy to prod them into taking cases.

How does Agatha Christie enter the picture?

JS: I’d taught liberal arts mathematics — math for poets — maybe ten times with temporary success but no retention.  Students would learn what was necessary to pass the course, and a year later they’d forgotten all of it.  That’s not surprising, because the typical liberal arts math course has no context that’s relevant for them.  They’re not math-oriented.  I know I had several history courses discussing the Battle of Azincourt, but I don’t remember anything about it because it has no context for me.

Agatha Christie’s best-known detective is Hercule Poirot, and one day I was in a library reading a collection of short stories she had written entitled The Labors of Hercules.  Christie had a background in the classics, and did something absolutely brilliant — she constructed a series of twelve detective stories featuring Hercule Poirot, each of which was modeled, in one way or another, around the Twelve Labors of Hercules in classical mythology.  I thought to myself — why don’t I do something like that for topics in liberal arts math?  Maybe the students would remember a few of the ideas because they’d have the context of a story from which to remember it.

Could you give an example?

JS: How about this?  Why don’t we take a story from the book, and present it the way Ellery Queen would have.  Ellery Queen always played fair with the reader, giving him or her all the clues, and after all the clues had been presented, EQ would write a paragraph entitled “Challenge to the Reader”.  EQ would tell the reader “Now you have all the clues.  Can you figure out whodunit?” — or words to that effect.

OK, here’s what we’ll do.  We’ll take The Case of the Vanishing Greenbacks, Chapter 2 in L.A. Math, and present the story up to the crucial point.  Then we’ll let the reader try to figure out whodunit, and finish the story next week.

Chapter 2 – The Case of the Vanishing Greenbacks

   The phone rang just as I stepped out of the shower. It was Allen.

“Freddy, are you available for an embezzlement case?”

My biggest success had been in an embezzlement case involving a Wall Street firm specializing in bond trading. Allen had given me a whopping bonus for that one, which was one of the reasons I could afford to take it easy in L.A. I had done well in a couple of other similar cases, and had gotten the reputation of being the go-to guy in embezzlement cases. It never hurts to have a reputation for being good at something. Besides, you don’t see many guys in my line of work who can read balance sheets.

I’ve always felt it’s important to keep the cash flow positive, and the truth was that I was available for a jaywalking case if it would help the aforementioned cash flow. But it never hurts to play a little hard-to-get.

“I can probably clear my calendar if it looks interesting.”

Allen paused for a moment, either to collect his thoughts or to take a bite of one of those big greasy pastrami sandwiches he loves. “I’m pretty sure you’ll find it interesting. It’s stumped some people in L.A., and I told them I had a good man out there. BTW, that’s you.”

It’s nice to be well thought of – especially by someone in a position to send you business. I knew that Allen’s firm, though headquartered in New York, had arrangements with other firms in other cities. I didn’t really care about the details as long as the check cleared – which it always had.

“I’m certainly willing to listen. What’s the arrangement?”

“Consulting and contingency fee. Fifty‑fifty split.”

That was our usual arrangement. Burkitt Investigations got a guaranteed fee, plus a bonus for solving the case. Allen and I split it down the middle.

“OK, Allen, fill me in.”

“Ever heard of Linda Vista, Freddy?”

Temporary blank. Movie star? Socialite? Then I had it. Linda Vista was a town somewhere in Orange County with a big art community.

For those of you not up on California politics, Orange County is a bastion of conservatism. You have Orange County to thank, or blame, for Richard Nixon and Ronald Reagan. But Linda Vista, which my fragmentary Spanish translates as “pretty view”, was different from your basic Orange County bastion.

The vista in Linda Vista was sufficiently linda that it had attracted a thriving artistic community.   There were plenty of artists in Linda Vista, and most of them were liberals.

As a result, Linda Vista was highly polarized. The moderates were few and far between. On the left, you had the artists, with their funky bungalows and workshops. On the right, you had the stockbrokers and real-estate moguls, living in gated communities so they wouldn’t have to have any contact with the riff-raff, except for the tradesmen delivering or repairing stuff. However, there were enough artists and hangers-on to acquire political clout – after all, it’s still one man-one vote in a democracy, rather than one dollar-one vote. Pitched battles had raged over practically every issue from A (abortion) to Z (zoning), and many of these battles had made state and even national news.

That’s all I knew about Linda Vista, other than not to try to drive down there at rush hour, which turned one hour on the 405 to more than twice that. The obvious question was: what kind of a contingency case had they got? So I asked it.

Allen filled me in. “The city is out a bunch of bucks, and each side is accusing the other of fraud and embezzlement. Because of the split in the political situation, the City Manager gave half the budget to the conservatives, and the other half to the liberals, letting each determine how to spend its half. Both sides claim to have been shortchanged.”

Allen paused to catch his breath. “I’ve got a friend who works in the City Manager’s office. I told him I had a good man out there who’d done a lot of first‑class work in embezzlement cases. Want to take a look at it?”

“Sure. How much time should I put in before I throw in the towel?” In other words, how much is the consulting fee?

“As much as you like.” In other words, since Allen’s meter wasn’t running, feel free to burn some midnight oil. “The consulting fee is $3,000, upped to ten if you figure it out and get proof.” You don’t have to be an expert at division to realize that I was guaranteed a minimum of $1,500 for the time I put in, and $5,000 if I doped it out. You also don’t have to be an expert at division to realize that Allen was getting the same amount for making a phone call. I decided to be reincarnated as an employer rather than an employee.

Allen gave me a brief description of the protagonists, and I spent a good portion of the evening with a pot of coffee and my computer, getting some background information on them. I’ll say one thing for the Information Age; it’s a lot easier to run a background check on people than it used to be. What with search engines and social networks, you save a lot on gas money and shoe leather.

The next morning I waited until after rush hour, and made the trek to Linda Vista. The City Hall was located in a section of town where the vista was a long way from linda, unless strip malls filled with 7‑11s and fast-food stores constitute your idea of attractive scenery. I found a place to park, straightened my coat and tie, and prepared for the interviews.

I was scheduled to have three of them. I had been hoping to arrange for longer interviews, but everyone’s in a rush nowadays, and I was getting a quarter-hour with each, tops. They’d all been interviewed previously – Allen had mentioned that this case had stumped others – and people are generally less than enthusiastic about being asked the same questions again. And again. The first interview was with Everett Blaisdell, conservative city councilman, who would explain why the conservatives happened to be short. The next was with Melanie Stevens, liberal city councilwoman, ditto. The last interview would be with Garrett Ryan, City Manager.

I have a bad habit. My opinion of members of groups tends to be formed by the members of those groups that I have seen before. Consequently, I was expecting the conservative Everett Blaisdell to look like a typical paunchy southern senator with big jowls. So I was a little surprised to discover that Everett Blaisdell was a forty-ish African-American who looked like he had spent years twenty through thirty as an NBA point guard.

He got right down to business. “I want you to know,” he barked, “that everything that we have done with our budget allocation has been strictly by the book. Our expenses have been completely documented.” He handed me a folder full of ledger sheets and photos of checks, which I glanced at and stashed in my briefcase.

Blaisdell was clearly angry. “The business community is the heart of Linda Vista, and it is ridiculous to suggest that it would act in a manner detrimental to its citizens. We are $198,000 short in our budget.”

You don’t expect NBA point guards to get out of breath too easily, considering the time they have to go up and down the court, but maybe Blaisdell wasn’t in shape. He paused, giving me a chance to get a question in edgewise. “Just what do you think has happened, Mr. Blaisdell?” I inquired mildly.

“I know what has happened. Melanie Stevens and her radical crowd have managed to get hold of that money. They want $200,000 to fund a work of so‑called art which I, and every right‑thinking citizen of Linda Vista, find totally offensive. It’s mighty suspicious that the missing funds, $198,000, almost precisely cover the projected cost of the statue.”

I was curious. “If you don’t mind my asking, exactly what is this statue?”

Blaisdell’s blood pressure was going up. “They are going to build a scale replica of the Statue of Liberty and submerge it in Coca‑Cola. You may know that Coca‑Cola is acidic, and it will eventually dissolve metal. They say that this so‑called dynamic representational art represents the destruction of our civil liberties by over‑commercialization. Well, let me tell you, we’ll fight it.”

He looked at his watch. “Sorry, I’ve got another appointment. When you find out what those scum have done with the money, let me know.” He walked me to his door.

It took a few minutes to locate Melanie Stevens’ office, as it was in a different wing of the building, possibly to minimize confrontations between her and Blaisdell. It was a bad day for stereotypes. My mental picture of Melanie Stevens, ultra‑liberal, was that of a long-haired hippie refugee from the ’60s. The real Melanie Stevens was a pert gray‑haired grandmother who looked like she had been interrupted while baking cookies for her grandchildren. She, too, was evidently on a tight schedule, for she said, “Sorry, I can only give you about ten minutes, but I’ve made copies of all our expenses.” More ledger sheets and photos of checks went into my briefcase.

“Let me tell you, Mr. Carmichael, that we could have used that $198,000. We planned to use it for a free clinic. I know exactly what has happened. Blaisdell has doctored the books. I’m sure glad that Ryan had the guts to ask you to look into it.”

“Blaisdell seems to think that your people are responsible for the missing funds,” I observed.

She snorted. “That’s just typical of what they do. Whenever they’re in the wrong, they lie and accuse the other side of lying. They rip off the community, and channel money into PACs. Political action committees. Or worse. Blaisdell knows he faces a stiff battle for re-election, and I wouldn’t be the least bit surprised to find that money turning up in his campaign fund.”

“He seems to think that you are going to use the funds for an art project, rather than a free clinic,” I remarked.

“He’s just blowing smoke. He knows quite well that the statue will be funded through private subscription.” She looked at her watch. “Let me know when you pin the loss on them.”

I left Stevens’ office for the last interview, with Garrett Ryan, whose anxious expression made it clear that he was not a happy camper. “Have you got any ideas yet?” he asked.

I shook my head. “I’ve just talked to Blaisdell and Stevens. They’ve each handed me files containing what they consider to be complete documentation. They’ve each given me a story asserting their own innocence, and blaming the other. I take it that the missing amount is $198,000?”

Now it was Ryan’s turn to shake his head. “No, each side says that it is missing $198,000. Quite a coincidence. And I’ll tell you, Mr. Carmichael, despite the animosity between them, I think that they are both honorable individuals. I find it difficult to believe that either would rip the city off.”

I focused on Ryan’s coincidence. “It’s funny that they are both short exactly the same amount. Perhaps you could tell me a little more about the budgetary process.”

“It’s really quite simple. Each resident of Linda Vista is taxed a fixed amount. Any complicated tax scheme would just result in a full employment act for accountants. The previous census resulted in a $100 assessment per individual. The population of Linda Vista increased by 20% since the last census. We didn’t need any increase in operating expenditures; under my guidance we’ve done a fiscally conservative and frugal job of running the city. As a result, the Council voted to reduce everybody’s taxes by 20%. Needless to say, this was a very popular move.”

“I’ll bet it was. Did everyone pay their taxes, Mr. Ryan?”

“Everybody. We’re very proud of that ‑‑ a 100% collection rate. Despite what you may have heard, the citizens of Linda Vista are very civic‑minded. Liberals and conservatives alike.”

I’ve spent enough time with balance sheets to know that accuracy is extremely important. “Was this population increase exactly 20%, or is that merely an approximate figure?”

Ryan consulted a sheet of paper. “Exactly 20%. I have a sheet of printout that gives information to four decimal places, so I can be quite sure of that.”

Just then a phone rang. Ryan picked it up, and engaged in some political doubletalk. After a few minutes he replaced the receiver. “Sorry, Mr. Carmichael. I’m behind schedule. Let me know if you make any progress.”   We shook hands, and I left.

A couple of hours later, I got home, having stopped for a bite but still avoiding rush-hour traffic. Pete noticed my presence, and asked, “So how’d things go in Linda Vista, Freddy?”

“I had a pretty interesting day. Want to hear about it?”

He nodded. I took about fifteen minutes to describe the problem and the cast of characters. “It looks like I’ll have to spend a day or so looking over the books.”

Pete shook his head. “It seems pretty clear to me.”

I’d seen it before — everybody’s a detective. Amateurs always think they know who the guilty party is, because it fits in with their preconceptions. I didn’t know whether Pete had cast Blaisdell in the role of a political fat-cat out to line his campaign war chest, or whether he was a conservative who saw Melanie Stevens as a radical troublemaker. Anyway, you’ve got to learn not to jump to conclusions in my line of work.

“You can’t do it like that, Pete. You’ve got to trace down the paper trails. I’ve done this lots of times.”

Pete grabbed a piece of paper, scribbled something on it, and sealed it in an envelope. “Five dollars will get you twenty that the name of the guilty party is inside this envelope.”

Pete needed taking down a peg. Maybe two pegs. Besides, I liked getting four‑to‑one odds on what was obviously an even‑ money proposition. “You’ve got a bet,” I said. We wrote our names on the envelope, and Pete put it on the table next to the HDTV.

“Whenever you’re ready, we’ll unseal the envelope.” I headed back to the guesthouse for a session with the books.

Challenge to the Reader: You have all the clues. Can you name the party responsible for the missing greenbacks? We’ll give you until the next blog to figure it out, when we’ll present the conclusion to the story.

Anna Frebel on the search for the oldest stars

Frebel jacketAstronomers study the oldest observable stars in the universe in much the same way that archaeologists study ancient artifacts on Earth. Stellar archaeologist Anna Frebel is credited with discovering several of the oldest and most primitive stars, and her book, Searching for the Oldest Stars is a gripping firsthand account of her work. Recently she took the time to answer some questions:

What is your main research topic and what is stellar archaeology?

AF: My work is broadly centered on finding the oldest stars in the universe and using them to explore how the first stars and the first galaxies formed soon after the Big Bang. This works because these ancient stars are about 13 billion years old and they are still shining. The universe itself, by comparison, is 13.8 billion years old. I find these ancient stars in the outskirts of the Milky Way galaxy, using a large telescope. I’m also researching how the chemical elements heavier than hydrogen and helium were first created in those early stars, which ultimately allowed Earth to form and to bring about life in the universe.

What is your biggest discovery?

AF: I have been fortunate enough to discover several “record holding stars”. In 2007, I found a 13.2 billion year-old star, which is incredibly old. This followed the 2005 discovery of the chemically most primitive star – a star of the second generation of stars to have formed in the universe. Since then, I have analyzed some incredible ancient stars in dwarf galaxies that orbit the Milky Way galaxy, and together with my team, we have recently beaten said 2005 record, which was enormously exciting.

Why do people say we are made from stardust?

AF: We humans are made from all sorts of different chemical elements, mostly carbon. We breathe oxygen and nitrogen, we wear silver and gold jewelry. All these elements were once, atom by atom, created inside different kinds of stars and their supernova explosions over the course of billions of years. Studying this evolution of the chemical elements in the universe with the help of ancient stars means that I’m literally studying the cosmic origins of the building blocks of life. So we really are closely connected with the universe, far more than we realize.

How did you decide to become a scientist?

AF: From a young age I knew I wanted to study stars. They were just so fascinating to me, these big spheres of gas, fusing new elements to gain energy to shine for eons in the sky. Fortunately, I received good advice during high school on how to become an astronomer. After studying physics until 2002, I turned to astronomy and the rest is history. Today, I take pride in sharing my story with young people and the general public by telling them what astronomers do on a daily basis, and how scientific results are achieved. I am passionate about conveying the importance of science literacy to the young and the young at heart while inspiring them with the beauty and mystery of the cosmos.

What kind of telescope is used for your astronomical observations?

AF: Astronomers use all kinds of different telescopes on Earth as well as from space to peer deep into the cosmos. It depends on the type of project and the brightness of the objects which telescope is best suited. Space observations are being carried out remotely, whereas ground-based observations are still done by the astronomer who has to travel to the telescope. More and more telescopes are becoming automated to enable remote controlled “office observing”.

Anna Frebel in front of the 6.5m Magellan Telescope in Chile.

Anna Frebel in front of the 6.5m Magellan Telescope in Chile.

Are you traveling to any telescopes?

AF: Yes, I regularly fly to Chile to the Magellan Telescopes to carry out my observations. These are some of the largest telescopes in the world and the dark night sky in the Southern Hemisphere is terrific for studying the cosmos. It’s the favorite part of my job and I love discovering new facts about the universe through these observations!

What does it mean when you say you’re going observing?

AF: To use the telescopes, you have to fly to Chile. First to Santiago, then to La Serena and from there is a 2-3h drive up the mountains of the Atacama Desert where the telescopes are. There are guest rooms there for the observers to sleep during the day and the observatory chefs are cooking delicious meals for everyone. Dinner is eaten together by all observers, including the technical staff. It’s a little community with the sole purposes of caring for the telescopes and obtaining exquisite astronomical observations all night long of a breathtaking sky.

What does a typical night at the telescope look like?

AF: All preparations for the night happen during the afternoon while it’s still light outside. After sunset, I usually choose the first targets from my list, which I begin to observe soon after dark. Each star is observed for 10-30 minutes. We immediately inspect each observation and then decide on the fly whether we need more data or not. If we have found an interesting old star we may choose to immediately observe it for a few more hours.

Did anything ever go wrong at the telescopes?

AF: Of course! Mostly when it’s cloudy because then we can’t observe any starlight. This can be very frustrating because it can mean that we have to come back to the telescope a year later to try again. Clouds spell bad luck. Other times, the air layers above the telescope are often not as smooth as is required. This makes the stars twinkle and appear less sharp, which means less good data and longer exposure times. And sometimes there are technical problems with the telescope too.

How do you get your telescope time? Can I go to your telescope and observe, too?

AF: To obtain telescope time, astronomers have to submit a proposal to a committee that selects the best projects and awards them the time. The proposal contains a detailed description of the project and the technical details on what information is being sought. Telescope use is restricted to professional astronomers because of the considerable expense. The cost is about USD 50,000 to 100,000 per night, depending on the telescope, and often paid by various institutions and universities who jointly operate observatories. While this is a lot of money, it’s actually not that much in comparison to many other research facilities.

Are there any special moments at the telescope that you remember in particular?

AF: Yes, going observing is always magical and memorable. Of course I particularly remember big discoveries and the excited nervousness of checking and checking whether we didn’t make a mistake and that the discovery was really what it appeared to be. Then, there have been the frustrating moments of sitting at the telescopes for nights on end listening to the rain and flying home empty-handed. I have been there when severe technical problems and even a bush fire prevented observing during clear nights. But I always associate observing with the most colorful sunsets, the calm and peaceful atmosphere up in the mountains, and of course the sleepless but exciting nights.

Anna Frebel is the Silverman (1968) Family Career Development Assistant Professor in the Department of Physics at the Massachusetts Institute of Technology. She is author of Searching for the Oldest Stars, and has received numerous international honors and awards for her discoveries and analyses of the oldest stars. She lives in Cambridge, Massachusetts.

New Physics & Astrophysics Catalog

We invite you to browse our Physics & Astrophysics 2016 catalog:

 

Interacademy Partnership Check out Doing Global Science, an introductory guide to responsible science in our globalized society. Written by a committee of leading scientists from all over the world, this text is required reading for anyone involved in scientific inquiry.
Thorne Modern Classical Physics is a graduate-level text and reference book for first-year students that covers statistical physics, optics, elastodynamics, fluid mechanics, plasma physics, and special and general relativity and cosmology.
Maoz

A. Zee has contributed another new title to our In a Nutshell series entitled Group Theory in a Nutshell for Physicists. He takes all the nuts and bolts of a mathematical subject and makes it accessible for physicists. PUP is also publishing the second edition of Astrophysics in a Nutshell by Dan Maoz this season, a work that has become a standard text in courses on astrophysics.

If you would like updates of new titles emailed to you, subscribe to our newsletter.

Finally, PUP will be at the American Physical Society March Meeting in Baltimore from March 14 to March 18.