Did you know you could turn a hyperbola into a loop?

Porter Square Books’ penchant for choosing surprisingly excellent books for it’s 2-shelf math collection has remained undimmed over the five years that I’ve known the place. A few months ago, I discovered a book called Elliptic Tales, written by a couple of professors at Boston College. Avner Ash and Robert Gross have taken upon themselves the difficult task of bringing the elegant theory of elliptic curves to the genre of popular mathematical literature.

The writers ask of their readers nothing more than a high school mathematics background, curiosity and the patience to process the beautiful but not immediately obvious results in number theory. It is remarkable to me that one can, in the authors’ opinion, start the book not knowing anything about complex numbers and finish it with a decent grasp of the Birch and Swinnerton-Dyer conjecture, an unsolved problem at the frontier of number theory. Obviously, one of the reasons for the authors’ success at communicating such a difficult topic is their choice of explanatory material. Often, there is more than one approach to a given result, and the writers have judiciously chosen the approach that allows them to reach readers with limited mathematical training, even though that may not necessarily be the most popular approach in the field. While explaining the projective plane, for instance, they do mention the geometric approach and provide a reference to Courant and Robbins’ What is Mathematics? *. However, they treat the geometric approach no further, instead choosing to explain the concept of the projective plane algebraically, using the first principles of rings, fields and abelian groups.

Equally important, I believe, is the fact that they are not shy about using mathematics “as is” when it is warranted. As was the case with most of Sarah and David Flannery’s engaging story In Code, the writers have enough faith that the interested reader will persevere and eventually understand the concepts that are best communicated concisely, systematically and quite precisely using mathematical notation. They are showing us the process by which real mathematicians approach difficult problems. There is another lesson in this, I think. Too often, books with the honorable intention of bringing math to popular audiences take the path of replacing the language of math by everyday metaphors, thinking that this will make for an easier learning curve. The result is often immediate, but deceptively so. Most readers can, in fact, grasp the metaphors readily and understand the patterns at play in the mathematical problem. However this victory is shortlived. The metaphor, by definition, turns out to be an inept analogy, and it can only get a reader so far, before it fails to provide any new insight. Here lies the math writer’s conundrum: if a reader is interested enough at this impasse she will take the trouble to consult a proper text and run with the subject for as long as she has the will and the means. On the other hand, if the reader has grasped the analogy but is still not intrigued enough, the book has failed its didactic mission. Worse still, the reader is now left with a paper-thin understanding of the subject, shored up by pretty English analogies that don’t apply beyond their specific use cases.

The difficult alternative is to instruct a reader, from the very beginning, with a judicious mixture of analogy and mathematical language. The learning curve is now steeper, but progress – at least in this blogger’s case – is surer and longer lasting. This alternative is harder to execute, it demands patience and application from the reader and therefore, it may result in fewer book sales. That is why I find myself full of admiration for Ash and Gross. This isn’t a mathematics textbook by any means, but they make no excuses for their math and that is the way it should be. Furthermore, when they choose to make analogies, they compare the difficult topic at hand to a simpler, more accessible topic that an average high-school student is already familiar with. Therefore, the analogies tend to share a common language, are more precise, and go a longer way in aiding and extending understanding than a random analogy from human emotions**.

I am a little less than half way through the book now, and much of it has been devoted to the following abstract conceit: Let us try to develop new mathematics that allows us to say that a line intersects a curve of degree k in k points. Among the concepts developed to enable us to think the above thought, are complex numbers and intersection multiplicities, which are not that hard to grasp. However, the development of the projective plane was tougher, and though I can safely say that I got the main ideas, I am still not comfortable in my head about the mysterious points at infinity. I’ve worked with coordinate transformations before, yet it completely blew my mind that a simple coordinate transformation like x -> x/z as z tends to zero causes such a profound change in the geometry of curves that I thought I was quite familiar with. It starts off being merely funny: a line and a parabola loop over at infinity under the transformation. But then, things become weird: The two disjoint elements of a hyperbola connect and loop over at infinity too! I’ve had many happy moments traversing this loopy hyperbola in my mind’s eye, and returning to the starting point. At the moment, I am coasting along more or less comfortably, through the material on abelian groups. Elliptic curves have made their brief introductory appearance, but they are still mysterious and I am waiting impatiently for the book to tell me how elliptic curves are related to abelian groups and the whole curves-and-intersections story in the first third of the book.

I recommend the book highly to anyone who likes mathematics (for its own sake).


 What is Mathematics?  is a superb book. I would classify it as a proper math “overview” textbook rather than a representative of the popular science genre.

** The use of stupid analogies is the biggest problem with the books of David Berlinski. If there was ever a clear example of vacuous  mathematical writing, his is it.

5-minute calculation

Of all the things I thought I would be blogging about, this did not cross my mind.

If you’ve watched videos online, you’ve doubtless been bombarded by the annoyingly tacky commercials for 5-Hour Energy, an energy shot that is supposed to contain B-vitamins, amino acids and nutrients, and is supposed to provide benefits that are not confirmed by the Food and Drug Administration.

Sometime in the past few weeks, the folks who run 5-Hour Energy have been flooding the internet with a less tacky commercial in which a polished-looking model describes the results of a survey of primary care physicians:

We asked over 3000 doctors to review 5-Hour Energy and what they said is amazing! Over 73% who reviewed 5-Hour Energy said they would recommend a low-calorie energy supplements to their healthy patients who used energy supplements. 73%!

[…stuff about the product’s low-calorie property and how millions use it …]

Is 5-Hour Energy right for you? Ask your doctor. We already asked 3000!

There are the usual advertising gimmicks at work here. At first glance you would think that 73% of the 3000 doctors, i.e., 2190 doctors, recommended the product. However, as the lady (honestly!) points out, they only recommended a low-calorie energy supplement, not necessarily 5-Hour energy. To find out how many doctors recommended the product, we have to read the fine print on the page, which is not at all difficult given how many times the advertisement shows up in videos of the Daily Show. This fine print states:

Of the 73% of primary care physicians who said they would recommend a low-calorie energy supplement to their healthy patients who use energy supplements, 56% would specifically recommend 5-Hour Energy …

This means that 56% of 2190, i.e., 1226 doctors truly recommend the product being advertised. That’s still a significant number, and a significant fraction of 3000, until you read more of the fine print about how the survey was conducted. Apparently, some doctors were interviewed online, and some were visited by 5-Hour Energy representatives (emphasis mine).

Two surveys were conducted to determine the opinions of primary care physicians regarding energy supplements and 5-Hour Energy: (1) An online survey of 503 participants, and (2) An in-person survey of 5-Hour Energy representatives of 2500 participants (50% of those approached)”. In both, participants agreed to review materials regarding 5-Hour Energy consisting of label and basic description of its ingredients.

Now, anyone with the slightest curiosity would wonder: What happened with the other 50%, i.e. 2500 doctors? Did they refuse the survey? If so, why? Did they not recommend 5-Hour Energy? And if you’re even slightly familiar with data collection and surveys, one would ask: How did you choose which 50% to report results on, and which 50% to reject? Why is your data-point selection criterion not given in the fine print. What is to stop me from believing that you reported the most favorable 50% and summarily rejected the other half?

At any rate, it is clear that the total number of physicians (assuming that they were truly primary care physicians) approached by the 5-Hour Energy representatives was not 3000, but roughly 500 + 2500 + 2500 = 5500. Of these, 1226 doctors, or 22% recommended 5-Hour Energy. Even though these folks choose their words carefully, the advertisement clearly wants you to believe that 73%, i.e., nearly 3 out of 4 doctors recommend the product, when in reality, slightly more than 1 out of 5 doctors do so.

If you want to nitpick further, you might ask: Why should I trust that one doctor out of 5 who makes a recommendation based on the “label and basic description of its ingredients”?. That is not how doctors recommend medicines in general. There has to be at least a proper double-blind study with placebos*. No wonder, the statements of 5-Hour Energy are not approved by the FDA. Still, it is apparently used 9 million times a week. That is a lot of suckers.

5-Hour Energy is owned by this guy, a businessman and philanthropist. He’s just doing what marketers in the lucrative supplement industry do over and over again. They want to mislead, so they choose their advertising monologues carefully while flashing the fine print just to keep their hands clear of the law. One wonders, do they need a 10-Hour-Sleep supplement to get through the night?

* In one of the comments below, a reader, Christy, has provided a link to the webpage containing more information about the test. It shows the label provided to the doctors being surveyed, and claims that a double-blind study was conducted in 2009 but that it is still in peer review. If you are familiar with studies of this sort, I would appreciate knowing from you  whether it is normal for peer review to take 3 years. [Edit added on August 13, 2012].
[Afterward: Just realized that I mentioned 5-Hour Energy so many times that the context-sensitive advertisement robot may want to advertise the product on this page. If you, faithful-reader-without-a-wordpress-account, find that this is indeed the case, do let me know in the comments, for that would be one messed-up irony in the age of the internet. This is when I really hate the fact that I still use the free version of wordpress.com, providing my implicit consent to the serving of ugly ads to non-wordpress netizens in order to pay Matt Mullenweg’s bills.]


A few weeks ago, I railed against David Berlinski’s A Tour of the Calculus – that pretentious, overwritten and exasperating treatment of mathematics left me disillusioned about the effectiveness of the popular maths genre. This view was obviously extreme, given that I had recently read two wonderful books: In Code by the charming father-daughter team of David and Sarah Flannery, and Simon Singh’s superbly researched Fermat’s Enigma. Still, Berlinski’s insufferable book had me hungering for an antidote to all the negativity.  I understand that a more scholarly reader would allow a writer his point of view, and try to see the world through his lenses. Indeed, I do not begrudge Berlinski his right to publish, but I don’t like the view through his lens; it is pseudo-philosophical and manipulative; served with layers of ostentation to give it the appearance of profundity; contrarian merely for the sake of being contrarian.

Happily, the book I read next was not just an antidote, but also an elixir. Marcus du Sautoy’s The Music of the Primes is a splendid yarn about the Riemann Hypothesis. In it, we marvel at the great Gauss as he makes an inspired connection between the distribution of primes and the logarithmic tables. We wince, as Riemann’s housekeeper burns his papers upon his death, and wonder whether an elusive, unpublished proof had gone up in flames. We smile at the rivalry of Gauss and Legendre. We wonder what it must have been like to walk through the halls of Cambridge and encounter Hardy, Littlewood, Russell, Whitehead and Ramanujan all in the space of an afternoon!

But, for all the history of mathematics and its colorful personalities, du Sautoy’s marvellous book holds us because he explains the mathematics lucidly. The analogies are simple and precise – a clock is invoked to explain modular arithmetic, the “music” in the title stands for frequencies associated with complex numbers, the probability of encountering primes is modeled after the tosses of a prime number coin, and so on. As someone who recently began working in applied cryptography, I was particularly thrilled to read du Sautoy’s explanation of the celebrated RSA algorithm, where he likens encryption to shuffling cards in a very large deck.  Yes, the book makes it very clear that the big unsolved problem is to prove that all the zeros of the zeta function have the same real coordinate. But, it also leaves you wanting for more information and more insight about Riemann’s landscape. Armed with the mathematical and historical context from this book, I feel really excited about reading a more technical book on number theory.

This is what a popular maths book should be.

Cloying Calculus

This is a short rant.

I started off liking David Berlinski’s A Tour of the Calculus. Then, his syrupy, metaphor-ridden, simile-mangled, history-bending style of describing math began to slowly grate on my nerves. If you want to make math accessible, by all means, cut through the jargon. Elucidate, present the concept in easier language, or in terms of easier concepts that we understand. But, don’t obfuscate math by comparing it with the human condition. As much as you like to appeal to the poets within us, a mathematical limit is not like romantic love – and describing it as such leaves the concept just as blithely amorphous as before. At best, such strategies can (temporarily) engage the attention of the non-mathematical within us;  they will not teach us any math at all.

Needless to say, I hated the book and hated myself for being impressionable enough to read all but the last three chapters. From the encomiums on the jacket, you find that The New York Times called it “Playful, witty, highly literate”. It is all of those. It is also a lot of bullshit.

There. I have exhausted my vitriol. Now I need some fresh air.