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*This is Jordan Ellenberg’s review of Andrew Hodges’s book “One to Nine: the Inner Life of Numbers” in the NY Times:*

My cousin, bound for a top liberal arts college in the fall, was amused when I told her I was reviewing a book about big ideas in mathematics, from the classical to the contemporary. “Don’t they already know everything about math?” she asked. “You know, there’s algebra … and then calculus … and that’s it, right?”

Andrew Hodges, a fellow at Oxford and the author of the lively new book “One to Nine,” would have been horrified, but not surprised. My cousin, in his view, is a victim of the pedagogical tradition that presents math as an eternally fixed array of computations, to be memorized and repeatedly executed without motivation or explanation. The result, he writes, is a “legacy of fear and anxiety generated by schools, which leaves most of their victims with a lifetime of mumbling apologetically about ‘my worst subject.’”

“One to Nine” offers a different model for teaching math — discursive rather than linear, topical rather than abstract and remote, and, above all, manically energetic rather than repetitive and plodding. The book is composed of nine chapters, each focused — very, very softly focused — on one of the first nine natural numbers. Chapter Four, for instance, starts out with the observation that four is a perfect square, and from there skips along to the construction of Latin squares, the irrationality of the square root of two, the definition of the logarithm (whose relation to “four” never comes entirely clear), complex numbers, and the even more exotic quaternions (a number system in which “numbers” are actually strings of four integers, and the product of two numbers depends on the order in which you multiply them!), the theory of four-dimensional spacetime and Einstein’s equation E=mc2 (squares again) before finishing with a short and speculative account of the theory of twistors, one of many competing candidates for the universe’s underlying geometry.

Catch all that? Hodges’ lightning pace allows him to cover in one little book the greatest hits of the last 3,000 years of math and physics, leaving plenty of room for jabs at the Bush administration and quotes from the Pet Shop Boys (these so frequent as to be tabulated in the index).

The overall effect is like that of a lecture by the type of professor who paces back and forth in front of the blackboard, with insistent voice and waving arms, and has trouble adhering to the ostensible syllabus for any extended period. Being this type of professor myself, I can attest that the style is popular with students. But it requires discipline to convey real information as well as enthusiasm.

Hodges often manages that trick. He ably explains the subtly distinct shadings of the word “probability” in statements like “There’s a 90 percent probability I’ll get a six in the next 12 rolls of this die,” “There’s a 90 percent probability of a catastrophic climate change in the next 50 years” and “There’s a 90 percent probability that the current warming of the earth is a result of human activity.” This is the kind of tacit knowledge about mathematical practice that’s unrelated to computations and hard to test, so it doesn’t show up in school math classes. But it’s crucial for arguing sensibly about the planet’s uncertain future. Hodges is also excellent at describing the ways in which mathematical thinking winds its way into every aspect of human life, from fashion to politics to finance, whether we’re aware of it or not. Like Molière’s Monsieur Jourdain, most people have been speaking math their whole lives.

The hyperactive style of “One to Nine” is less well suited to more conventional mathematical material. Georg Cantor’s notoriously mind-boggling hierarchy of infinities is covered in just three pages; the proof of the irrationality of the square root of two, a theorem so profound and startling in its time that its discoverer is traditionally held to have been drowned in the sea by his scandalized Pythagorean colleagues, gets just one. And it’s hard to imagine a novice reader getting much from exposition this compressed:

“Latin squares can be used for devising a duty roster for n dirty jobs in a houseshare of n people. Or, for comparing the effect of n drugs on n animals in some doubtless vital trial. Who said mathematics wasn’t useful? Similar ideas lead to the error-correcting codes which make it possible for computers to communicate reliably. Or, for that matter, to football leagues, speed-dating nights and the plot lines for ‘Desperate Housewives.’”

Too much of the book is like this passage, which is not exactly math, but what Stephen Colbert might call “mathiness”: a series of fervent gestures that gives the impression that mathematical ideas are being expressed, but doesn’t actually deliver the goods. Readers will enjoy sprinting through “One to Nine,” and they’ll certainly learn that there’s much more to the subject than the algebra and calculus taught in high school. But they might not be able to explain exactly what.