Imagining Numbers

peterwayner writes "One mathematician I know told me that the most important lesson he learned was how to read a math book. It did no good, he said, to just start plowing through the theorems because that brought confusion. The key was to skim the book five or six times to get an idea of what the writer was trying to do. Then, and only then, was it possible to figure out the equations. This is what Barry Mazur tries to do in his book Imagining Numbers . There are some equations, graphs and diagrams, but first and foremost he offers plenty of poetry, philosophy and history to lay a foundation for understanding imaginary numbers." Peter's review continues below -- despite its complicated, abstract subject matter, he says that it's "simple enough to be accessible to most who will be interested in it." Imagining Numbers author Barry Mazur pages 267 publisher Farrar, Straus and Giroux rating 8 reviewer Peter Wayner ISBN 0374174695 summary How to imagine imaginary numbers like the square root of minus fifteen.

Much of modern mathematical literature is structured with crisp, scripted precision. First there is theorem one, then theorem two, which leads to theorem three, which could only be followed by theorem four, and so on until we reach theorem n. If you want to learn the mathematics of complex numbers (a +bi), then classic texts (this or this) will get you there.

Some may like this logical progression, but it leaves others cold in the same way that crisp, modern architecture by Mies van de Rohe leaves some craving a more layered, fractured, ornate, organic and just plain fun place to live and work. Less isn't more, as Robert Venturi said, less is a bore.

If you happen to feel a chill when churning through an assembly line of theorems, you might enjoy the treatment of Mazur, a professor at Harvard who seems to spend as much time reading poets like Rilke or Stevens as he does examining old mathematical texts. Mazur is not the kind of machine that turns coffee into theorems-- he's too busy stopping to smell the rhetorical flourishes.

The book isn't aimed at mathematicians per se. The publisher, Farrar, Strauss and Giroux specializes in mainstream literature and that's probably the best pigeonhole for this book. Mazur wants the reader to understand how to think about imaginary numbers, not evaluate some integrals -- and that reader could really be anyone with the desire to think about mathematical things. The book is simple enough to be accessible to most who will be interested in it.

In many ways, Mazur attempted a much harder task than just teaching complex analysis. It's one thing to learn how to find the roots of polynomials, but it's another thing to try to help people get a feeling or an intuition for the square root of minus fifteen. Integers are easy to understand and even feel by counting out things, but imaginary numbers don't seem to exist. Mathematicians have spent many years trying to find the best metaphors and structures to understand how to find answers for all polynomials and it's never been an easy struggle.

The best part of the book is, without doubt, the historical treatment of how other mathematicians confronted the question of irrational and complex numbers. These ideas have always been hard to grasp and it took time to evolve the most compact and consistent nomenclature.

If you're interested in mathematics as more than just a mechanism that churns out answers, you'll probably enjoy the book. It's a light, friendly, philosophical expedition looking for a way to make imaginary numbers work in our minds.

Peter Wayner is the author of Translucent Databases , a book on how to imagine databases that hold no information yet still do useful work. You can purchase Imagining Numbers from bn.com. Slashdot welcomes readers' book reviews -- to see your own review here, read the book review guidelines, then visit the submission page.

Comments

Re:Hands on is the best for those who can

[zzyzx]

"IMHO, assuming you have access in school to the resources: the best way to understand concepts like imaginary numbers is through hands on lab work."

Spoken like a physicist. To a mathematican, the best way to understand imaginary numbers is to say something like, "It annoyed people that the equation 'x^2 = -1' didn't have a solution. They just made up an answer to give them something to play with. Oh it also turns out that this models real world stuff for some reason, but that's not very important."

Ugh

[hal200]

Frankly, I'm about halfway though this book and at times, it's all I can do to keep from tossing it in the trash bin in disgust.

The author seems to be incapable just getting to the subject and explaining himself in a clear and consise manner. Instead, he embarks on these long, florid poetry-filled diatribes about the imagination, and a yellow tulip.

In the few places where he's actually able to keep himself on topic for more than a page, the historical description of the search for imaginary numbers is actually an interesting story in and of itself.

Why he feels the need to expound on it with inapropriate references to poetry and half-baked philosophies on the nature of imagination is beyond me. I'm not against the poetry per se, it's just that there are many occasions where I'll read a passage, hit the poetry, sit back and think, "What the hell does that have to do with the subject?" Even when there is a conceptual link, most of the time, it's very weak. (Of the I'm talking about imagination, and the word imagine is in the poem level)

Frankly, it's been a very dissapointing read. If you're looking for an interesting math book (some people would consider that an oxymoron), I'd recommend David Berlinski's "A Tour of the Calculus" or either of Simon Singh's excellent books ("Fermat's Enigma" and "The Code Book").