Imagining Numbers: Particularly the Square Root of Minus Fifteen

I cannot think of another book quite like this one, and I am an avid reader of nonfiction. Perhaps the closest is the work of Chet Raymo; but he focuses on astronomy and its poetic significance. This book, by contrast, is really a kind of playful meditation on imagination. I'll admit, I'm not sure what the author's point about imagination was. There may be some philosophy of mind and literary criticism in the background, and I don't know about those things enough to understand his points or to say anything about them. However, I did enjoy that part of the book, despite my general unfamiliarity with that world. In particular, this book inspired me to get online and find out a bit about the way the Ottomans used tulips in their decorations, and soon I found out about carnations and so on as well. Besides that, I was exposed to some poets that I'd heard of, but never read, such as Wallace Stevens, and several others that I'd never heard of. But the math part was lovely. And what I mean is if you are willing to pay attention to the words, and think along with the author of this book, you will come to understand, to feel, imaginary numbers much more vividly than you probably ever have (unless you are a mathematician, of course). Now imaginary numbers, and roots in general for that matter, are not the most straightforward things in math, yet many of us encounter them in high school, along with trig functions, polar coordinates, and other things we didn't understand. What makes this book unique is that the author moves so slowly through his presentation of these things--one element at a time, with pauses for reflection, wafting back to his thoughts on poetry and imagination, including details of historical mathematicians, with their struggles and uncertainties and discoveries and triumphs--that as you move along, your ability to imagine the math increases just as slowly, page by page, until by the end you are comfortable with the idea of using polar coordinates to multiply complex numbers. You have learned to imagine a new dimension of numbers. And, perhaps, witnessed some of the elegance that makes mathematical thought so beautiful to those of us who enjoy it mroe than, say, poetry. I hope that my ability to appreciate poetry increases over the years, because I'd still like to enjoy that field of beauty and human accomplishment as well. This book whet my lips for that. On the other hand, it also refreshed in me some mathematical intuitions that have been dormant for a decade, and I once again much enjoyed the beauty and accomplishment in this field. That makes it a truly rare book, one that I floated through quite joyfully. On the other hand, if either the math or the poetry is too much for you, you might tire of this book. Just a fair warning. This is a book that often rewards, even requires, a few moments of reflection.

Pythagoras is supposed to have said that all things are numbers, and from his time onwards, people have found that mathematics has been surprisingly supple and fitting in explaining the physical universe. If something is mathematically true, then it is among the most trustworthy concepts we can count on in this uncertain world. Yet mathematicians have had to incorporate more inclusive number systems, some of which they have originally found intimidating or even revolting. In Imagining Numbers (particularly the square root of minus fifteen) (Farrar, Straus, and Giroux), Harvard mathematician Barry Mazur has given a poetic and absorbing illustration of what it is to imagine mathematically. It isn't a book for mathematicians, but it has wonderful ideas about mathematics and what it is that mathematicians spend their time doing. Readers will need to do a few calculations, but mercifully few; the endnotes sometimes take a stronger mathematical background, but the actual mathematics within the text is unintimidating. Some numbers just seem to be part of us; even babies seem to know the small ones. But big ones, or fractions, or irrationals, take a bit of imagination to understand. When negative numbers were discovered (or invented), mathematicians could use them practically in calculations, even though they were originally called _fictae_ or fictions. But the square root of a negative number doesn't make much intuitive sense. Think of a square with an area of negative nine; it then has a side equal to the square root of negative nine, which isn't three or negative three. Mazur explains, "This has more the ring of a Zen koan than of a question amenable to a quantitative answer." The square roots of negative numbers would not stay impractical like a Zen koan, however. By the 1700s, mathematicians were solving equations that called for such numbers as answers. René Descartes dismissed them by terming them "imaginary numbers," and the name has stuck, even though they are really no more imaginary than negative numbers or irrationals. Mazur does not mention that these less-than-real, more-than-real numbers have been put to practical work in the real world; they have proved unimaginary enough to be useful in understanding electrical circuits, signal processing, and holography. The complex plane, with real numbers along the horizontal axis and imaginary ones on the vertical (beautifully developed here), is where the Mandlebrot set resides, producing all the resultant hallucinatory colors of pictures of fractals. Mazur has given a history of the idea of imaginary numbers, but he has also tried to explain mathematical imagination in general. He uses many examples from poetry and literature, so a reader who does not know numbers but has some idea about literary images will feel at home. Literary analogies abound here, and Mazur winds up comparing them to mathematical analogies, such as how an algebraic context throws light on a geometric one. D

...this is the wonderful, brilliant book he might write. This is a book about, yes, imaginary numbers--for example, the square root of -15--but also about art, literature, and poetry. Is "imagining" more than visualization? Yes, says Mazur, and one example is Gregor Samsa, Franz Kafka's protagonist in "The Metamorphosis," who awakes one morning to find that he has been turned into an insect. Nabokov's attempt to visualize and determine the type of bug Samsa had become is clearly off the mark, not contributing much to understanding Gregor's condition. One has to imagine the situation on a deeper level. Mazur shows how this kind of understanding is also necessary for mathematicians, beginning with ancient Greeks, Indians, rensaiisance Italians, and others, and working to the present. The Greeks, for example, were incapable of thinking about numbers with powers beyond 2 and 3, because they liked to visualize squares and cubes, but a number to the 4th meant nothing to them. Yet such numbers can be manipulated arithmetically (if that's the correct phrase). Mazur goes back and forth from the numbers work--trying to imagine imaginary numbers--to examples in art and literature, so we inumerate numbnuts never get too tuckered out with the math. I was asked to leave math class by 11th grade (okay, I wasn't exactly "asked"; it was clear there was no option here), and yet I could follow the mathematics in this book. For anyone whose first loves are literature and art, and wish to become acquainted with harder stuff, like math, THIS is the book. And you will learn not only math, but a condensed history of the field, meeting such luminaries as Bhaskara and Cardano. One of the things I liked about "Imagining Numbers" was entering the mind of a major-league mathematician, to find out what he thinks about. And what seems to interest Mazur are very basic problems, not highfalutin equations, but how we perceive things, and how we count, and what a square root is, what the rules are and whether there are any rules, and how ambiguous this can all be. Of course, you could just rent "A Beautiful Mind," and learn that mathematicians have visual hallucinations rather than auditory ones when they become schizophrenic, and then are cured by sleeping with Jennifer Connoly. But I think Mazur's book might be more on target.

This isn't a book for people whose sole focus is mathematics. In fact, it's a book for those who are interested in the imagination and all of its works: poems, novels, paintings, music, and yes, mathematical concepts and ideas. The central question of the book is simply "what happens when we imagine something?" By way of shedding some light on that question, Mazur explores the slow, tentative process by which mathematicians came to feel that they had an adequate picture of what such a number as the square root of -15 actually is.There is a lot of good history of mathematics here. Mazur has done his homework, and at times he departs from the received wisdom among historians because his reading of the primary sources has convinced him otherwise. He displays his erudition as lightly as possible, however, which makes it easy to miss the fact that some of the interpretations are in fact novel. Folks interested in the history of how complex numbers came to be accepted as honest-to-goodness numbers should definitely read this book.And finally, this is a book that gives us a chance to see a great mind in action. It feels as if we have been invited to the author's house and we are sharing in a relaxed and rambling after-dinner conversation in which Mazur, one of the world's greatest living mathematicians, explains to his guests how it is that imagining numbers is like imagining the yellow of a tulip. Anyone in his right mind, had they a chance to actually go to Mazur's house and have this conversation, would be crazy to miss the opportunity. We can't have Mazur in person, but here he is on the page, and it's a pleasure to get to know him.

I would just like to point out a few things, common sense:This book is a popular-science book about mathematics, and as such it is not supposed to be a comprehensive and rigorous introduction to the subject matters. Readers dissatisfied for this reason deserve to be -- if you buy a bike, expecting it to act like a car, well, it won't. Point two: Take all the 'mathematical arguments' the reviewers below present with A GRAIN OF SALT and with extreme suspicion. They don't know what they are talking about... just to refute the argument of the previous reviewer:The "a + bi" is an adequate representation, and the i of course should be included, because otherwise "a + b" is indistinguishible from adding to scalars in R. Another possible representation is the ordered pair (a,b) given that what this means is understood from the context. But in either case, the reviewer is wrong.Actually, this brings us to an important point the book makes: Learn how to think about mathematics -- as long as you get the concept right and can express it in a clear manner to others under any formalism, it does not matter how you do it.. if one prefers, a complex number can be defined as any vector that is the Complex Vector space, the respective field being R. Now this looks different from the other definition but is actually saying the same thing - but with different words, which may (or may not) provide deeper insight into the subject matter. and Common sense should tell you it is more likely an event that a distinguished mathematics prof. at Harvard knows his math better than your average book reviewer - so my final suggestion would be: be wary of reviews, especially of math books, including this one!