Customer Reviews

Imagining Numbers: (particularly the square root of minus fifteen)

5 star:		(5)
4 star:		(3)
3 star:		(4)
2 star:		(2)
1 star:		(4)

 

 

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.

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. Deep structural analogies have always brought impressive understanding in diverse mathematical fields, as mathematicians have striven to make the analogies into equalities. Readers who stick to Mazur's rich and happy exposition may not start using imaginary numbers practically, but they will gain insight into just why Mazur loves doing mathematics and how imagination can be extended in to previously forbidding numerical territory.

I read this book during the leisure time of a vacation, when I could have spent hours on tangents if the situation called for it. In fact, I did take the time to do some of the calculations suggested in the text. Unfortunately, despite its moments of brilliance, I did not find the book in general suitable for leisurely contemplation, but found myself racing toward the conclusion and both relieved and disappointed when I reached it.

Mazur is trying very hard to reach the liberal arts audience. To that end, he throws in piles of philosophical speculations, with copious references to classical works. That in itself is not a fault; but the execution of it is awkward. He slavishly alternates mathematical teaching with philosophical speculation, thus destroying any sense of continuity in the narrative. I found myself skipping the "liberal arts" portions so I could continue the thread of mathematical reasoning without interruption.

He is at his best when he introduces the complex plane. Its connection with rotation is beautifully made, and is the one piece of "new" information I took from the book. I wish he had emphasized more the "Fundamental Theorem of Algebra," which shows how inclusion of imaginary numbers completes the theory of solutions of algebraic equations, but I won't quibble about that.

In the end, I must conclude that Mazur's goal of helping us imagine what must have been in the minds of the inventors/discoverers of imaginary numbers is a failure. Less philosophy and more history would have been a better path to that end. I would love to read what mathematicians themselves were thinking and saying about this new theory as it was being developed, but there is precious little of that.

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!

The author is trying to bridge the gap between the "two cultures" -scientific minded, and the literary minded. He is trying to target the literary camp in particular.

He gently tries to introduce the reader to complex numbers by use of examples. We can 'imagine' positive integers, and we can imagine negative ones too. At least, we aren't bothered by not really knowing them; we can find physical analogies for them.

Mazur tries to do the same with imaginary numbers. I think he did an okay job. I can imagine adding them now, and multiplying them, and even taking their square roots. He does, however, stop short of raising a number to an imaginary exponent. The imagery is simply transformations on the plane.

By reading this book, it is immediately apparent that the author has an encyclopedic knowledge. But, this is the problem however.

He's all over the place with analogies. We have drawings of cockroaches, passages about a particular tulip versus an idealized tulip, talks about Allah. None of it has anything to do with imaginary numbers, nor imagining them. Instead, these images are used to describe how ideas come into fruition. He tries to say something like, "hey ideas take time to bubble up into consciousness, we have traces of it in the atmosphere. Later, we can feel it and know it's there. Finally we get a handle on it, and it becomes concrete."

Looking briefly through this book right now, I notice these irrelevant imageries don't take much book space, but they are so oddly out of place, they take up a majority of my impression of the book.

I can't say this book is a complete waste of time. I enjoyed his explanation of the basics of algebra, and why we can't divide by zero, and why a negative times a negative is a positive. In fact, it's the best explanation I've read so far.

Also, the history of the emergence of complex numbers is abbreviated, but informative. However, things are just watered down and lost by these crazy tulip analogies about how ideas become concrete. This book is so-so. I feel that if someone wants to know the history of imaginary numbers and how to think about them, they could probably find a better book.

If there is a second edition, I think Barry should expand his bookkeeping example as an introduction to algebraic rules. Then cut to the chase, show us to grasp imaginary numbers, think of them as points on the plane, and operations on them as transformations and vector addition. He can later discuss how this mental model of imaginary numbers came to be, and these tulip images won't stick out so sorely.

This would be much like how people view a great painting or a magnificent edifice. Rarely is anyone privileged to see a magnificent work in progress. And those who do rarely grasp or appreciate the beauty that is forming before their eyes. Rather, after appreciating the final work, we then watch a documentary on how such and such a building was built.

Barry would do better to follow this formula, instead of immersing us in a work in progress, and more-or-less, confusing his readers.

Finally, I hope Barry uses his tremendous intellect to show how imaginary numbers relate in the day to day. And not via electrical engineering! Imaginary numbers are used in electricity, but since electricity is hare to grasp, real world examples using electricity would be confusing.

What I'm getting at is this: We can find uses for negative numbers in the day to day: walk 3 north, 4 south, and you'll be 1 south. Perhaps there is something quite simple for complex numbers too.

If in succeeding with that last point, then we may not be so bothered by not grasping imaginary numbers, because we have a physical analogy of them, and then we can pretend to know what they are, just as we do the integers.

I enjoyed this book and read it to the end, but it had its frustrations. In particular, assuring the reader they won't need more than limited high school math doesn't take into account that they may have forgotten it. The briefest refresher (in the back?) about manipulating equations in algebra would have helped a lot.
Also, saying "try it yourself" and then not going over it in case the reader was hopelessly lost was a bit annoying.
Still, any attempt to rejoin the arts and sciences is a good thing; and most of the book was comprehensible.

As an engineer, I really wanted to like a book that would claim it could help you visualize an abstract concept like imaginary numbers in a way that gives an intuitive feel to their form and purpose. The idea of "how such an experience compares with the imaginative work involved in reading and understanding a phrase in a poem" sounded like it could liven up the delivery as well. However, while occasional flashes of insight can be found here, I felt plodding through the other 180 pages of his boorish "I'm so highly educated" prose was not worth the price of admission. The same amount of cash would be better spent on the work from which Mr. Mazur paraphrased all but his "yellow of the tulip", "Number: The Language of Science" by Tobias Dantzig (1930), a wonderfully inspired and inspiring look at mathematical history and its discoveries (which includes graphing in the complex plane). Mr. Mazur even wrote the seven page (!!) forward to the 2005 reprint (doesn't this guy ever stop with the incessant multi-language quotations?).

The book is ultimately about how to understand/visualize imaginary numbers and its operations (addition/multiplication). (By visualize I mean visualizing these numbers, operations in the "complex plane".) In my view the book could have been better had Mazur included a section or two on his personal view of what "mathematical imagination" consists of. Such inclusion wouldn't have done any harm to the over all pedagogic tone of the book, and would have added value for the mathematics community as well, especially given the fact that Mazur -- recipient of the Steele/Veblen/Cole prizes -- would have so much to offer in this regard. Without such inclusion the book is still beautiful and enjoyable, but for readers already acquainted with the notion of the complex plane, it has not much to offer other than 1) the various connotations of [literary] imagination, and 2) the historical development of notions of solutions to algebraic equations.

This book is good but needed some more editing. Incredibly Plato's diagram showing a proof of the Pythagorean theorem is missing entirely from page 9--whoops! So I looked it up on the internet and then drew the picture in myself. Actually that helpful to assist me in understanding the proof. (Also on page 148 the denominator in all the calculations is missing.)

I like the way the writer presents these mathematical ideas in prose along with the accompanying algebra. There are no infinite sequences nor calculus here that would be too difficult for someone without much training in math. In fact the whole point of the book is to show how mathematicians working with limited tools (i.e. early algebra without benefit of future discoveries) were stuck when they came upon the square root of a negative number. What does this mean? The author explains that concept using algebra, the number line, triangles, and nothing too advanced. The whole goal is solve this riddle as a historical puzzle then show what is meant by the imaginary numbers.

The writer tries to mix poetry into the narrative but the transition from poem back to math is often abrupt and one is left wondering what one section had to do with the next. Still I enjoy the references to Kafka and the poets. Also in writing about circular reasoning the writer gets, well, circular and the section on "Bombelli's Puzzle" is, well, puzzling.

Still this book is a good one since I find reading about math most fun when there is some English text mixed in with all the heuristic symbols to give one time to catch one's breath before diving into yet another difficult proof.

...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.