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30 of 33 found the following review helpful:
Not for math geeksApr 06, 2003
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.
19 of 23 found the following review helpful:
Not So Imaginary NumbersJun 20, 2003
By R. Hardy
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.
13 of 16 found the following review helpful:
A disjointed bookAug 01, 2003
By Dan Taflin
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.
13 of 17 found the following review helpful:
Comments about the bookMar 19, 2003
By Royal Tenenbaum
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!
27 of 37 found the following review helpful:
Who is this book written for?Sep 02, 2003
By Godfrey T. Degamo
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.
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