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

by Barry Mazur (Author)

22 customer reviews

ISBN-13: 978-0374174699

ISBN-10: 0374174695

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How the elusive imaginary number was first imagined, and how to imagine it yourself

Imagining Numbers (particularly the square root of minus fifteen) is Barry Mazur's invitation to those who take delight in the imaginative work of reading poetry, but may have no background in math, to make a leap of the imagination in mathematics. Imaginary numbers entered into mathematics in sixteenth-century Italy and were used with immediate success, but nevertheless presented an intriguing challenge to the imagination. It took more than two hundred years for mathematicians to discover a satisfactory way of "imagining" these numbers.

With discussions about how we comprehend ideas both in poetry and in mathematics, Mazur reviews some of the writings of the earliest explorers of these elusive figures, such as Rafael Bombelli, an engineer who spent most of his life draining the swamps of Tuscany and who in his spare moments composed his great treatise "L'Algebra". Mazur encourages his readers to share the early bafflement of these Renaissance thinkers. Then he shows us, step by step, how to begin imagining, ourselves, imaginary numbers.

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

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Product Details

Hardcover: 270 pages
Publisher: Farrar, Straus and Giroux (2003)
Language: English
ISBN-10: 0374174695
ISBN-13: 978-0374174699
Product Dimensions: 7.9 x 5.3 x 0.9 inches
Shipping Weight: 11.2 ounces
Average Customer Review: 3.1 out of 5 stars See all reviews (22 customer reviews)
Amazon Best Sellers Rank: #1,472,942 in Books (See Top 100 in Books)
- #1158 in Books > Science & Math > Mathematics > History
- #11652 in Books > Textbooks > Science & Mathematics > Mathematics
- #19196 in Books > Textbooks > Humanities > Literature

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Customer Reviews

3.1 out of 5 stars

5 star		27%
4 star		18%
3 star		18%
2 star		14%
1 star		23%

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Most Helpful Customer Reviews

32 of 35 people found the following review helpful

Not for math geeks

By A Customer on April 6, 2003

Format: Hardcover

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.

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20 of 24 people found the following review helpful

Not So Imaginary Numbers

By Rob Hardy HALL OF FAMETOP 500 REVIEWER on June 20, 2003

Format: Hardcover

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.Read more ›

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16 of 19 people found the following review helpful

A disjointed book

By Dan Taflin on July 31, 2003

Format: Hardcover

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.

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Most Recent Customer Reviews

Mazur has an excellent style of writing and brings it to life

It's a fascinating book. The concept of imaginary numbers and the fact that they have broad application in real life has always fascinated me and Mr. Read more

Published 9 months ago by Michael Enright

Between poetic passion and mathematical elegance

Between poetic passion and mathematical elegance, there must be something commensurable, that Mr. Mazur approached very close to... You know, what is alive in your soul. Read more

Published 10 months ago by Amazon Customer

disappointing

Mr Mazur writes intelligently and well, but for me the book failed to live up to some of the other reviews and descriptions. Read more

Published 19 months ago by Phillip Thompson

Convoluted prose obscures the main ideas

I read this book when I was 18, largely because of Barry Mazur's reputation as a mathematician. His attempts to compare complex numbers to poetry obscures the main algebraic and... Read more

Published on November 4, 2012 by JL

Well written in places but needed tighter editing

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! Read more

Published on February 10, 2010 by Walker E. Rowe III

Bad Poetry + Pretentious Prose + Tired Examples = Wasted Hours

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... Read more

Published on July 7, 2007 by J. Schneeg

An interesting ramble

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... Read more

Published on February 1, 2007 by Sophie G

Entertaining

ITs an entertaining book on math, particularly the concepts on imaginary numbers. Very easy reading and a real page-turner. Excellent for mentor reading.

Published on January 30, 2007 by Farseem Mohammedy

a truly unusual book

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... Read more

Published on October 21, 2005 by Wyote

Great insight giver.

I never really grasped the meaning of SQ-1. After reading this book, I can say that now I have a much better grasp. Read more

Published on May 7, 2004 by Hugh Effow