Unknown Quantity: A Real and Imaginary History of Algebra [Paperback]

John Derbyshire (Author)

Book Description

Release date: May 29, 2007

For curious nonmathematicians and armchair algebra buffs, John Derbyshire discovers the story behind the formulae, roots, and radicals. As he did so masterfully in Prime Obsession, Derbyshire brings the evolution of mathematical thinking to dramatic life by focusing on the key historical players. Unknown Quantity begins in the time of Abraham and Isaac and moves from Abel?s proof to the higher levels of abstraction developed by Galois through modern-day advances. Derbyshire explains how a simple turn of thought from ?this plus this equals this? to ?this plus what equals this?? gave birth to a whole new way of perceiving the world. With a historian?s narrative authority and a beloved teacher?s clarity and passion, Derbyshire leads readers on an intellectually satisfying and pleasantly challenging journey through the development of abstract mathematical thought.

Editorial Reviews

From Publishers Weekly

This book's title is deceiving, for Derbyshire offers a very real and very entertaining survey of the development of algebra. "Real" and "imaginary" refer to types of numbers, and Derbyshire (Prime Obsession) opens with a basic primer on the various flavors of numbers and polynomials before looking at algebra's development over 3,000 years. As he explains how algebraic notation wended its way from Sumerian scratches on clay to such contemporary mathematical structures as Calabi-Yau manifolds (used by Andrew Wiles to solve Fermat's Last Theorem), Derbyshire introduces readers to the colorful figures who made contributions: Hypatia, whose death in Alexandria at the hands of an angry Christian mob marked the end of mathematics in the ancient world; 19th-century mathematician Hermann Grassmann, who published a 3,000-page translation of the ancient Hindu text the Rig Veda after his work on vector spaces was ignored; and Emanuel Lasker, more famous as the longest-reigning world chess champion than for his contributions to ring theory. This book will appeal to readers who relished the rigorous mathematical discursions interspersed with informal historical vignettes of David Berlinski's A Tour of the Calculus, but less mathematically inclined readers more interested in the history of science will also enjoy it. (May)
Copyright © Reed Business Information, a division of Reed Elsevier Inc. All rights reserved. --This text refers to the Hardcover edition.

Review

[A] very entertaining survey of the development of algebra. (Publishers Weekly)

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Product Details

• Paperback: 374 pages
• Publisher: Plume (May 29, 2007)
• Language: English
• ISBN-10: 0452288533
• ISBN-13: 978-0452288539
• Product Dimensions: 8 x 5.4 x 0.9 inches
• Shipping Weight: 12 ounces (View shipping rates and policies)
• Average Customer Review: 4.6 out of 5 stars  See all reviews (45 customer reviews)
• Amazon Best Sellers Rank: #210,311 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

117 of 118 people found the following review helpful

5.0 out of 5 stars Abstraction brought down to Earth June 1, 2006

By Mike Birman TOP 500 REVIEWERVINE™ VOICE

Format:Hardcover|Amazon Verified Purchase

Those of us who read and enjoyed Prime Obsession (even the title has a delicious tabloid flavor, reminiscent of Basic Instinct or Fatal Attraction) may have been most amazed at the very idea of popularizing something as arcane and difficult as the Riemann Hypothesis. What made that book work so well was Derbyshire's brilliant alternation between historical narrative and description with chapters that served as a mathematical primer on number theory and other background material. The mathematically challenged reader could peruse these more technical chapters or leave them be by choice: there was still much knowledge to be gained in either case. For the more mathematically sophisticated, a complete reading of the book served as a reasonably deep (if popularized) analysis of the famous Riemann Hypothesis. Short of tackling H. M. Edward's Riemann's Zeta Function, the classic discussion and much more difficult, Derbyshire provided the most cogent introduction to the RH.

Unknown Quantity is similarly constructed, with historical and biographical material alternating with chapters Derbyshire once again describes as mathematical primers. Although trained as a molecular biologist, I have a fairly strong background in mathematics. I still found much to learn. Especially interesting is the material on Vector Spaces and Algebras, the introduction to Hamiltonian Quaternions, Rings and Fields (with the vista of Abstract Algebra just over the hill) and a short introduction to Algebraic Geometry. I found even more to enjoy. The historical and biographical threads make the unfolding mathematics that much clearer and easier to visualize, hence more enjoyable. Derbyshire has produced another superb book that makes mathematics live and breath. To breath life into abstraction is a great gift.... I reread Prime Obsession and will do the same for this newest work. If you find mathematics at all amenable to your taste, I urge you to sample this book. I look forward to being pleasantly surprised by the topic of his next work.

Mike Birman Read more ›

94 of 95 people found the following review helpful

5.0 out of 5 stars ALGEBRA THEN AND NOW May 16, 2006

By Mead C. Whorton Jr.

Format:Hardcover|Amazon Verified Purchase

John Derbyshire's Prime Obsession, the story of the Riemann Hypothesis,was a mathematical tour de force but Mr. Derbyshire has done it again. He has written an extraordinary book which traces the history of algebra from its beginnings in the Fertile Crescent nearly four thousand years ago to such modern day abstractions as Category Theory. To assist the reader who has never encountered higher undergradate mathematics or who has forgotten the content of courses taken long ago, Mr. Derbyshire has provided well written, concise MATH PRIMERS on such diverse topics as Cubic and Quartic Equations, Roots of Unity, Vector Spaces and Algebras, Field Theory, and Algebraic Geometry. These Primers are scattered through the text and serve as guide-posts for the reader as she/he treks through the historical development of Algebra. If you have ever wondered how Algebra began and what groups, rings, fields, vector spaces, and algebras are then purchase this book. The author has also done a wonderful job of bringing alive the many men and women who, through the centuries, created modern day abstract algebra. This is not a light read but the prose and logic are superb. The reader who is willing to invest the time to complete this book will emerge all the richer for completing a thrilling intellectual adventure of the highest order.

45 of 45 people found the following review helpful

5.0 out of 5 stars Another great read from Derbyshire June 20, 2006

By R. Lighthizer

Format:Hardcover

Mathematics is not a topic that is easy to read or write about.

How lucky we are, then, that John Derbyshire has chosen once more to grace us with his talent for writing clear, concise, coherent prose on higher math.

In Unknown Quantity, Derb has again achieved the near-impossible feat of writing an approachable, relatively easy-to-read book on mathematics.

Reading Mr. Derbyshire's mathematical writings allows one to experience some of the awe and majesty of the deepest, most esoteric reaches of higher mathematics. In giving the common reader this chance, he does a service both to mathematics by allowing those who would rarely even hear about such topics to learn something of them and also to the reader by allowing him for a moment to feel smarter than he probably has any reason to.

I cannot disagree with others who found Prime Obsession to be the better read, however this should not be taken as a serious criticism of Mr. Derbyshire or Unknown Quantity. Prime Obsession was helped by its more limited focus - not that the author had any shortage of interesting and enlightening information and insight to share.

Unknown Quantity's goal of presenting a readable, reasonably approachable history of algebra is definitely met, but it would probably require a book several times the length of this one to properly explore all the intricacies of the story with the thoroughness that Mr. Derbyshire could. That book might not be as broadly marketable but I feel it would be gladly received by those of us who have discovered Derb's genius.

If you have any interest in math or the history of human thought, you cannot go wrong with Unknown Quantity.

39 of 40 people found the following review helpful

4.0 out of 5 stars A fine biographical history of algebra May 24, 2006

By Steven W. Erbach

Format:Hardcover|Amazon Verified Purchase

I had high hopes for UQ. My hopes were not dashed, but I wasn't as uplifted and exalted as I was with Derbyshire's excellent "Prime Obsession". If one comes upon these two books for the first time, one should definitely read UQ first.

As in "Prime Obsession", Derbyshire writes very appealingly about the history of the times and about the mathematicians themselves. The biggest issue is that the book is too small for such a huge subject. It's only 320 pages long with 32 pages of notes.

Derbyshire's portraits of algebraists in his book are uniformly delicious. His bio of Alexander Grothendieck reminded me of the life of former world chess champion, Bobby Fischer. Grothendieck was as unworldly, uninformed, naively opinionated, anti-American, and brilliant as Fischer. We find him now holed up in a remote village in the Pyrenees, where "he is known to come up with ideas like living on dandelion soup and nothing else."

Or Solomon Lefschetz, the algebraic geometer, who lost both his hands in an industrial accident. He was "energetic, sarcastic, and opinionated", and something of a character. His most famous quote: "It was my lot to plant the harpoon of algebraic topology into the body of the whale of algebraic geometry."

I think that Derbyshire had to edit severely. His introduction to "Unknown Quantity" says that it was "written for the curious nonmathematician." Perhaps he should have said, "written for the college math major who decided not to pursue a career in mathematics." I studied math in college but I didn't get a degree.... I mention this because I was disappointed by the blandness with which he writes of the "simple substitution" one can make if one only notices the "simple algebraic fact" that turns a general cubic equation into a depressed cubic equation. It's something I never encountered in high school or college. Granted, that was 35 years ago now. I tried for quite some time over a period of three days to derive the "simple algebraic fact" for myself before moving on with the rest of the Cubic and Quartic Equations chapter, but I couldn't. And this was only page 58!

Derbyshire's math primer interludes are designated with initials. So, instead of referring to a section in "Cubic and Quartic Equations" as section 4.7 as he does for all the non-primer chapters, he uses abbreviations: section CQ.7 for "section 7 of the Cubic and Quartic Equations chapter". A bit annoying, actually. If one wishes to brush up on a concept by re-reading, one has to refer first to the table of contents to find it.

However, there are plenty of interesting things to learn for the "curious nonmathematician". For example, the complex cube roots of the number 1. I found this fascinating. I didn't do much with complex numbers in school. Derbyshire whetted my appetite for them in "Prime Obsession" and UQ sated me!

I loved matrices and determinants in high school. They made complete sense to me. The section on the discovery and application of matrices was a comfortable interlude of re-discovery. So was the discussion of Boolean algebra. A few of the chapters weren't. I felt myself to be way outside the book's target audience even though I'm a "curious nonmathematician".

UQ is not as sprightly as "Prime Obsession". The jokes are there, just more widely spaced and drier. The best chuckle I had was in one of the notes in which the author wrote of a video "demonstrating one of the 20th century's most fascinating discoveries in topology: how to turn a sphere inside out." He recalls that he "used to bring it out and play it to dinner guests as a conversation piece, but this was not an unqualified social success."

Derbyshire continues his practice (begun in "Prime Obsession") of collecting all of the footnotes into a set of endnotes at the back of the book. As I said, the best laugh of the book is in one of those notes, but if you don't care to keep flipping back there to find the right note, you might miss it. At least the notes are numbered consecutively; the numbering doesn't start over again with each chapter. But I sure enjoy sidebars and footnotes as part of the text.

Part of what made "Prime Obsession" so vivid were the illustrations. Algebra uses symbols that don't lend themselves to illustration very readily, except where geometric figures can be plotted from algebraic equations. But even in his chapter on geometry Derbyshire's use of illustrations is stingy.

UQ did prove useful right away, though. In skimming through references to help my 15-year-old son find material for a paper on Archimedes, I found a mention of Archimedes' Cattle Problem in the 11th edition of the Encyclopaedia Britannica. The invaluable Mathworld web site contained a more detailed article. I worked on the problem for some time while reading UQ, but I didn't pursue it to a solution. I now realize, though, that both Newton's method and the use of determinants would have solved it...both of which are covered in detail in UQ!

"Unknown Quantity" is a fine historical and biographical treatment of algebra, with engaging writing and plentiful -- though brief -- explorations of a multitude of algebraic topics. There's plenty of meat here for the "curious nonmathematician". Read more ›