Unknown Quantity: A Real and Imaginary History of Algebra


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 ...

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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.

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Editorial Reviews

From Barnes & Noble
In Unknown Quantity, John Derbyshire sets out to construct an accessible survey of algebra's development over 3,000 years. With a keen sense of readers' fear of formulas, he navigates a path from Sumerian and Babylonian clay bookkeeping to the manifolds used by Andrew Wiles to solve Fermat's Last Theorem.
From the Publisher
[A] very entertaining survey of the development of algebra. (Publishers Weekly)
American Scientist
A reader who wants to learn some theory of equations and modern algebra in a relatively painless way will find this book attractive. The explanations of many algebraic topics are accessible and clear, especially those of the following: how Vieta's new general symbolic notation demonstrates the relations between roots and coefficients of polynomials; symmetric polynomials and solvability; the roots of unity; how Paolo Ruffini's proof of the unsolvability of the quintic worked; how studying permutations of roots of polynomial equations gave rise to group theory; how linear transformations and matrices are related; the nature of quaternions and octonions; what an invariant is; an introduction to algebraic geometry; what a vector space is; and what the differences are between groups, division rings and fields. Those explanations also make clear why mathematicians cared about these problems and how these concepts were used. The anecdotes...certainly make the book fun to read. -- Judith V. Grabiner
New Scientist
The story of algebra is the story of civilization itself. Unknown Quantity buzzes with rivalries, frustrations, and breakthroughs ... a first-rate account.
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 2006 Reed Business Information.
Library Journal
National Review columnist Derbyshire follows up Prime Obsession with a similar book on the historical development of algebraic principles. As a mathematician, linguist, systems analyst, and critic, he interweaves historical insight and biographical sketches into a book that is both compelling and easy to follow. The story line delves into algebraic principles concentrating on mathematical abstractions, historical narratives, and the development of mathematical ideas. Derbyshire moves quickly through the contributions of select mathematicians (not a complete who's who in algebra), including Diophantus, Descartes, Bernhard Riemann, and David Hilbert. The text-complete with mathematical primers, solved problems, figures, and historical vignettes-is written at a high school level for a general audience interested in recreational mathematics. Recommended for all school and public libraries and mainly undergraduate academic libraries.-Ian Gordon, Brock Univ. Lib., St. Catharines, Ont. Copyright 2006 Reed Business Information.
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Product Details

  • ISBN-13: 9780452288539
  • Publisher: Penguin Publishing Group
  • Publication date: 5/29/2007
  • Edition description: Reprint
  • Pages: 392
  • Sales rank: 958,327
  • Product dimensions: 5.50 (w) x 8.10 (h) x 0.90 (d)

Meet the Author

John Derbyshire is a mathematician and linguist by education, a systems analyst by profession, and a celebrated writer in his spare time. His work appears frequently in National Review and The New Criterion. Born and raised in England, he has made his home in the United States for the past fifteen years.

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Table of Contents

Math primer : numbers and polynomials 7
Pt. 1 The unknown quantity
1 Four thousand years ago 19
2 The father of algebra 31
3 Completion and reduction 43
Math primer : cubic and quartic equations 57
4 Commerce and competition 65
5 Relief for the imagination 81
Pt. 2 Universal arithmetic
6 The lion's claw 97
Math primer : roots of unity 109
7 The assault on the quintic 115
Math primer : vector spaces and algebras 134
8 The leap into the fourth dimension 145
9 An oblong arrangement of terms 161
10 Victoria's Brumous Isles 177
Pt. 3 Levels of abstraction
Math primer : field theory 195
11 Pistols at dawn 206
12 Lady of the rings 223
Math primer : algebraic geometry 241
13 Geometry makes a comeback 253
14 Algebraic this, algebraic that 279
15 From universal arithmetic to universal algebra 298
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Sort by: Showing all of 4 Customer Reviews
  • Posted August 15, 2010

    more from this reviewer

    I Also Recommend:

    An attempt to relate the history of algebra to a lay audience.

    John Derbyshire's Unknown Quantity: A Real and Imaginary History of Algebra attempts to relate the history of algebra to a lay audience. As the algebra becomes increasingly abstract, the attempt becomes less successful.

    Derbyshire's approach is to introduce new topics with brief mathematical primers, then discuss the mathematicians who made the discoveries and the historical context in which those discoveries are made. Since he assumes little mathematical training on the part of the reader, the mathematical primers tend to be nebulous. For instance, he gives examples of vector spaces without actually defining what a vector space is. Strangely enough, when he discusses mathematical concepts in the chapters, Derbyshire gives a clearer picture of the key ideas.

    Derbyshire divides the book into three parts. The first part, which is accessible to anyone who has completed high school algebra successfully, describes the historical foundations of algebra, the search for solutions by radicals of polynomial equations, the conceptual hurdles (such as the introduction of negative numbers, irrational numbers, and complex numbers) that had to be overcome for algebra to advance, and the introduction of symbols. This part of the book is clearly written and informative.

    The second part of the book, which is accessible to readers with some exposure to college mathematics, demonstrates how the unsuccessful search for a solution by radicals to the general quintic polynomial equation led to the development of abstract algebraic structures such as groups. Derbyshire also discusses quaternions, linear algebra, and how algebra was put to use in logic. Readers with the necessary mathematical background will find this section illuminating but may be put off by Derbyshire's use of imprecise analogies rather than definitions in his mathematical primers and his indulging in the use of nonstandard terminology since he likes the way it sounds. Also, Derbyshire sometimes chooses to emphasize certain mathematicians since there are interesting anecdotes to be told about them rather than focusing on the ones who made the most important contributions to algebra.

    The final part of the book discusses algebraic structures such as groups, rings, and fields before delving into algebraic geometry and the push into abstraction that followed. The beginning of this section is accessible to those who have studied college mathematics. By the end of the book, Derbyshire finds himself addressing topics requiring knowledge of graduate level algebra that he admittedly does not possess. Consequently, he ends up telling the reader more about Alexander Grothendieck's life and political activism than he does about Grothendieck's contributions to algebraic geometry. The reader would have been better served had Derbyshire simply told the reader that modern developments in algebra were too abstract for him to explain to a lay audience.

    Readers with a deep foundation in mathematics may wish to read Isabella Bashmakova and Galina Smirnova's Beginnings & Evolution of Algebra or V. S. Varadarajan's Algebra in Ancient and Modern Times instead. I use "or" here in the inclusive sense.

    Lay readers with an interest in the history of algebra will find that their choices are much more limited. Derbyshire's book, while not ideal, is worth reading through his discussion of groups, rings, and fields. After that, it breaks down.

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  • Posted May 30, 2009

    Buckle Up

    Great ride and fairly rapid so for some a quick review of Alg II
    might avoid any time consuming stumbles on this grand trek.

    0 out of 1 people found this review helpful.

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  • Anonymous

    Posted June 19, 2010

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    Posted April 18, 2011

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