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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 ...
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.
|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|
Posted August 15, 2010
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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.
Posted May 30, 2009
Posted June 19, 2010
No text was provided for this review.
Posted April 18, 2011
No text was provided for this review.