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Unknown Quantity: A Real and Imaginary History of Algebra

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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

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.

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.

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".

I learned the Modern Algebra 28 years ago in the very university of "The Last Theorem of Fermat" in Toulouse, France (Classe Préparatoire aux Grandes Écoles, Lycée Pierre de Fermat - Mathématiques Supérieures et Mathématiques Spéciales). These were the 'darkest' years of my study life when we slogged for 2 years learning the abstract Modern Algebra and Analysis. The French are "Maths lovers", given 100+ streets in Paris are named after their mathematicians.

I remembered the Maths were taught in the form of arcane and boring Axioms/Theorems. starting from Set Theory (Ensemble), Group (Groupe), Ring (Anneaux), Field (Corps), Vector Space (Éspace Vectorielle), Affine Space(Éspace Affine), Matrix, Topology, etc. The toughest Grandes Écoles Entrance Exams (Concours) demanded the students master these maths abstract concepts in order to solve difficult maths questions in long-hour written and oral Papers. Many bright top students, after scoring brilliant results to enter the prestigious École Polytechniques (the one which failed twice Évariste Galois!), shied away from Maths in their life later because of this "Maths Phobia". What a shame and waste of maths talents.

After reading The "Unknown Quantity", I always ask "If only these Maths were taught in the similar interesting way", we could have actually loved and enjoyed it in our entire life.

Derbyshire has introduced many 'revolutionary' Maths teaching ideas:
1) Group, Ring (Ideal) and Field are presented in a non-traditional reversed order of all Maths text books. He said: "Field is a more common place kind of thing than a Group, and therefore easier to comprehend." I agree 100% when I read this book without any difficulty to follow.
2) Many enlightening 'tips' e.g. NZQRC (Nine Zulu Queens Rule China), help my teenage children grasp instantly the intrinsic Number Theory over a dinner talk.
3) 'Vector Space' was presented in a refreshing manner, without bothering us with the difficult theorem, which helps us understand the linear (in)dependence, hence linear algebra and its importance in application.
4) Chapter 8 "The Fourth Dimension" on Hamilton's Quaternions (1,i,j,k) and the intriguing story of the discovery (page 151) at Brougham Bridge on one Monday, 16th Oct, 1843.
5) Why x is the predominant used unknown variable in equations (Chapter 5, Page 93), because the french printer ran short of letters (y and z are commonly used in French language).
6) The reason behind the eccentric choice of letters (a,h,b,g,f,c, skipping i and e) for coefficients in conic equation: ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 was uncovered in the matrix on Page 245 (another maths tip: "All hairy guys have big feet") and Page 248 (homogeneous coordinates).
7) The Yin-Yang view of Geometry vs Algebra. Geometry is for Space and Algebra for Time (Sequence of transformations).

I had spent my entire 1 week holidays in end December till 1 Jan 2007 reading this book. No regret of time wisely spent. I urge all who are "curious nonmathematicians" to follow me.

This book has cured my 28-year Modern Maths Phobia!

Cornelius

Fascinating History of Algebra

"Unknown Quantity: A Real and Imaginary History of Algebra" by John Derbyshire

Readers who enjoyed "Prime Obsession" will find "Unknown Quantity" irresistible. In this very readable text John Derbyshire covers the broad history of modern algebra. The history starts four thousand years ago in Egypt and Mesopotamia.

The author tells the lives of the men and women who created modern algebra. Their stories are fascinating.

The people who make up the history of algebra include (from the photographic plates after page 184):
01 - Otto Neugebauer - found algebra in old Babylonian tablets
02 - Hypatia
03 - Omar Khayyam - wrote poetry and tackled the cubic equation
04 - Girolamo Cardano - found a general solution for the cubic
05 - Francois Viete - separated things sought from things given
06 - Rene Descartes - algebrized geometry
07 - Sir Isaac Newton - saw symmetry in solutions
08 - Gottfried von Leibniz - found relief for his imagination
09 - Joseph-Louis Lagrange - carried symmetry forward
10 - Paulo Ruffini - believed the quintic was unsolvable
11 - Augustin-Louis Cauchy - made an "arithmetic" of permutations
12 - Niels Abel - proved Ruffini right
13 - Evariste Galois - found permutation groups in equations
14 - Arthur Cayley - abstracted the group idea
15 - Ludwig Sylow - delved into the structure of finite groups
16 - Camille Jordan - wrote the first book on groups
17 - Sir William R. Hamilton - found a new algebra
18 - Herman Grassman - explored vector spaces
19 - Bernard Riemann - launched two geometric revolutions
20 - Edwin A. Abbot - took us to Flatland
21 - Julius Plucker - based his geometry on lines not points
22 - Sophus Lie - mastered continuous groups
23 - Felix Klein - mastered the group-ification of geometry
24 - Henri Poincare - algebraized topology
25 - Eduard Kummer - used algebra on Fermat's Last Theorem
26 - Richard Dedikind - discovered ideals
27 - David Hilbert - a geometry of tables, chairs and beer mugs
28 - Emmy Noether - pulled it all together
29 - Solomon Lefschetz - harpooned a whale
30 - Oscar Zariski - refounded algebraic geometry
31 - Saunders Mac Lane - attained a higher level of abstraction
32 - Alexander Grothendieck: - as if summoned from the void

Just as before, the author takes a field of mathematics interesting for expert and layman alike. This is a very fresh perspective on the history of algebra.

See Also:
Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

I thoroughly enjoyed and highly recommend "Unknown Quantity."

"Unknown Quantity" is a history of algebra presented in Derbyshire's chatty, sometimes quirky style. It is compulsively readable. I kept stopping to remind myself I was reading a HISTORY OF ALGEBRA--and enjoying it! The math is kept at a decently low level. If you can get the idea of a polynomial (Derbyshire explains it up front), that will get you through most of the book. The explanation of groups is painstaking and pretty good, and I am fairly sure I got the main idea about rings and fields, too--though I'm afraid he lost me with "p-adic numbers." There is a strongly geometric angle in the later part of the book. This gives you some attractive diagrams--look at 13-2, the "ampersand curve," or 13-3, which beautifully illustrates the meaning of "variety"--but I found myself wondering whether the author was falling back on geometry because the algebra was just getting too hard to present to his chosen audience ("the curious nonmathematician"). When Derbyshire gets to the 20th century he pretty much gives up on trying to explain what the algebraists are doing, and concentrates on the personalities and the historical background. From the little he does tell us about 20th-century algebra, it's hard to see that he had much choice. I still have no idea what Alexander Grothendieck did in his day job, but he sure sounds like a fascinating character. Best chapter: "The Leap into the Fourth Dimension." I now understand what multi-dimensional spaces are all about and why people started thinking about them. Best chapter title: "Lady of the Rings" (that's Emmy Noether, another fascinating character). Best photograph: Hypatia--is this the first algebra book to include a picture of a naked woman? Best footnote: No. 142. (As with "Prime Obsession," the footnotes are wonderfully readable.) Still, "Unknown Quantity" shows even more clearly than "Prime Obsession" did that Derbyshire is at heart a novelist. I wish he'd give us another novel. Anyone who can make the history of algebra interesting is a born story-teller!

If you're like most folks, you took algebra in high school and within a week you learned to stop wondering "why" something was like it was or how "they" figured out some rather abstract idea. Instead, you just memorized the definitions and the "tricks" that were presented as "teaching", without ever really understanding the "why" of it all. We tend to forget that it took a lot of people thousands of years of thought to produce what we have today between the covers of a basic algebra book. And the subject is taught in such a way that we feel slightly guilty if we don't immediately "get" some idea or concept that actually was a long time in the making. The problem only compounds as we take more math courses, since algebra is a key language for all of them. What to do?

Read John Derbyshire's book! Take it slowly, with a pencil and pad next to you; don't just read the book, digest it! His thoughtful inclusions of somewhat detailed explanations of building block material is especially helpful. Even if you're past the point of needing to know anything about math, let alone algebra, it'll be good exercise for the brain and it'll help with other tasks that you do use every day -- like trying to remember someone's name or phone number right after you heard it! I took my first algebra course almost 50 years ago (can it be?) and I often found myself saying "if only they had presented it this way back then, I could have actually LEARNED something!. The "teachers" I had for those first courses in high school were uniformly bad, so I shouldn't have expected them to convey what they didn't have: understanding.

Incidentally, if you know someone just starting out with high school math, this book would be a nice gift to help fill the gaps. Who knows? It might even motivate them to go further. In a society when mathematical literacy is becoming increasingly critical to personal and societal success, the schools are doing an ever-poorer job of meeting the need. Books like this help fill that gap.

Highly recommended.

This book is partly a history book and partly an Algebra book. The first part of the book is about 70% history and 30% algebra, but the distribution gradually shifts so that the last third of the book is about 70% modern algebra and 30% history of the development of this subject. In addition to the narrative, there are six Math Primer chapters, covering:
· Numbers and polynomials
· Cubic and Quartic equations
· Roots of Unity
· Vector Space and Algebra
· Field Theory
· Algebraic Geometry

The back cover of the book says "For armchair mathematicians and algebra buffs, ....". I believe this to be a very accurate description of the book. Those who love algebra will love this book, those who like algebra will probably like this book, but those who are math phobic are likely find this book not to their liking. Those who like history, but not math, are also forewarned that this may not be the book for them.

I have a typical engineer's mathematical background (basic algebra (including vectors and tensors), calculus, differential equations and advanced calculus), and since I also have an interest in history I read this book hoping to learn more about modern algebra and its development. While modern algebra is a very esoteric subject, it has become the language for quantum mechanics, relativity theory and the melding of the two (in string theory, M theory and quantum loop gravity). This book succeeded in educating me about these subjects, but frankly the math in the last third of the book was way over my head and I found myself just skimming over much of the last 75 pages. All in all, I am glad that I invested my time with this book. Derbyshire is a very good writer and did a good job of giving me an idea of what modern algebra is all about and I plan to re-read much of the Math Primer chapters as they are good introductions to many of the facets of modern algebra.

Polymath John Derbyshire has written this history of algebra, from ancient Babylon to today.

Derbyshire looks at the various stages in the increase of complexity of algebra, from the discovery of zero, the solution of cubic and quartic equations, the development of negative numbers and vectors, up through the latest developments in algebra today. He goes into some detail about some of the mathematicians of the past who made significant discoveries and contributed to the development of this discipline.

Be forewarned--you might need an undergrad degree in math to get the most out of this book. I did well in my algebra classes in school, and was able to follow along very well for the first third of the book. By the middle third, I was only able to get the general gist of what Derbyshire was writing about, and by the last third I was baffled when he was trying to describe advanced areas of algebra that are very abstruse.

Still, it was interesting to read about how in the last several decades several branches of mathematics are blending together, and anyone who has read Derbyshire's articles for National Review magazine or listened to his "Radio Derb" audio show at the same magazine's Internet site knows that the author has the ability to interest the reader over a wide range of topics.