Unknown Quantity: A Real and Imaginary History of Algebra

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."

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" people, 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 Polytechnique (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 Zuru Queens Rule China), helps 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 transformation). 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 ar

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.

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.