Unknown Quantity: A Real and Imaginary History of Algebra

I really relate to the Derb. I'm an old-school tech-nerd (BS Engineering, 1973, learned the slide rule as a freshman; read Scientific American Magazine--especially Martin Gardner's "Mathematical Games" column in middle & high schools, etc.).Why do I enjoy reading this book (I do keep it around to read a random page now and then)? Derb successfully --Creates a sense of the humanity of the individuals who, despite a tumultuous and dangerous world, were inspired to create the Mathematics we have today (by the chance at fame and fortune, or the joy of intellectual accomplishment).Explains the Mathematics needed to understand the history in a very simple way with the humility of a student/amateur who understands much of the subject but is still awed by the subject and respectful of the masters.Tells a good story. He is an accomplished fiction writer after all.Most of us encounter Algebra as a necessary evil, unfortunately few of tread far into the subject to appreciate Algebra for its aesthetics and revelations of aspects of human personality and intelligence. Derb is a pleasant, friendly guide for latter journey.

Unfortunately but inevitably John Derbyshire's "Unknown Quantity" is nowhere near as good as his previous pop math book "Prime Obsession", but in saying this I am praising the latter and not disparaging the former. "Unknown Quantity" still rates the full five stars."Prime Obsession" is laser focused on one problem (Riemann's hypothesis on an important characteristic of the zeta function) and on one man (Bernhard Riemann). Derbyshire then expands on the mathematical topic twice, first by explaining the context in which the hypothesis mattered (its relationship to prime numbers) and finally by taking a general look at the branch of mathematics called analysis. He does the same thing with the human side of his story: he presents Riemann's friends and colleagues and then he presents earlier and later analysts. This makes for a near perfect book, both topically and dramatically."Unknown Quantity" on the other hand gives us a more general history of its subject, which is algebra. Derbyshire takes us from its beginnings in ancient Babylon and describes in detail how to decipher a sample problem found on cuneiform tablet. He then goes to Alexandria where Diophantus used the first notation using something like "x" to represent unknown quantities. It didn't catch on, and Arab mathematicians developed the field during the middle ages using word problems again.Early in the Renaissance Italians then took over the search for roots and found methods to solve roots of third and fourth degree polynomials. Descartes then invented analytic geometry and standardized the usage of x-y-z as our notation for unknown quantities.Newton contributed to the field, but his invention of calculus (yes, yes along with Leibniz) diverted attention away from algebra for two centuries. It was in the nineteenth century that algebra as we know it today took off. Mathematicians were getting comfortable with the square root of minus one and the Complex field of numbers extended from the Real field. An Irishman named Hamilton worked hard at extending fields again from Complex numbers to the theory of Quaternions. Quaternions never caught on but as a by-product, Hamilton laid down ideas that directly lead to the development of vector spaces and linear algebra.Along with this, later mathematicians developed group theory, ring theory, and field theory. A group is simply a set and an operation on that set that meets certain requirements. Rings and fields are groups with more requirements. These theories form algebra as it is studied today. In a nutshell, Derbyshire presents an excellent history of the origin and development of much of algebra and of most of the important figures that have contributed to the field. But of course, to be general he's had to sacrifice focus.Perhaps instead of a single chapter, Derbyshire could have written his whole book around the story of Evariste Galois. Galois, a politically radical young Frenchman, died fatally wounded in a duel at twenty years of age. Despite his youth, he solved a problem that ultimately changed algebra from simply searching for roots of polynomials to the more abstract pursuit of studying groups and permutations. There's plenty of drama in >>that<< story! But it has also been done to death. Derbyshire made the right choice by giving us a less exciting but more informative book.Vincent Poirier, Tokyo

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.

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.

The sample of reviews on this site is small, but it has an unusual distribution - five out of six reviews (including this one) give 4 stars. Thus it qualifies as a book which almost all readers found excellent without giving it their ultimate approval. And this is exactly the right conclusion. It is very well done, and loaded to the gunwales with interesting material - both historical and mathematical. But there is a missing ingredient.Derbyshire's brilliant earlier book on Riemann (actually the only popular book on the subject that deserves to stay in the literature) was focussed by the clarity, and the difficulty, of its goal: to get the reader to understand the Riemann Hypothesis and the state of recent research. In slightly disappointing contrast, Unknown Quantity flows nicely along on a current of Whiggish historical progress, chugging downstream from Ahmes and Diophantus to the broad modern concept of algebra. Inevitably, this is less exciting.The feature that should nevertheless make this book a continuing success is that Derbyshire provides unusually direct historical explorations of the way the mathematics was created. So, for instance, his exposition of Abel's proof of the unsolvability of the quintic is not a discussion of some later simplified proof of the same result. Moving on to Galois, we get a real sense of how the same result appears in a more general setting. All of this done using essentially elementary tools (with extensions in the useful "primers" that are dotted about the book).The UK paperback edition has a curiously meaningless cover illustration (a sort of sprocket replacing the iconic "x" that adorns the US version). And there is a tantalising list of illustration credits; tantalising because only one of the illustrations has made it into this edition.

I am currently reading this ( over half-way through ) and it is superb:A roughly chronological blend of prose and clear mathematical description.It is sufficiently skilful to "do its best" for laymen like me without in anyway beingtiresome or condescending.Unsuited to ( and not intended for ) doctoral-level mathematicians.

Having read Prime Obsession by the same author I was expecting a a good read with this book and I was not dissapointed. John Derbyshire is in my view one of the best writers of popular maths books around. A primer chapter introduces some of the more demanding topics and certainly serves to make the book more informative and interesting. There is a greater depth to some of the historical facts surrounding the main players and at last we have an author who explains what some of the women mathematicians actually did that made them famous - Emmy Noether being the most recent. I will be looking out for any more books by this author.

Derbyshire strikes a good balance of mathematical history and maths, if the target audience is a semi-technical one.Personally, I would have preferred more mathematical specifics, but Im sure these are available in other texts, and he does provide a good bibliography to delve into for further reading.

This is an engaging and thoroughly competent history. John Derbyshire both knows and loves the subject, and it shows. His pleasure at breaking off into long but fairly gentle computations is everywhere apparent, and his readiness to tell the truth as he sees it regarding the achievements of past civilizations is reassuring. He cares, and so the reader cares too.