Unknown Quantity, by John Derbyshire

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

I haven't submitted a book entry in nearly a month because it's taken me this long to finish this book even though it has a mere 320 pages.

Derbyshire's book is about the history of algebra, and also the stories of algebraists. The first few chapters are easy: how systems of expressing numbers evolved over the centuries; how Diophantus and al-Khwarizmi paved the roads for the development of algebra; etc. I remember in my childhood working on Diophantus's famous riddle (it's really just a simple one equation with one unknown that any Grade Six kid can do). What's interesting is how concepts that small children nowadays take for granted were completely rejected by the most learned men in ancient days. Zero came into being long after the natural numbers. Negative numbers were discarded because they didn't make any sense. Imaginary numbers were again ignored for as long as possible. It's so intuitive now to use letters, in particular, the letter x, to represent unknowns, but the ancients had to use cumbersome language to refer to any unknown quantity. Simplicity and clarity of presentation are paramount in mathematics. Imagine having to do arithmetic using Roman numerals, or calculus using Newton's notations. Ugly!

The rest of Part I of the book describes the quest to find general solutions to quadratic, cubic and quartic equations. Everyone knows the importance of quadratic equations. The cubic, a reluctant nod. But the quartic? Really, who cares about formulas for solving the quartic? For higher degree polynomial equations, numerical methods are sufficient in most cases. However, as an intellectual exercise, it is still something challenging to work on.

Part II of the book deals with more advanced topics in algebra developed from the late 16th century to the early 18th. The Fundamental Theorem of Algebra was stated by Descartes, and proven by Gauss. Euler worked on the problem of the general quintic, and Abel proved its unsolvability. The study of n-dimensional spaces was developed. Now we're into first- and second-year university algebra. Since I remember my vector spaces, basis, and matrices well, I'm still doing OK.

The last part of the book gets really hairy. Some of the topics are taught in first-year algebra classes: rings, fields, groups, and modular arithmetic. I remember finding those things a challenge back then. Reading about them again has clarified things. Also, since my first year in university, I've read a lot more on rings and fields, so it's natural that I should understand them better. Same with non-Euclidean geometry. When I took a third-year course on non-Euclidean geometry, I could do the problems but my mind rebelled against the concepts. Over the years, the theorems of non-Euclidean geometry started to make more sense in my head. However, when it comes to Galois Theory, Noetherian Rings, and topology, I'm as lost as I've always been. This, I've come to realize, is my personal limit of understanding abstract mathematical ideas.

At the end of the book, there is a nice little summary on the distance between algebra and the practical world:

"The very earliest algebra arose ... from practical problems of measurement, timekeeping, and land surveying.

"From the invention of modern literal symbolism in the decades around 1600 to the late 18th-century assault on the general quintic equation, the new symbolism was widely used to tackle practical problems.

"The growth of pure algebra in the 19th century, however, was so abundant that the subject raced ahead of any practical applications to dwell almost alone in a realm of perfect uselessness.

"The 20th century, for all its trend to yet higher abstraction, saw the gap close somewhat. All the new mathematical objects discovered in the 19th century have found some scientific application, if only in speculative theories."