Unknown Quantity

Title: Unknown Quantity – A Real and Imaginary History of Algebra

Author: John Derbyshire

Publisher: Plume, 2006 (First)

ISBN: 978-0-452-28853-9

Pages: 320

Mathematics is one subject which is best avoided by most popular science writers. The apathy of the audience plays a large part in this conceivable reluctance. Stephen Hawking himself says that he could have sold double the number of copies of his magnum opus ‘A Brief History of Time’ if he had omitted the only equation E = mc² contained in the book. So much for the poor opinion science writers cultivate about the general readers. Presumably, John Derbyshire and the publisher Plume Books don’t share this misconception and the result is a thoroughly enjoyable and infotaining book on mathematics. Unfortunately, it deals only with a topic within the vast ocean of math, algebra. The author is a mathematician and linguist and the celebrated author of Prime Obsession, a mathematical biography of Bernhard Riemann. He traces the history of algebra from the mists of prehistory to ultramodern concepts which is just finding acceptance among scientific community.

Mathematical ability was possessed by mankind from prehistorical times. People used it to quantify game of prey, and to distribute it among the tribe in a fair manner. However, the sense of arithmetic became abstract much much later. When the concept of, say, threeness as in the case of three lambs got detached from bondage to the physical entity and ended up in a symbol representing threeness, the seed of mathematics was born. Every primitive society had concepts of their own, but we find formulae of sufficiently advanced level by 18^th century BCE Babylon. Ancient Babylonians were very good astronomers and clay tablets of computations have come down to our time. We must keep in mind that these computations have no resemblance to modern symbolism. In fact, they didn’t even use letter symbols to represent unknown quantities as algebra does today. The first stirrings of thought in this direction was made by Greek thinker Diophantus who used primitive symbols in computations and is considered to be the father of algebra.

After the flowering of classical Greek period, Europe relapsed into the darkness of Middle Ages. The spirit of scientific inquiry and reason were kept alive by Arab scholars who flourished during the early Islamic period in Baghdad and Isfahan. In fact, the term ‘algebra’ was derived from the title of the book, ’al-Kitab al-Mukhtasar fi hisab al-jabr wa’l-muqabala’ written in the 9^th century. The scholar Abu Jafar Muhammad ibn Musa al-Khwarismi, who was the author, also gave the term algorithm through a poor rendering of his own name to Latin. When Renaissance finally dawned, Europe caught up for the lost time and surpassed all others in ingenuity of thought. General solutions of cubic and quartic equations (of powers three and four of the unknown quantity) were found in the 15^th and 16^th centuries. Descartes provided the basis for modern symbolism. Algebra really took off from solving equations to new realms in 19^th century when Niels Abel proved that there is no algebraic solution to the general quintic equation (of power 5) in 1826.

As the author himself admits, the handling and processing of numbers alone is quite dry and devoid of colour, which adds zest to life. Math is not without such flamboyant characters, the most notable being Evariste Galois, the French mathematician who was the proponent of group theory that went on to become the essence of many other fields. Galois was a rebel who lost his life in a duel with an opponent probably waged for an unrequited love. Anticipating his defeat and sure death the next morning, Galois was reported to have scribbled a few notes on a piece of paper which paved the way for the development of group theory. That century also saw the re-emergence of geometry in the guise of algebraic geometry. Derbyshire also lists the brand new areas that have grown up in math like category theory, motivitic cohomolgy and others which are still not understood by people who are not in the possession of a higher mathematical degree.

The book is superbly conceived and delightfully presented, at least in the first three quarters of the text. Considering the nature of the subject, Derbyshire has worked wonders in presenting the concepts in such a way as to be comprehensible to any class of readers and with the right mix of history which is equally important to do justice to the title of the book. The illustrations are ample and the brief mathematical prefaces which the author terms ‘primers’ serve their purpose well. He also ensures that the readers stay with him on the same page, literally! The biographical sketches add interest to the narrative.

Even with all this, there is no denying that reading becomes tardy when the historical account reaches the 19^th century. From here onwards, the concepts become profound and turns unintelligible for those who have no background of higher mathematics. However, this is quite understandable and does not diminish the charm of the work.

The book is highly recommended.

Rating: 3 Star