Unknown Quantity: A Real and Imaginary History of Algebra

An excellent general history of algebra

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. ...