4: Algebra
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The first rule of intelligent tinkering is to save all the parts.
Paul R. Ehrlich (1932– )
Many important aspects of serious mathematics have their roots in the world of arithmetic. However, when we implement an arithmetical procedure by combining numbers with very different meanings to produce a single numerical output, it becomes almost impossible to see how the separate ingredients contribute to the final answer. In other words, calculating exclusively with numbers contravenes Paul Ehrlich’s “first rule of intelligent tinkering”. This is why in Chapters 1 and 2 we stressed the need to move beyond blind calculation, and to begin to think structurally - even when calculating purely with numbers. Algebra can be seen as a remarkable way of “tinkering with numbers”, so that we not only “keep all the parts”, but manage to keep them separate (by giving them different names), and hence can see clearly what contribution each ingredient variable makes to the final output. To benefit from this feature of algebra, we need to learn to “read” algebraic expressions, and to interpret what they are telling us - in much the same way that we learn to read numbers (so that, where appropriate, 100 is seen as 102, and 10 is seen as 1 + 2 + 3 + 4).
Before algebra proper was invented (around 1600), the ability to extract the general picture lying hidden inside each calculation was restricted to specialists. The ancient Babylonians (1700–1500 BC) described their general procedures as recipes, presented in the context of problems involving particular numbers. But they did this in such a way as to demonstrate convincingly that whoever formulated the procedure had managed to see “the general in the particular”. The ancient Greeks used a geometrical setting to reveal generality, and encoded what we would see as “algebraic” methods in geometrical language. In the 9th century AD, Arabs such as Al-Khwarizmi (c.780–c.850), managed to encapsulate generality using a very limited kind of algebra, without the full symbolical language that would emerge later. The abacists, such as Paolo dell’Abbaco (1282–1374) who featured in Chapter 3, clearly saw that the power and spirit of mathematics was rooted in this generality. But modern algebraic symbolism - in particular, the idea that to express generality we need to use letters to represent not only variables, but also important parameters (such as the coefficients a, b, c in a general quadratic ax2 + bx + c) - had to wait for the inscrutable writings of Viète (1540–1603), and especially for Fermat (1601–1665) and Descartes (1596–1650) who simplified and extended Viète’s ideas in the 1630s.
Within a generation, the huge potential of this systematic use of symbols was revealed by the triumphs of Newton (1642–1727), Leibniz (1646–1716), and others in the years before 1700. Later, the refinements proposed by Euler (1707–1783) in his many writings throughout the 18th century, made this new language and its discoveries accessible to us all - much as Stevin’s (1548–1620) version of place value for numbers made calculation accessible to Everyman.
Our coverage of algebra is necessarily selective. We focus on a few ideas that are needed in what follows, and which should ideally be familiar - but with an emphasis that may be less familiar. When working algebraically, the key mathematical messages are mostly implicit in the manipulations themselves. Hence many of the additional comments in this chapter are to be found as part of the solutions, rather than within the main text.