5 - A little mathematics

Summary

If a thing is worth doing, it is worth doing badly.

G.K. Chesterton

The only mathematical operations we have needed so far have been addition, subtraction, division, multiplication, and an extension of the last: the operation of taking the square-root. [The square-root of a number multiplied by itself is equal to the number.] We have also had the help of a friendly work-horse, the computer, which gave insight into formulas we produced. That insight was always particular to the formula being evaluated; the general rules of thumb we used in the last chapter were obtained by nothing more or less than sleight of hand. To do better, there is no way of avoiding the mathematical operation associated with taking arbitrarily small steps an arbitrarily large number of times. This is the essential ingredient of what is called ‘calculus’ or ‘analysis.’ But don't be alarmed: our needs are modest. We shall be able to manage very well with only some information about the exponential and the logarithm. You will have heard, at least vaguely, of these ‘functions’ – to use the mathematical expression for a number that depends on another number, so that it can be represented by a graph – but rather than relying on uncertain prior knowledge, we shall learn what is needed by empirical self discovery.

Powers

The constructions we need are very natural generalizations of the concept of ‘powers’ which you will find that you are quite familiar with from what has gone before.