Sir Isaac Newton is certainly one of the greatest scientists to have ever lived. He is generally reckoned to have been one of the three most outstanding mathematicians of all time, along with Archimedes and Gauss, and his discoveries in physics are unrivalled in their width and influence. What was Newton's secret? How did he achieve as much as he did? Obviously there is no simple answer, but Newton had one secret, which he guarded jealously, and which he believed to be vital. It was ‘Data aequatione quotainque fluentes quantitoe involuente fluxions invenire et vice versa’ or in English ‘solve differential equations’.

Nowadays this ‘secret’ is entirely unremarkable; we are all aware that many processes and phenomena in the world are governed by differential equations. The very fact that Newton's secret is now common knowledge clearly indicates its worth and power. Of course his secret was rather hard won; he did have to invent differential equations before pronouncing his dictum concerning solving them!

In this chapter we shall see what the microcomputer can do for those intending to follow Newton's advice. Our eventual viewpoint will be considerably more modern than Newton's. It turns out that in certain circumstances solving differential equations is not as useful as watching them.

Differential equations and tangent segments

Much of science is devoted to the problems of predicting the future Differential behaviour of some physical system or other. Often the underlying equations and physical law will describe the rate at which the system evolves; what tangent segments we require is a description of how it evolves.

One of the most important and useful ways in which mathematics can help us to solve problems is by the solution of equations. ‘Let x be the length of the piece of string; then x satisfies the equation x2 – 2x - 3 = 0 and solving the equation gives x = 3.’ We are sure that you have solved many problems using equations; unfortunately all but the simplest equations cannot be solved exactly.

There are two reasons for this. In the first place even for a quadratic equation, unless the solutions are rational numbers (as in the above example), there is a square root such as √2 to be evaluated, and this cannot be done exactly. The decimal expansion does not terminate or recur, so we must be satisfied either with the formal ‘√2’ or with an approximation to so-many decimal places.

The second reason is more profound. Exact formulae analogous to the famous quadratic formula do exist for equations of degrees 3 and 4 – of course these formulae involve cube roots and so on, so are open to the same difficulty as we noted above for quadratics. On the other hand no algebraic formula exists at all for equations of degree 5 or more! In a precise sense, the equation x5 - 6x + 3 = 0 cannot be solved algebraically at all. This is a difficult statement and has an even more difficult proof, in which computers won't help in the least.

As we mentioned in the first chapter, general questions concerning whole numbers, rather than merely mechanical computations, can be extremely difficult to answer. Indeed this higher (as opposed to basic) arithmetic is one of the most difficult and fascinating branches of mathematics, and has been extensively studied by many of the greatest mathematicians. These mathematicians often spent an enormous amount of time and energy checking their conjectures with specific numbers and sifting through mountains of experimental data. (Mathematics is an experimental science after all. It differs from other experimental sciences, however, because often the amount of experimental evidence prior to the proof of a result might be very slim, and after the proof experimental evidence is strictly irrelevant, although psychologically reassuring.) Anyway, the computational feats of mathematicians of old are often extremely impressive, bearing in mind that they only had paper and pencil (or papyrus and stylus or whatever) to hand. What an advantage we have! Indeed what might Gauss have achieved with his own micro? Let us start our investigations. The emphasis throughout this chapter will be on the building blocks of arithmetic: the prime numbers.

Prime numbers

Recall that a natural number x is prime if it is only divisible by 1 and itself (in other words cannot be written as a product of natural numbers a and b unless a or b is 1). It is fairly clear that any number can be factored as a product of prime numbers and the resulting factorization is unique up to the order of the factors.