THE WONDER OF MATHEMATICS

At various times in my life I have been a mathematics teacher. Ad hoc surveys of my pupils have shown that most children believe that mathematics is "important". But other adjectives - surprising, relevant, beautiful - get a decided thumbs-down.

That is a pity, since there is much in the subject that is both surprising and relevant. Take the humble right-angled triangle, for instance. It has been known since the time of the Ancient Greeks that the sum of the squares of the two smaller sides equals the square of the hypotenuse (Pythagoras' Theorem). Many people also know the easy solution (3,4,5) and the less-easy one (5,12,13). What is surprising about Pythagoras is that, of all the zillions of possible right-angled triangles, there is both a solution with whole numbers and that solution involves SMALL numbers.

The latter fact is very relevant. We know that the square-based Pyramids are extremely precise, and that their right-angles are almost exactly 90 degrees. How would an Egyptian overseer be able to build such a structure, even if he knew his Pythagoras? Plastic set-squares didn't exist in the time of Ramases II. But the overseer could make a rope out of papyrus, with knots at each end and 11 knots in the middle, each knot the same distance from its neighbours. Then he would put a stick in the middle of the fifth knot, one in the middle of the eighth knot and one in the middle of the first and thirteenth knots together. Hey presto, he now has a right-angled triangle and can start building a stone building that doesn't fall down. If there were no whole number solution to Pythagoras, or if there were one and the numbers were large (and why shouldn't they be large?), then the whole history of architecture would have been very different. Humans would probably have gotten there in the end, but it would have taken a much longer time.

Another surprising feature of mathematics is the large number of equations that are linear. By linear, I mean one in the form y = ax; double x and you double y. Linear equations are the simplest of all, easy to work with. You don't need a computer to find a solution.

Two of the most important equations for the whole development of science are linear. One of Newton's laws of motion is F = ma; force equals mass times acceleration. The other is Ohm's law V = IR, voltage equals current times resistance. A priori there is no reason why either of these equations should be linear; and if we looked at them in terms of probability, then the chances would be that they were not. Yet the world would probably have turned out very differently if Newton and Ohm had had to wrestle with non-linear relationships. Again, humans would probably have gotten there in the end, but it would have taken a much longer time.

Mathematics is of course entirely man-made, so it is not as if these finds are part of nature. However, it does seem as if nature is prodding us along in our endeavours, saying "find an easy relationship, guys, and I'll make sure that it works in practice". If true, that would be the most surprising and relevant fact of all. Beautiful even.

Walter Blotscher