Know your Numbers

This is not a website offering a course in elementary mathematics, needful though such a course is (interested readers might try the Khan Academy); my concern is rather with the underlying rationale of mathematics, the theory of numbers and numbering. However, “If you want to be a blacksmith, go and work at the forge” as the saying goes and one cannot really understand numbers without actually using them and getting to know them well. Hate them and they will hate you. Once you have overcome a likely initial repulsion, you will find that numbers have a fascination all of their own. It is very easy to pick out patterns in batches of numbers that crop up and mankind is, above all, a pattern loving animal — indeed the search for pattern has probably done more for progress than the tool-making capacity. Fibonacci Numbers, Pascal’s Triangle, Primes and so on  retain their fascination nearly a thousand years since they were first discovered and people are still finding new things in them today.

As I have mentioned elsewhere, it was pattern that made me start studying mathematics in the first place : at school I was in the second set and just scraped through what was then Maths ‘O’ level. From then on I made haste to abandon such rubbish for good although I did at one time think I ought maybe to study mathematics simply in order to better attack it. But one day I came across in a book that had nothing to do with mathematics the statement that if you added together three successive odd numbers starting at unity, you get a square number. This worked for 1 + 3 + 5 = 9 = 3²  and, seemingly, for longer strings of odd numbers. Even better, if you select three consecutive odd numbers at random you end up, if not with a square number the next best thing, the difference between two squares. Thus, 11 + 13 + 15 = 39 = 64 ‒ 25 = 8² ‒  5².  That there was such an unexpected association between odd numbers and squares struck me as being little short of miraculous and from then on I became increasingly hooked on mathematics,  number theory in particular. The very first article I published in  M500, the magazine of the Mathematics Department of the Open University, was entitled precisely Sums of Odd Numbers.
Nearly three thousand years ago early Greek mathematicians messing around with pebbles and counters discovered the so-called figurate numbers — which is why we still to this day talk of ‘squares’, ‘triangular numbers’ and ‘odd’ and ‘even’ numbers. Before being marks on paper, numbers were objects and before becoming abstract entities that ‘obey the Axioms of Fields’, numbers were, well, what most of us think of as numbers. Arithmetic goes back to the Ancient Babylonians and Egyptians and scribes five thousand years ago were passing around mathematical puzzles just like people today passing round Sudokus. Contemporary pure  mathematicians look down on the integers, considering them small beer compared to ‘irrationals’, complex numbers and the like; it is even said that they make mistakes with the change though this is probably affectation. Mathematicians behave like this because they  despise everyday physical reality and since the ‘natural’ numbers are about the closest one ever gets in mathematics to the real world, they rate the lowest. The positive integers are the proles, disdained precisely because the modern mathematical elite can’t do without them but don’t want to admit it. As it happens,  the great classical mathematicians were usually excellent calculators: Gauss, sometimes called the ‘Prince of Mathematicians’, was a calculating prodigy and Euler spent his last years doing abstruse computations in his head since he became blind. More recently, the brilliant Indian mathematician, Ramanujan, spent his best years trawling through seas of numbers in the days when the computer was not even a pipe-dream, working with a slate board and chalk (because he found paper and ink too expensive) on his parents’ verandah near Madras far from the madding crowd of Oxford and Cambridge.

The best initial advice I can give to anyone who wants to understand and enjoy numbers is to use them on a more or less daily basis and do stints of mental arithmetic. Several excellent little books on mental arithmetic exist and I will mention one in due course but at the beginning you can just take whatever comes to hand in the nearest charity shop. Even when using a calculator it is well worthwhile making an estimate mentally first  because you can then spot at once if the answer the calculator comes up with is plausible or not. It is also well worth experimenting with ‘primitive’ methods such as finger counting or Russian Peasant Multiplication (about which I have already written). If professional mathematicians had given more thought to the diverse ways of writing numbers, they would have grasped much more quickly what seems painfully obvious today, namely that numbers can be written in any base, our chosen base, ten, being a consequence of our having five fingers and two hands. When one or two mathematicians at the end of the nineteenth century first put forward the notion of alternative bases, they were met with incredulity and scorn from within the profession itself.

After mental arithmetic comes written longhand computation. It pays to be proficient even in long division because the principle is very useful for dealing with polynomials.     When doing mathematics, it is essential to have plenty of space and one of the most useful practical tips I ever got from a tutor was to work with an A4 sheet in the landscape not portrait position, i.e. arrange the sheet so that it is wider than it is high. I had in fact already found something even better than this  (and which also reduced my expenditure), which was to  buy rolls of plain, so-called ‘lining paper’ (wallpaper without a pattern) and spreading this across a table, clamping down the ends at each side. You can now really spread yourself, use different coloured marker pens, have various calculations going on at the same time and so on. Results and formulae are then transferred to a Notebook or computer and you can tear off the sheet and throw it away. It is amazing how much paper a mathematician requires and few  mathematicians work directly on a computer. Recently, I  came across something to replace Ramanujan’s slate and chalk, namely the medium  size ‘whiteboards’ now available cheaply in hardware stores and which allow you to simply wipe off calculations with a Kleenex. This, along with the ‘biro’ pen that writes in real ink, are two recent inventions I would not like to be without.

One cannot get on these days without a pocket calculator but using one is not without its dangers. Apart from it being very easy to touch the wrong  button, everything is in decimal and this gives people get the impression that a perfectly straightforward fraction such as a third (1/3) is ‘really’ an unending expansion which goes on for ever 0.33333333….  In many cases, we actually need to see what is happening on the level of ordinary fractions (which are simply numbers with a temporary base other than ten), or transfer to another base such as 2.

As for the full-size computer, I personally never use it at all for doing mathematics, only to check results I’ve got out already. One of the present-day big mathematics programmes will most likely contain all the mathematics you are likely to need in your lifetime and everything you are likely to discover unless you are a second Euler, but you will learn nothing about mathematics by just clicking  the right buttons to get the ‘right’ answer. The entire joy of mathematics lies in discovery and you will never discover anything if you just look up the subject on Wikipedia. After spending one or two sometimes fruitless mornings on some problem, I do have recourse to Wikipedia but only when I’ve already got out some results for myself, or am completely flummoxed. It is gratifying to find that one has unwittingly been treading, as it were, in the footsteps of some great name from the distant past. Just last week investigating Unit Fractions, I was gratified to see that I eventually hit upon the same all-round formula for reducing any fraction to a series of unit fractions as Leonardo of Pisa (Fibonacci), the greatest mathematician of the Middle Ages, did working in 1203 with quill and parchment by the light of a candle. (He also gave several other artful methods and generalised results in a way that I never managed to.) A friend of mine, a retired civil engineer, Henry Jones, discovered for himself the so-called parametric equation of the ellipse — a result taught in all schools today and already known to Copernicus (though Jones was not aware of this). In its way Jones’s independent discovery of this formula is as great an achievement as Copernicus’s. A further boon of discovering something for yourself rather than reading it in a book, is that you are more likely to see a practical application : Jones proceeded to invent a compass that could draw ellipses, a substantial achievement at the time though subsequently rendered obsolete by computer aided design.

All in all, the trouble with professional mathematicians today is not that they know too little, but that they know too much. This website is in any case not for them but is aimed at the small band of people who practise mathematics simply for enjoyment. Out of all branches of mathematics the most accessible and the most satisfying for a beginner is Number Theory. It is the branch of mathematics where you can make little (or perhaps not so little) discoveries for yourself very rapidly, since, although the higher reaches of the subject still exercise the best brains, the fundamentals are easy to learn and easy to experiment with.

In this ongoing blog, eventually to be turned into a book, I intend to intersperse various fairly simple problems with theoretical and philosophical observations and give the answers in a subsequent post. They are not really so much problems as invitations to explore areas that are of interest or that I have found so.

All this does not rule out doing the necessary spadework including such things as learning off by heart times tables, lists of Fibonacci Numbers, primes and so on : this sort of drill is inevitable in every art or activity and is quite acceptable, even stimulating, if undertaken in small doses. A final word of advice to the aficionado : don’t go at a problem  hammer and tongs for hour after hour. It has been shown scientifically that it is best to take short breaks between stints of study, usually after about forty minutes at most. Hardy, a distinguished mathematician in his day, said that he could never work more than four hours at mathematics in a day (a whole morning) and he then did physical activity in the afternoon (cricket in his case). I can’t manage more than three hours at most before getting hopelessly muddled. Drop it, do something completely different and come back to it afresh.      S.H.  11/3/12