PLAYING THE NUMBERS

Dick Pountain/ Idealog 291/ 6th October 2018 13:41:14

Most people encounter turning points in their lives, and one of the earliest in mine came while deciding to apply for university: I loved both chemistry and maths, and had the A level passes to study either, so I could quite easily have become a mathematician, but at that time the lure of smells and bangs was just a bit stronger. However I’ve never lost my love of mathematics, especially number theory, and still do some for fun from time to time. Back in the sixties I spent months playing at topology, in particular notations for classifying knots. More recently – as I’ve mentioned here before – I wanted to hear what the prime numbers sound like, which lead me into a whole computer music composition project.

A subject of particular fascination for me has always been irrational numbers. If you remember your school maths, these are numbers that cannot be expressed as fractions – that is as the ratio of two whole numbers like ½ or 3793/8331. Examples are pi and , which can only be represented by infinitely-long, non-repeating strings of decimal (or binary, hex, octal etc) digits, which are of course impossible to complete. And the weirdest thing about irrational numbers is how many of them there are....

You can present all the numbers as a line that stretches off forever to left and right, on which every point represents a different number. You can tick off the whole numbers along this line, like the marks on a ruler, and in between each mark there will be infinitely many rational numbers, fractions whose decimal representation comes to an end like 4.567757, and infinitely many irrational numbers. The scary thing is that there are ‘more’ (whatever that means) irrationals than anything else, because if you choose a point at random on the line, there’s a 100% chance it will be irrational! I know, it made me feel seasick too when I learned this, from an article in Quanta Magazine called ‘Why Mathematicians Can’t Find the Hay in a Haystack’ (https://www.quantamagazine.org/why-mathematicians-cant-find-the-hay-in-a-haystack-20180917). Almost all the numbers we use in everyday life are rational, whole numbers like ‘12 eggs’ or fractions like £4.56, but these are like needles in an infinite haystack of irrationals, impossible to locate by random search. And worse still, the hay itself is impossible to describe in finite space.

Rational numbers are easier to handle, and of course they’re all we can deal with on a computer that has both finite memory and finite word-length. But even rational numbers can boggle the mind. Another of my passions is computer graphics, and images on a computer screen are really nothing but tables of numbers, displayed as colour intensities by the graphics hardware. Suppose your computer has a 1920 x 1080 HD display and 32-bit colour, then every line on that screen could be represented by a 61,440-bit long binary number, and every possible image by a 66,355,200-bit number (61440*1080).

So, you could write a program that counts from 0 to 66,355,200, converts each number to a bitmap and displays it, and it would eventually show everything that could be possibly seen on that screen, like those proverbial monkeys on typewriters that might produce Shakespeare. It wouldn’t take infinite time either: if you showed each picture for a second you’d be done in two years, though I wouldn’t sit up waiting for the Mona Lisa to appear.

Actually the more I think about this, the more I want to set it up and present it to some gallery as an art project - people might sit in front of it hoping to see something recognisable, like the librarians in Jorge Luis Borges’ Library of Babel searching for a legible word.

Maths, when properly appreciated, is like a good roller coaster that induces vertigo, nausea and pleasure all at the same time. For me, the ultimate boggle is Cantor’s ‘diagonal’ proof that there are infinitely more real numbers, rational and irrational, than there are whole numbers. Build a table, infinitely deep and infinitely wide, of all the real numbers (which will take a while). Traverse this table diagonally, taking first digit of line one, second digit of line two and so on, add one to it and string these all together into a new infinitely-long number. This new number can’t be anywhere in the table because it differs from every number in the table in one place, so infinity squared isn’t big enough to hold them all – they are uncountable. First encountered at school, this still hurts my head a little even now. Perhaps I made the right choice after all: in chemistry you can only poison yourself or blow yourself up.