THE NUMBER OF EVERYTHING

Dick Pountain/Idealog 240 /09 July 2014 11:45

My interest in computing has always been closely connected to my love of maths. I excelled in maths at school and could have studied it instead of chemistry (where would I be now?) My first experience of computing was in a 1960 school project to build an analog machine that could solve sixth-order differential equations. I used to look for patterns in the distribution of primes rather than collect football cards - you're probably getting the picture. I still occasionally get the urge to mess with maths, as for example when I recently discovered Mathlab's marvellous Graphing Calculator for Android, and I'll sometimes scribble some Ruby code to solve a problem that's popped into my head.

Of course I've been enormously pleased recently to witness the British establishment finally recognising the genius of Alan Turing, after a disgracefully long delay. It was Turing, in his 1936 paper on computable numbers, who more than anyone forged the link between mathematics and computing, though it's for his crucial wartime cryptography that he's remembered by a wider public. While Turing was working on computable numbers at King's College Cambridge, a college friend of his David Champernowne, another rmathematical prodigy, was working on something rather different that's recently come to fascinate me. Champernowne soon quit maths for economics; studied under John Maynard Keynes; helped organise aircraft production during WWII; in 1948 helped Turing write one of the first chess-playing programs; and then wrote the definitive book on income distribution and inequality (which happens be another interest of mine and is how I found him). But what Champernowne did back in 1933 at college was to build a new number.

That number, called the Champernowne Constant, has some pretty remarkable properties, which I'll try to explain here fairly gently. The number is very easy to construct: you could write a few million decimal places of it this weekend if you're at a loose end. In base 10 it's just zero, a decimal point, followed by the decimal representations of each successive integer concatenated, hence:

0.12345678910111213141516171819202122232425262728293031....

It's an irrational real number whose representation goes on for ever, and it's also transcendental (like pi) which means it's not the root of any polynomial equation. What most interested Champernowne is that it's "normal", which means that each digit 0-9, and each pair, triple and so on of such digits appear in it equally often. That ensures that any number you can think of, of whatever length, will appear somewhere in its expansion (an infinite number of times actually). It's the number of everything, and it turns out to be far smaller (if somewhat longer) than Douglas Adams' famous 42.

Your phone number and bankcard PIN, and mine, are in there somewhere, so it's sort of like the NSA's database in that respect. Fortunately though, unlike the NSA, they're very, very hard to locate. The Unicode-encoded text of every book, play and poem ever written, in every language (plus an infinite number of versions with an  infinite number of spelling mistakes) is in there somewhere too, as are the MPEG4 encodings of every film and TV programme ever made  (don't bother looking). The names and addresses of everyone on earth, again in Unicode, are in there, along with those same names with the wrong addresses. Perhaps most disturbingly of all, every possible truncated approximation to Champerknowne's constant itself should be in there, an infinite number of times, though I'll confess I haven't checked.  

Aficionados of the Latin-American fiction will immediately see that Champernowne's constant is the numeric equivalent to Jorge Luis Borges' famous short story "The Library of Babel", in which an infinite number of librarians traipse up and down an infinite spiral staircase connecting shelves of random texts, searching for a single sentence that makes sense. However Champernownes' is a rather more humane construct, since not only does it consume far less energy and shoe-leather, but it also avoids the frequent suicides -- by leaping down the stairwell -- that Borges imagined.

A quite different legend concerns an Indian temple at Kashi Vishwanath, where Brahmin priests were supposed to continually swap 64 golden disks of graded sizes between three pillars (following the rules of that puzzle better known to computer scientists as the "Tower of Hanoi"). When they complete the last move of this puzzle, it's said the world will end. It can be shown that for priests of average agility this will take around 585 billion years, but we could remove even that small risk by persuading them to substitute instead a short Ruby program that builds Champerknownes' constant (we'll need the BigDecimal module!) to be left running on a succession of PCs. Then we could be absolutely certain that while nothing gets missed out, the end will never arrive...