Whilst reading von Baeyers ‘Information’ recently, I came across the following fun mathematical tidbit which I thought was worth sharing. Mainly because I couldn’t find many references to it elsewhere on the ‘net.

In the chapter on “Randomness”, von Baeyer introduces several definitions of the term “random”, iteratively showing how each is slightly flawed. Considering a binary sequence of digits, the first definition describes a random number as one in which there is no pattern to the series of 1’s and 0’s. However a sequence such as 000110000100 is not random as it has an unequal proportion of the binary digits. A slightly improved definition is one which states that the numbers of each digit are approximately equal. But not only that: there combinations of the two digits (00, 01, 10, 11) must also occur in roughly equal proportions. And so on for combinations of three, four, five digits. Sequences that meet this restriction are apparently known as “normal numbers”.

The first explicit (rather than theoretical) example of a normal number is Champernowne’s Constant which was produced (discovered?) in 1933. David Champernowne pointed out that if one starts with zero, then one then string together all possible pairings, then all eight triples, an so on you end up with a number which must, by construction, contain all possible patterns, and is therefore “normal”.

Von Baeyer then points out that this number in its binary form is “a fabulous object. Using Morse code, or some other translation of zeroes and ones into typographical symbols, it can be transformed into a string of letters, spaces and punctuation marks. Since every conceivable finite sequence of words is buried somewhere in the string’s tedious gobbledygook, every poem, every traffic ticket, every love letter and every novel ever written, or ever to be composed in the future is there in that string…You may have to travel out along the string for billions of light years before you find them, but they are all in there somewhere….” (pp101-102).

So who needs a million chimpanzees with typewriters? Distributed computing project anyone?

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