Amazing Normal Numbers

For any randomly chosen decimal number, we might expect that all the digits, 0, 1 , … , 9, occur with equal frequency. Likewise, digit pairs such as 21 or 59 or 83 should all be equally likely to crop up.  Similarly for triplets of digits. Indeed, the probability of finding any finite string of digits should depend only on its length. And, sooner or later, we should find any string. That’s “normal”!

This is obviously not true for all numbers; for example, the number 0.11111111… with only ones is clearly not normal. But there are compelling arguments that most numbers are normal, although I am unsure whether this has been rigorously proved.

Normal Numbers

To give a precise definition, a decimal number N is normal if its digits follow a uniform distribution: that is, all digits are equally likely, all pairs of digits are equally likely, all triplets of digits are equally likely, and so on (strictly, we should say normal in base 10). We would expect each of the strings 0, 1, 2,…, 9 in a normal number to occur one tenth of the time, each of the strings 00, 01, … , 98, 99 to occur one hundredth of the time, and so on.

It is difficult to prove whether a given number is or is not normal. For example, we don’t know if the mathematical constants π and e are normal or not. But normal numbers can be constructed. One of the first shown to be normal was the Champernowne constant, devised in 1933 by the English economist and mathematician David Champernowne while he was an undergraduate in Cambridge. It is written simply asChampernownewhere we write all the counting numbers in order after the decimal point (the spaces are for clarity). It is obvious that any string of digits must occur in C, since every string is also a whole number. In fact, every string must occur an unlimited number of times.

Encoding Strings

Now consider the word “IRELAND”. We can encode this by assigning numbers to the letters of the alphabet, A=01, B=02, … , Z=26, yielding the string [09 18 05 12 01 14 04]. This string occurs in C so we can say that the word IRELAND is encoded in the number.

But a similar argument holds for a longer string, such as the Bible or the complete works of Shakespeare. The (very long) string corresponding to the Bible occurs infinitely often in C. And also strings corresponding to translations of The Good Book into every known language. And every other book that has ever been published is hidden within it.

But there is more:  your genetic makeup is determined by the structure of your DNA, and this can be expressed as a string of digits. So you are encoded in C, and so am I, and so is everyone else who has ever lived. Indeed, if the Universe is finite, it can be represented by a number, unimaginably large but finite nonetheless. So the total state of the Universe is in C.

One more thing: the winning numbers of next week’s lottery are most certainly contained (infinitely often) in C. But there is a slight difficulty: how to get them out!


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