Almost every number is normal, yet we can barely name one
Mathematicians proved over a century ago that nearly all real numbers are normal, meaning their digits look perfectly random, with every string of digits turning up equally often. Yet nobody has managed to prove it for pi, e or the square root of 2. Being almost everywhere, it turns out, is not the same as being easy to find.
A number is simply normal in a base if each digit appears with equal long-run frequency, so in base 10 every digit shows up a tenth of the time. Full normality asks more: every block of two digits, every block of three, and so on, must appear with its fair share. In binary that means 0 and 1 each half the time, each of the four pairs a quarter of the time, and so on. Picture an endless run of coin tosses: long streaks of tails will occur, but no pattern is favoured over any other of the same length.
The concept came from Émile Borel in 1909, who proved that almost all real numbers are normal. The non-normal ones are uncountably many, yet together they take up no room on the number line, a set of measure zero. They are also everywhere: every fraction is non-normal, because its digits eventually repeat, and so is any number whose decimal expansion never uses the digit 1. Borel's proof was not constructive, though. Wacław Sierpiński pinned down a specific normal number in 1917, and in 2002 Becher and Figueira showed one can be computed digit by digit.
Examples that are normal in a single base can be built by stringing numbers together. Champernowne's constant, made by writing the natural numbers one after another after a decimal point, is normal in base 10. Copeland and Erdős showed in 1946 that concatenating the primes works too, and Besicovitch had done the same for the squares in 1935. Davenport and Erdős extended it in 1952 to the values of any suitable polynomial.
A number can be normal in one base and not in another. Only one that passes in every base counts as absolutely normal. Chaitin's constant qualifies but cannot be computed, which leaves the familiar constants as tantalising suspects.
Source: Normal number
