Write out all positive integers in order after a decimal point: 0.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15… This is the Champernowne constant. Its decimal expansion contains every finite sequence of digits somewhere, and every block of k digits appears with exactly the frequency 1/10ᵏ.
First 1000 digits – digit 1 appears most due to numbers 1-9, 10-19... Distribution normalises as n grows.
D. G. Champernowne constructed this number in 1933, as an undergraduate at Cambridge, to provide the first explicit example of a normal number in base 10. A normal number is one in which every block of k digits appears with frequency 1/10ᵏ. Champernowne proved his constant is normal, a feat that remains impossible for naturally occurring constants like π or e.
In the first 100 digits, digit 1 appears 14 times. The imbalance disappears as more digits are included.
Kurt Mahler proved in 1937 that C₁₀ is transcendental. The number 0.1234567891011… is one of the rare constants we can compute to any precision trivially, yet whose decimal expansion encodes every possible finite text, every number, every piece of information ever written, somewhere in its digits.
Selected 2-digit diagonal pairs in the first 10,000 digits of Champernowne's constant. Each pair appears close to 1% of the time. Full normality emerges at much larger scales.