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.
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 a normal number, each digit occurs with frequency 10% (red line). Digit 1 is overrepresented early because all two-digit numbers start with 1-9, but the balance evens out over the long run.
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.
A normal number has each k-digit block appearing 1/10^k of the time. For 2-digit blocks: each should appear ~1%. Here all 10 selected pairs cluster near the 1% line (red). Over longer runs, all 100 pairs converge exactly to 1%.