This Number Is Randomn't

In 1933, a 20-year-old Cambridge undergraduate named David Champernowne wrote down this number:

0.12345678910111213141516…

You construct it by concatenating all positive integers after the decimal point. The digits are completely deterministic — given any position n, you can compute the n-th digit in seconds. Zero randomness anywhere in the construction.

And yet, Champernowne proved this number is "normal" in base 10. Every digit appears with frequency 1/10. Every two-digit string appears with frequency 1/100. Every length-k string appears with the right frequency for a random sequence of digits. Statistical tests cannot distinguish Champernowne's digits from the output of a fair die.

Four years later, Kurt Mahler proved something stronger: the number is transcendental. It cannot be the root of any polynomial with integer coefficients. The proof uses Liouville's theorem on rational approximation — the digit structure provides rationals that approximate the number too well for it to be algebraic.

This video unpacks: what "normal" means (Borel 1909), why Champernowne is normal, why normal doesn't mean random, Mahler's transcendence proof, the rational approximation trick, and why the same recipe gives us the Copeland–Erdős constant (concatenating primes).

The deeper lesson: π, e, √2 — we don't know if any of them are normal. But by careful construction, we can write down a number that demonstrably is.

Chapters
00:00 The number
01:50 What's normal?
04:00 Champernowne is normal
06:00 Normal doesn't mean random
08:00 Mahler's transcendence proof
10:20 The approximation trick
12:00 Other constructed transcendentals
13:30 What Champernowne means

References
- Champernowne (1933), "The construction of decimals normal in the scale of ten"
- Mahler (1937), "Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen"
- Copeland & Erdős (1946), "Note on normal numbers"
- Borel (1909), "Les probabilités dénombrables et leurs applications arithmétiques"

#math #numbertheory #transcendental #normalnumbers #manim

Видео This Number Is Randomn't канала Euclidia