Concept

Normal number

A number whose digits in a given base contain every block of digits as often as a random sequence would. Almost every number is normal, but no familiar constant such as the square root of two or pi is known to be.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

$How long four orbits of the doubling map spend in each quarter. Grouped bars for four starting points of the doubling map, each group showing the fraction of time the orbit spends in each quarter of the unit interval, against a dashed line at one quarter.$

Almost every orbit is fair

Double a number and keep the fractional part, and do it again, and again. Where the orbit goes is written in the number's binary digits, and for almost every starting point it spends exactly a quarter of its time in each quarter of the interval. The proof is a short argument about Fourier coefficients being pushed to infinity — and it leaves room for a set of exceptions with no length at all, which includes every fraction and, for all anyone can prove, every number anyone has ever named.

dynamics · Iteration $Base-φ digits of two fractions and two irrational numbers. Four rows of 72 base-φ digits each, for 1/3, 2/7, √2 − 1 and 1/π; the two fractions repeat with periods 8 and 16, the two irrational numbers show no period.$

Fractions repeat and roots look random

Almost every number between 0 and 1 has base-φ digits with the frequencies Parry's measure predicts. Which particular numbers do? Every fraction, provably, does not: its expansion repeats from the first digit, with a period equal to the period of the Fibonacci numbers modulo its denominator. And √2 − 1 and 1/π, computed exactly to twelve thousand digits, match every predicted frequency to within sampling error — which proves nothing about them at all.

dynamics · Symbolic dynamics

The cosines of the powers of two

Start at cos 1 and apply x ↦ 2x² − 1 over and over. The map is chaotic and exactly solvable: the k-th point is cos(2ᵏ), the cosine of two to the k radians. Iterate it in floating point and the computed orbit leaves the true one after 51 steps. Read the binary digits of 1/(2π) instead and the true orbit can be followed for as long as digits can be computed — here 262,080 steps — and at that length it is as fair as any orbit can look. Whether it stays fair is a question about those digits, and nobody can answer it.

dynamics · Iteration

One number, fair in base 2 and gappy in base 6. α binary: 000010101011100011100011100011110110100001001011…; base 6: 013014043000333425113050213000000124355504543223…; zero runs after 3^k: 81:16, 243:42, 729:121.

Fair in base two and not in base six

Almost every number has digits that look like fair coin tosses in every base, and almost no number anyone can name is known to. Stoneham's number, 1/(3·2³) + 1/(9·2⁹) + 1/(27·2²⁷) + …, is one of the few: its binary digits are provably fair. Written in base 6 the same number has a run of 121 zeros starting at the 730th digit, and a longer one at every power of three after — because in base 6 each of its terms is a terminating fraction, and the gaps between them never fill.

dynamics · Symbolic dynamics

Named alongside it

The objects these essays reach for when they reach for this one.

Binary expansion Doubling map Ergodic theorem Invariant measure Beta expansion Chebyshev polynomial Decimal expansion Equidistribution Ergodicity Fibonacci numbers Fourier series Golden ratio

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