Concept

Decimal expansion

The writing of a real number as a sequence of digits in base ten, or in any other base. Its digits record repeated multiplication by the base, which is why questions about digits are often questions about a dynamical system.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

A square's worth of points, on a line. A unit square with a point marked, the decimal places of its two coordinates woven into one number, and that number marked on a line beneath.

A line with as many points as a square

Interleave the decimal places of two numbers and one number comes out; take every other place back and the two return. The square has no more points than the segment, and dimension turns out to be invisible to counting.

logic · Cardinality

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.

Bijection Cardinality Continuity Counterexample Dimension Doubling map Equidistribution Homeomorphism Interleaving Invariance of domain Multiplicative order Normal number

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