NumeralSystemShape

What a system's numerals look like, as a Dictionary: whether it is a bijection, which integers it spells, and with which digits.

NumeralSystemShape(system)Bijective, Integers, and when they apply Digits (one set, or one per place), Width and Rule

Typed signatures

Type	Package	Applies
(any) -> dictionary	numerals	always

Domain: Numeral systems

Details

• Digits is one set when every place shares it, and a list — most significant first — when places differ, as in a mixed radix or a residue system
• Rule names a constraint no per-place bound captures: Zeckendorf's no two adjacent ones, Ostrowski's ceiling rule
• A residue system whose moduli share a factor still spells every integer below $\prod m_i$, but not uniquely: Bijective is False

Examples

Basic #

4 and 6 share a factor, so this is not a bijection

live ↗

In$\mathrm{NumeralSystemShape}(\mathrm{ResidueNumerals}(\bigl\lbrack4, 6\bigr\rbrack))$Out$\left\lbrace \text{Bijective}→\mathrm{False},\;\text{Integers}→0..23,\;\text{Digits}→\bigl\lbrack0..3, 0..5\bigr\rbrack,\;\text{Width}→2\right\rbrace$

Scope #

live ↗

In$\mathrm{NumeralSystemShape}(\mathrm{ZeckendorfNumerals})$Out$\left\lbrace \text{Bijective}→\mathrm{True},\;\text{Integers}→ℕ,\;\text{Digits}→0..1,\;\text{Rule}→ \text{no two adjacent ones}\right\rbrace$

ordinary base 2, in the same shape every other system reports

live ↗

In$\mathrm{NumeralSystemShape}(\mathrm{PositionalNumerals}(2))$Out$\left\lbrace \text{Bijective}→\mathrm{True},\;\text{Integers}→ℕ,\;\text{Digits}→0..1\right\rbrace$

every integer, negatives included, with no sign

live ↗

In$\mathrm{NumeralSystemShape}(\mathrm{BalancedNumerals}(3))$Out$\left\lbrace \text{Bijective}→\mathrm{True},\;\text{Integers}→ℤ,\;\text{Digits}→(-1)..1\right\rbrace$

About

Examples	4

See also: IntegerDigits, FromDigits