@enumeratio/numerals

Numeral-system extensions for @cortex-js/compute-engine. compute-engine's IntegerDigits and FromDigits already do fixed radix, so this package adds no head for that case — it widens the base slot to accept a system value, the way Wolfram's IntegerDigits[n, MixedRadix[…]] does: FactorialNumerals, ZeckendorfNumerals, BalancedNumerals(b), NegativeNumerals(b), BijectiveNumerals(k), MixedRadixNumerals([…]), PrimorialNumerals, CombinatorialNumerals(k), ResidueNumerals([…]), AdicNumerals(b, prec?) and PositionalNumerals(b) are all just values passed where an integer base would go. An integer base still goes to the native handler, unchanged. Builds on @enumeratio/residues for the residue system and gcd.

AdicNumeral (singular) is a separate value head: the $b$-adic number itself, with its own arithmetic (Add/Multiply/Negate/Divide/Power widened to carry it), infinite to the left rather than the right.

The old one-word-per-flavour names (Factoradic, Zeckendorf, BalancedRadix, Radix, …) still evaluate to their …Numerals spelling — see NUMERAL_ALIASES.

Usage

import { declareNumerals } from "@enumeratio/numerals";
import { declareAdic } from "@enumeratio/numerals"; // AdicNumeral alone, without the systems

declareNumerals(ce); // declares AdicNumeral too

./src is exported for workspace packages needing the plain TypeScript.

IntegerDigits(42, AdicNumerals(10, 6))   // the 10-adic digits of 42, width 6
IntegerDigits(-3, BalancedNumerals(3))   // balanced ternary spells the sign in the digits

Heads

Numeral systems (values for the base slot)

• Radix / PositionalNumerals — ordinary base $b$, as a system value
• BalancedNumerals / BalancedRadix — signless negatives, digits $-(b-1)/2 \ldots (b-1)/2$
• NegativeNumerals / NegativeRadix — signless negatives over base $-b$
• BijectiveNumerals / BijectiveRadix — digits $1\ldots k$, no zero (spreadsheet columns)
• FactorialNumerals — place $k$ holds at most $k$ (the Lehmer code)
• PrimorialNumerals — factoradic with primes
• CombinatorialNumerals / CombinatorialSystem — a strictly decreasing $k$-tuple (subset unranking)
• ResidueNumerals / ResidueSystem — independent residues, no place values at all
• ZeckendorfNumerals — Fibonacci place values, no two adjacent ones
• OstrowskiNumerals — digits over a continued fraction, generalises Zeckendorf
• AdicNumerals — fixed-width $b$-adic digits, negatives are all nines

b-adic values

• AdicNumeral — the $b$-adic value, exact or truncated to a precision
• AdicExpansion / AdicDigits — its digit string
• AdicValuation / AdicUnitPart — the power of $b$ dividing it, and what's left after dividing it out
• AdicSqrt / HenselLift — square roots and Newton lifting in $\mathbb{Z}_b$

Digits and strings

• IntegerDigits / FromDigits — compute-engine's own heads, widened to take a system in the base slot
• RealDigits — digits of a rational, including the repeating part
• DigitCount / DigitSum / IntegerLength / IntegerReverse — per-digit statistics and reversal
• IntegerString / RomanNumeral — string spellings
• NumberExpand / NumeralSystemShape / Fraction — expanding a number over a system, and describing a system's shape

See also

See Numeral systems for why a numeral system is a bijection between integers and digit strings, and a table of every system this package adds. See b-adic numbers for how AdicNumeral and AdicNumerals relate, and where they part ways off the non-negative integers.

Builds on @enumeratio/residues (residues) for ResidueNumerals and gcd. @enumeratio/number-theory (number-theory) depends on this package.

Pages

• Numeral Systems
• b-adic Numbers

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• Source: packages/symbols/arithmetic/numerals
• Depends on: boxes, engine, residues, structures
• In development, also: config, manifest, utils