AdicNumeral

A $b$-adic number: the rational $x$ read in $\mathbb{Z}_b$ (or $\mathbb{Q}_p$ for prime $b$), exact when built from a rational, $+ O(b^{prec})$ when capped. The ring operations work on it.

AdicNumeral(b, x)exact: $x$ any rational whose denominator $b$ can invert

SageMathQp(b, 60)(QQ(x))Rustadic(b, x)

AdicNumeral(b, x, prec)$x$ known modulo $b^{prec}$, normalised to its representative

AdicNumeral(b, x, prec?)A $b$-adic number: the rational $x$ read in $\mathbb{Z}_b$ (or $\mathbb{Q}_p$ for prime $b$), exact when built from a rational, $+ O(b^{prec})$ when capped. The ring operations work on it.

Typed signatures

Type	Package	Applies
(integer, rational | value, integer?) -> value	adelesreplaces numerals	on ProfiniteNumber
(integer, rational | value, integer?) -> value	numerals	always

Domain: Numeral systems

Details

• Two values share the head. An EXACT adic is a rational, and its expansion can be produced to any depth (it is eventually periodic). A CAPPED adic is known only modulo $b^{prec}$ — what a Hensel lift produces, and what any arithmetic with a capped operand yields; the precision of a sum is the weaker operand's, of a product $\min(v_1 + p_2, v_2 + p_1)$
• Prime $b$ gives the field $\mathbb{Q}_p$: any non-zero divisor works and the result may have negative valuation (digits past the point). Composite $b$ gives the ring $\mathbb{Z}_b$ — no field, zero divisors, and division only by units (coprime to $b$)
• A rational the base cannot expand — $1/2$ in $\mathbb{Z}_{10}$ — leaves the call standing
• A rational operand beside an adic one is read in the same base, so AdicNumeral(10, 1/3) * 3 is AdicNumeral(10, 1). Adics of different bases never combine
• Not a compute-engine number type: the value is a function expression, and Add, Multiply, Negate, Divide, Power are wrapped to recognise it (Subtract reaches them by canonicalisation)
• Default precision for anything unbounded — Hensel lifting, AdicSqrt — is 20 digits

Examples

1 more kept as data, checked against the oracles

Basic #

$…6667 × 3 = 1$: $1/3$ is a 10-adic integer

live ↗

In$3\mathrm{AdicNumeral}(10, \frac{1}{3})$Out$\mathrm{AdicNumeral}(10, 1)$

$…999 + 1 = 0$, carrying forever

live ↗

In$\mathrm{AdicNumeral}(10, -1)+1$Out$\mathrm{AdicNumeral}(10, 0)$

Scope #

capped: the representative modulo $10^8$

live ↗

In$\mathrm{AdicNumeral}(10, \frac{1}{3}, 8)$Out$\mathrm{AdicNumeral}(10, 66\,666\,667, 8)$

prime base: $\mathbb{Q}_5$ is a field, so the point moves

live ↗

In$\frac{\mathrm{AdicNumeral}(5, 3)}{5}$Out$\mathrm{AdicNumeral}(5, \frac{3}{5})$

A negative power inverts exactly: $3^{-1}=1/3$ in $\mathbf{Z}_{10}$

live ↗

In$\frac{1}{\mathrm{AdicNumeral}(10, 3)}$Out$\mathrm{AdicNumeral}(10, \frac{1}{3})$

Properties #

Adding two capped numerals keeps only the weaker operand's precision

live ↗

In$\mathrm{AdicNumeral}(10, \frac{1}{3}, 8)+\mathrm{AdicNumeral}(10, \frac{1}{3}, 5)$Out$\mathrm{AdicNumeral}(10, 33\,334, 5)$

Multiplying by $p^2$ raises the absolute precision by 2 along with the value

live ↗

In$25\mathrm{AdicNumeral}(5, 3, 6)$Out$\mathrm{AdicNumeral}(5, 75, 8)$

Possible issues #

composite base: 2 is not a unit of $\mathbb{Z}_{10}$, so this declines

live ↗

In$\frac{\mathrm{AdicNumeral}(10, 3)}{2}$Out$\frac{\mathrm{AdicNumeral}(10, 3)}{2}$

Numerals over different bases never combine and stay unevaluated

live ↗

In$\mathrm{AdicNumeral}(7, 2)+\mathrm{AdicNumeral}(5, 2)$Out$\mathrm{AdicNumeral}(7, 2)+\mathrm{AdicNumeral}(5, 2)$

1 / 2

The exact zero has infinite valuation

live ↗

In$\mathrm{AdicValuation}(\mathrm{AdicNumeral}(10, 0))$Out$∞$

About

Examples	12
Checked against	SageMath 1 differ7 errorRust 1 agree7 error
Crosswalk	SageMathRust

Implementation

mappedwolframpackages/oracle/src/mappings.ts

No p-adic or b-adic numeral type in Wolfram -- no prior art to map against. Wolfram Function Repository searched too; only ExponentLift turned up (a p-adic valuation helper via the lifting-the-exponent lemma), not a numeral value type.

mappedrustpackages/oracle/src/mappings.ts

The adic crate is p-adic only: our composite bases (10-adic) have no counterpart there.

mappedsagepackages/oracle/src/mappings.ts

Sage's Qp is prime-p only too: composite bases have no counterpart there.

See also: AdicExpansion, AdicValuation, AdicSqrt, HenselLift, IntegerDigits