AdicValuation

The $b$-adic valuation $v_b(x)$ — the power of $b$ dividing $x$ — with its companions: the norm $|x|_b = b^{-v}$ and the unit part $u$ in $x = b^v · u$.

AdicValuation(x)$v_b(x)$; PositiveInfinity for zero

Rustadic_valuation(x)

AdicNorm(x)$b^{-v_b(x)}$: small when highly divisible by $b$

Rustadic_valuation(x)

AdicUnitPart(x)$x / b^{v_b(x)}$, a unit of $\mathbb{Z}_b$

Rustadic_valuation(x)

Typed signatures

Type	Package	Applies
(value) -> integer | signed_infinity	numerals	always

Domain: Numeral systems

Details

• For composite $b$ this is the largest $k$ with $b^k | x$, which is not a valuation in the strict sense (it is not additive: $v_{10}(2) + v_{10}(5) = 0 ≠ v_{10}(10)$) — but it is what the expansion's leading zeros count
• A capped zero is $O(b^{prec})$, so its valuation is reported as the precision: all that is known
• Wolfram's IntegerExponent[n, p] is the valuation on integers

Examples

11 more kept as data, checked against the oracles

Basic #

live ↗

In$\mathrm{AdicValuation}(\mathrm{AdicNumeral}(5, 75))$Out$2$

The valuation of zero is PositiveInfinity, as the signature above says

live ↗

In$\mathrm{AdicValuation}(\mathrm{AdicNumeral}(5, 0))$Out$∞$

live ↗

In$\mathrm{AdicNorm}(\mathrm{AdicNumeral}(5, 75))$Out$\frac{1}{25}$

live ↗

In$\mathrm{AdicUnitPart}(\mathrm{AdicNumeral}(5, 75))$Out$\mathrm{AdicNumeral}(5, 3)$

Scope #

negative in $\mathbb{Q}_5$

live ↗

In$\mathrm{AdicValuation}(\mathrm{AdicNumeral}(5, \frac{3}{25}))$Out$-2$

About

Examples	16
Checked against	SageMath 14 agreeRust 15 agree
Crosswalk	Rust

Implementation

mappedrustpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

See also: AdicNumeral, AdicExpansion