P-adic number¶

An element of the completion of the rational numbers under the non-Archimedean absolute value determined by a prime p.

Version: v1 · 2026-09-08 · History
Domain-specific #: 5942
Origin domain: number theory
Subdomain: number theory

Core Idea¶

The prime p is fixed, expansions extend toward increasing positive powers, convergence is governed by divisibility and the resulting field is complete but ordered unlike neither the reals nor ordinary digit intuition. Rationals close under a metric in which numbers are near when their difference is divisible by a high power of p, producing infinite base-p expansions and ultrametric geometry. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application¶

P-adic number belongs to number theory and is useful where the analyst can specify the typed number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the prime p, p-adic valuation and absolute value, metric and Cauchy completion, digit expansion and uniqueness convention, arithmetic and carries, ultrametric inequality, embedding of rationals and convergence examples are explicit. The scope is broad within that domain but bounded by the need for the prime p, p-adic valuation and absolute value, metric and Cauchy completion, digit expansion and uniqueness convention, arithmetic and carries, ultrametric inequality, embedding of rationals and convergence examples are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the prime p, p-adic valuation and absolute value, metric and Cauchy completion, digit expansion and uniqueness convention, arithmetic and carries, ultrametric inequality, embedding of rationals and convergence examples are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to P-adic number. P-adic number compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the prime p, p-adic valuation and absolute value, metric and Cauchy completion, digit expansion and uniqueness convention, arithmetic and carries, ultrametric inequality, embedding of rationals and convergence examples are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of number theory because they reuse the typed number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Rationals close under a metric in which numbers are near when their difference is divisible by a high power of p, producing infinite base-p expansions and ultrametric geometry., and type the carrier, state every parameter and convention in the definition, test that the prime p, p-adic valuation and absolute value, metric and Cauchy completion, digit expansion and uniqueness convention, arithmetic and carries, ultrametric inequality, embedding of rationals and convergence examples are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction P-adic number Domain-specific

Parents (1) — more general patterns this builds on

P-adic number is a kind of Metric Prime

The proposed strict upward parent is prime:metric.

Hierarchy path (1) — routes to 1 parentless root

P-adic number → Metric → Function (Mapping)

Neighborhood in Abstraction Space¶

P-adic number sits in a crowded region of the domain-specific corpus (8^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Number-Theoretic Sequences & Classes (37 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08