Non-integer base of numeration¶
Represent numbers positionally with a real or complex radix that is not an integer, making admissible digits, expansion algorithms, and nonuniqueness depend on the radix.
Core Idea¶
A non-integer positional representation evaluates a digit string as a sum of integer digits times powers of a non-integer radix β. A β-transformation repeatedly separates an admissible digit and rescales the remainder; its orbit determines the expansion, while algebraic properties of β control finiteness, periodicity, and uniqueness. 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.
The load-bearing residual is not the broad topic of number representation. It is positional place value with a non-integer radix and radix-dependent admissibility dynamics.
Scope of Application¶
Non-integer base of numeration belongs to number representation and is useful where the analyst can specify a radix β with magnitude greater than one, a digit alphabet, positional indices, and finite or infinite digit strings, then evaluate digits from the declared alphabet and powers of one fixed non-integer radix converge to the represented number under a stated selection convention. The scope is broad within that domain but bounded by the need for digits from the declared alphabet and powers of one fixed non-integer radix converge to the represented number under a stated selection convention. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making digits from the declared alphabet and powers of one fixed non-integer radix converge to the represented number under a stated selection convention the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Non-integer base of numeration can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
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 Non-integer base of numeration. Non-integer base of numeration 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: a radix β with magnitude greater than one, a digit alphabet, positional indices, and finite or infinite digit strings. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express digits from the declared alphabet and powers of one fixed non-integer radix converge to the represented number under a stated selection convention independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of number representation because they reuse a radix β with magnitude greater than one, a digit alphabet, positional indices, and finite or infinite digit strings, A β-transformation repeatedly separates an admissible digit and rescales the remainder; its orbit determines the expansion, while algebraic properties of β control finiteness, periodicity, and uniqueness., and declare β and digits, verify convergence and each digit-selection step, identify finite versus infinite representation, and test alternate strings before claiming uniqueness. A theorem, diagnostic, or modeling warning can travel when those roles remain literal.
Relationships to Other Abstractions¶
Current abstraction Non-integer base of numeration Domain-specific
Parents (1) — more general patterns this builds on
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Non-integer base of numeration is a kind of Encoding And Decoding Prime
The proposed strict upward parent is
prime:encoding_and_decoding.
Hierarchy path (1) — routes to 1 parentless root
- Non-integer base of numeration → Encoding And Decoding → Transformation → Function (Mapping)
Neighborhood in Abstraction Space¶
Non-integer base of numeration sits in a moderately populated region (56th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Numeral Bases & Arithmetic Functions (8 abstractions)
Nearest neighbors
- Complex-base system — 0.91
- Champernowne constant — 0.89
- Persistence of a number — 0.89
- Signed number representations — 0.88
- Base36 — 0.87
Computed from structural-signature embeddings · 2026-09-08