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.

Version: v1 · 2026-09-08 · History
Domain-specific #: 5784
Origin domain: number representation
Subdomain: beta and complex radix expansions

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.^[1] 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. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if mixed bases are used by position, β is actually an integer, an arbitrary polynomial expression is called a numeral, or uniqueness is assumed from decimal intuition. This gives the entry an operational identity rather than merely a historical label.

A useful analysis keeps three layers separate. The constitutive layer says what must be true: digits from the declared alphabet and powers of one fixed non-integer radix converge to the represented number under a stated selection convention. The evidential layer asks what observation or proof warrants the claim: declare β and digits, verify convergence and each digit-selection step, identify finite versus infinite representation, and test alternate strings before claiming uniqueness. The use layer asks what reasoning becomes available once the identity is established: studying beta shifts and symbolic dynamics, coding, quasicrystal coordinates, and arithmetic of algebraic radices. Conflating the layers is the most common source of scope inflation.

Structural Signature¶

Carrier: a radix β with magnitude greater than one, a digit alphabet, positional indices, and finite or infinite digit strings
Inputs or antecedent state: radix, digit set, sign and complex convention, expansion direction, greedy or alternative digit-selection map, admissibility, convergence, and equivalence of strings
Constitutive operation: 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.
Invariant: digits from the declared alphabet and powers of one fixed non-integer radix converge to the represented number under a stated selection convention
Recognition test: declare β and digits, verify convergence and each digit-selection step, identify finite versus infinite representation, and test alternate strings before claiming uniqueness
Output or consequence: studying beta shifts and symbolic dynamics, coding, quasicrystal coordinates, and arithmetic of algebraic radices
Failure boundary: mixed bases are used by position, β is actually an integer, an arbitrary polynomial expression is called a numeral, or uniqueness is assumed from decimal intuition

What It Is Not¶

It is not the whole field of number representation. The field contains many questions and methods that do not instantiate Non-integer base of numeration.
It is not its most familiar example. For the golden ratio φ, powers satisfy φ²=φ+1, creating finite expansions and multiple strings for some values. exhibits the structure, but the example is evidence for the abstraction rather than its definition.
It is not the neighboring catalog concept Factorial number system. The factorial system uses a different place-value weight at every position; a β-expansion uses powers of one fixed, possibly non-integer radix.
It is not a claim that every boundary case has one uncontested classification. a qualified variant may preserve the core while changing notation, parameterization, or implementation, so the constitutive condition must decide the boundary
It is not an unrestricted metaphor for any process that seems similar. Outside number representation, the vocabulary and validity conditions do not transfer literally.

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.^[2]

Definition and recognition. Determine whether a proposed instance satisfies the constitutive conditions rather than merely sharing terminology.
Construction or evolution. Track how radix, digit set, sign and complex convention, expansion direction, greedy or alternative digit-selection map, admissibility, convergence, and equivalence of strings are converted, constrained, or organized by 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..
Comparison. Compare instances using carrier, defining parameters, convention, scale, scope, evidence, limiting cases, and implementation, without treating convenience measures as the definition.
Boundary analysis. Diagnose cases where a qualified variant may preserve the core while changing notation, parameterization, or implementation, so the constitutive condition must decide the boundary and state which convention or theorem controls the decision.
Downstream reasoning. Use the established identity to support studying beta shifts and symbolic dynamics, coding, quasicrystal coordinates, and arithmetic of algebraic radices while preserving the assumptions under which the inference is valid.

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. The disciplined statement is: given radix, digit set, sign and complex convention, expansion direction, greedy or alternative digit-selection map, admissibility, convergence, and equivalence of strings, the structure counts as Non-integer base of numeration exactly when digits from the declared alphabet and powers of one fixed non-integer radix converge to the represented number under a stated selection convention.

This format also separates identity from measurement. Empirical, computational, or documentary proxies support recognition only under declared validity and uncertainty assumptions; formal cases require proof rather than measurement. Measurements can be noisy, implementations can approximate, and proofs can use equivalent characterizations; none of those facts licenses changing the object being measured. When reports disagree, first check scope and convention, then data or proof, and only then interpret the disagreement as substantive.

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.

The compression has a price. A single label can hide standard, generalized, restricted, approximate, computational, and historically variant formulations of Non-integer base of numeration. Good use therefore carries a small declaration of assumptions alongside the name. The abstraction manages complexity when it reduces the state space of the question while keeping the failure boundary visible; it mismanages complexity when the label substitutes for that boundary analysis.

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.
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. This step prevents the canonical example from becoming the definition.
Derive consequences. From digits from the declared alphabet and powers of one fixed non-integer radix converge to the represented number under a stated selection convention, infer studying beta shifts and symbolic dynamics, coding, quasicrystal coordinates, and arithmetic of algebraic radices. Record each assumption used so that a later change of setting does not silently preserve an invalid conclusion.
Test adversarial cases. Examine a qualified variant may preserve the core while changing notation, parameterization, or implementation, so the constitutive condition must decide the boundary and a mixed-radix time notation using bases 24 and 60 is not a single non-integer-base system. A robust identity explains why the first is convention-sensitive and why the second is outside the class.
Compare and refine. Use carrier, defining parameters, convention, scale, scope, evidence, limiting cases, and implementation to compare legitimate instances, and refine the model when discrepancies reflect hidden variation rather than failure of the abstraction itself.

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. For example, the distinction between constitutive identity and a convenient observable transfers from For the golden ratio φ, powers satisfy φ²=φ+1, creating finite expansions and multiple strings for some values. to Greedy β-expansion iterates x↦βx−floor(βx) on the unit interval and records the removed digits..^[3]

Transfer outside the home domain is weaker. The skeletal pattern—type a carrier, apply a constitutive relation, preserve its invariant, and derive only qualified consequences—may suggest an analogy, but the domain-specific mechanisms, admissible evidence, and consequences do not come along automatically. The safe transfer procedure maps each role explicitly, checks the invariant again, and refuses the name when only a superficial resemblance remains.

Examples¶

Canonical¶

For the golden ratio φ, powers satisfy φ²=φ+1, creating finite expansions and multiple strings for some values. The algebraic radix relation permits a block of digits to be rewritten without changing the evaluated sum. This example is canonical because every role can be inspected: the carrier is a radix β with magnitude greater than one, a digit alphabet, positional indices, and finite or infinite digit strings; the operative rule is 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 invariant is digits from the declared alphabet and powers of one fixed non-integer radix converge to the represented number under a stated selection convention; and the result supports studying beta shifts and symbolic dynamics, coding, quasicrystal coordinates, and arithmetic of algebraic radices.^[1] Changing incidental notation or scale leaves the structure intact, while removing digits from the declared alphabet and powers of one fixed non-integer radix converge to the represented number under a stated selection convention destroys the classification.

Mapped back: 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. → digits from the declared alphabet and powers of one fixed non-integer radix converge to the represented number under a stated selection convention → studying beta shifts and symbolic dynamics, coding, quasicrystal coordinates, and arithmetic of algebraic radices

Applied / In Practice¶

Greedy β-expansion iterates x↦βx−floor(βx) on the unit interval and records the removed digits. The dynamical orbit encodes admissibility and frequency properties of the resulting digit sequence. The applied case is not licensed merely by vocabulary. It qualifies because the same recognition test—declare β and digits, verify convergence and each digit-selection step, identify finite versus infinite representation, and test alternate strings before claiming uniqueness—can be run and because the same failure boundary—mixed bases are used by position, β is actually an integer, an arbitrary polynomial expression is called a numeral, or uniqueness is assumed from decimal intuition—remains meaningful.^[2] The case also shows why practical outputs should report assumptions, resolution, and uncertainty instead of a naked label.

Mapped back: declared instance → recognition test → boundary check → qualified use

Structural Tensions¶

T1: Axiomatic identity vs. operational recognition. The defining conditions may be exact while empirical or computational recognition is approximate. Neither pole can be removed without changing the analytical task. Diagnostic: Can the reviewer state both the exact condition and the evidence used to infer it?
T2: Local roles vs. global consequence. The mechanism is enacted through local relations, but the abstraction is usually valued for a global classification or prediction. Neither pole can be removed without changing the analytical task. Diagnostic: Does the claimed global result actually follow from the declared local conditions?
T3: Ideal form vs. finite representation. Theory states a clean invariant while data structures, measurements, or proofs expose only finite representations. Neither pole can be removed without changing the analytical task. Diagnostic: Would increasing resolution converge toward the same classification?
T4: Canonical convention vs. legitimate variants. A standard formulation supports communication, while variants may preserve the same core under changed assumptions. Neither pole can be removed without changing the analytical task. Diagnostic: Which role is invariant across variants, and which convention-specific conclusion changes?
T5: Compression vs. hidden assumptions. The name compresses a complex argument but can conceal prerequisites. Neither pole can be removed without changing the analytical task. Diagnostic: Can each downstream inference be traced to an explicit assumption?
T6: Autonomous residual vs. reduction to catalog neighbors. The candidate uses broader structures but adds an identity-bearing residual. Neither pole can be removed without changing the analytical task. Diagnostic: After subtracting the proposed parent and named neighbors, does the constitutive residual still support independent diagnostics?

Structural–Framed Character¶

The entry is structurally mixed but domain-framed. Its portable skeleton is type a carrier, apply a constitutive relation, preserve its invariant, and derive only qualified consequences. Its identity-bearing terms—Non-integer base of numeration, carrier, parameter, relation, invariant, boundary, evidence, and application—derive their meaning from number representation and cannot be replaced by generic systems language without losing the tests that distinguish valid from invalid instances.

This mixed character explains why the abstraction is reusable inside the domain yet does not meet the Prime bar. The structure organizes reasoning, but its claims still depend on domain-specific objects, evidence, and intervention semantics.

Structural Core vs. Domain Accent¶

The structural core consists of a carrier, 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., a recognition invariant, and a consequence. That skeleton may resemble patterns elsewhere, especially type a carrier, apply a constitutive relation, preserve its invariant, and derive only qualified consequences. The domain accent is not decorative: Non-integer base of numeration, carrier, parameter, relation, invariant, boundary, evidence, and application determine what counts as an admissible carrier, a valid transition, and successful evidence.

The abstraction therefore remains domain-specific. A cross-domain reuse that preserves only words such as 'balance,' 'cut,' 'sequence,' 'loss,' or 'simulation' is metaphor. Literal transfer requires the original role structure and diagnostics, which in this case remain anchored in number representation.

The proposed strict upward parent is prime:encoding_and_decoding. The numeral system literally encodes values as digit strings and decodes by weighted summation; non-integer radix dynamics supply the residual. This is a proposal-only workspace relationship: the accepted Prime supplies a genuinely instantiated structural prerequisite or superclass, while Non-integer base of numeration adds domain-specific constraints.

The entry does not collapse into that parent because positional place value with a non-integer radix and radix-dependent admissibility dynamics It also declines a nearby thematic catalog node: the neighbor does not literally subsume the constitutive identity of Non-integer base of numeration. This explicit assert-and-decline pattern keeps the proposed DAG narrow and prevents a merely thematic edge.

The prospective workspace queue contains one strict upward edge to prime:encoding_and_decoding. No live DAG mutation is authorized.

Relationships to Other Abstractions¶

Current abstraction Non-integer base of numeration Domain-specific

Parents (1) — more general patterns this builds on

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 (56^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Numeral Bases & Arithmetic Functions (8 abstractions)

Nearest neighbors

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

Not to Be Confused With¶

Mixed-radix numeral system. Varies integer place bases.
Complex-base system. May overlap when the radix is complex, but non-integer real bases are a major separate case.
Factorial number system. Uses factorial weights.
Continued fraction. A nested quotient representation, not positional powers.
Canonical number system. Imposes algebraic digit and finiteness properties beyond non-integer radix alone.

References¶

[1] Alfréd Rényi, ‘Representations for Real Numbers and Their Ergodic Properties,’ Acta Mathematica Academiae Scientiarum Hungaricae 8, 477–493 (1957), DOI 10.1007/BF02020331. registry ↩a ↩b

[2] William Parry, ‘On the β-Expansions of Real Numbers,’ Acta Mathematica Academiae Scientiarum Hungaricae 11, 401–416 (1960), DOI 10.1007/BF02020954. registry ↩a ↩b

[3] Christiane Frougny and Boris Solomyak, ‘Finite Beta-Expansions,’ Ergodic Theory and Dynamical Systems 12(4), 713–723 (1992), DOI 10.1017/S0143385700007090. registry ↩