numeral system
Enable an AI agent to recognise a numeral system, assess whether a representation is valid and interpretable under its rules, and determine which reading, writing, comparison or conversion actions are justified.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
Researched by: Codex + Grok
Purpose and description
Enable an AI agent to recognise a numeral system, assess whether a representation is valid and interpretable under its rules, and determine which reading, writing, comparison or conversion actions are justified.
A numeral system is a conventional mapping from written or spoken numeral tokens to abstract numbers, together with the composition rules that determine how those tokens combine to denote a value.
It can be Identify candidate numeral systems for a representation and request context when identification is ambiguous.; Validate a numeral against a selected convention profile and explain rule violations.; Interpret a numeral or generate a representation for a supported number.; Compare numerical values while preserving distinctions between written forms.; Normalise or convert a numeral when equivalence is established, recording any approximation or lost representational information..
Distinguishing features
It supplies conventions connecting signs and arrangements to numbers; a number itself is independent of those conventions.
It governs numerical representation; a font or character encoding alone does not determine numerical value.
Its scope is expressing numbers; a mathematical notation may additionally express operations, relations and propositions.
A string using numeral signs may instead be an identifier: preserving leading zeros or forbidding arithmetic can indicate identifier use rather than a numerical interpretation.
A shared digit inventory does not establish a shared system: positional weights, grouping rules or value assignments must also agree.
Scope
+ Numeral signs, their numerical roles and accepted variants
+ Composition rules, including positional, additive, subtractive or combined mechanisms
+ Interpretation of numerals and the range of numbers expressible
+ Validity, ambiguity and preferred representation conventions
+ Conditions for reading, generating, normalising and converting numerals
- Numbers and number structures independently of their representation
- General writing systems, typography and character encoding standards
- Natural-language number names and grammatical number
- Measurement units, currencies and quantity semantics
- Arithmetic algorithms and calculating devices beyond their numeral interpretation requirements
Characteristics
- Representation mechanism
- Positional, additive, subtractive, multiplicative-additive, other documented mechanism, or combination Determines how component signs contribute to the represented number.
- Radix or position-weight rule
- Fixed radix with specified value; mixed radix with specified sequence; other weight rule; not applicable; unresolved Prevents an agent from assuming decimal interpretation or uniform position weights.
- Sign inventory
- Links to recognised signs, value assignments, modifiers and accepted visual variants Supports recognition while distinguishing numerical roles from appearance.
- Representable number domain
- Explicitly described number classes, bounds and exclusions Establishes whether a requested number can be expressed under the selected conventions.
- Zero convention
- Zero numeral, positional placeholder, both, neither, or context-dependent Distinguishes representing zero from preserving an empty position.
- Interpretation determinacy
- Unique under selected conventions; context-dependent; ambiguous; unresolved Determines whether automatic reading is justified.
- Representation preference
- Unique normal form; multiple preferred forms; accepted forms without preference; unresolved Controls whether rewriting an equivalent numeral is warranted.
- Applicable convention profile
- Links to documented historical, regional, disciplinary or implementation conventions Keeps an agent from combining incompatible rules.
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 6 bundles · 11 layers · 11 findings · 21 questions.
System identity and variants Establishes which numerical representation conventions the agent is applying.
The same signs can receive different interpretations across systems and convention profiles.
Identifying conventions
Records the features that distinguish this system from plausible alternatives.
System recognition
Records diagnostic combinations of signs, composition rules and contextual evidence.
- Which combination of signs and composition rules identifies this numeral system rather than a look-alike? definition
- What additional context is required when a numeral is valid in several systems? boundary
Convention profiles
Separates documented variants without assuming they are independent registry things.
Profile applicability
Records where particular numerical conventions apply and the evidence supporting that selection.
- Which sources establish the rules of each historical, regional or specialised profile? provenance
- Which differences change numerical interpretation, and which merely change the written appearance? boundary
Signs and composition Captures the elements of numerals and the rules for combining them.
Recognising individual symbols is insufficient to determine whether their arrangement forms a numeral.
Numerical signs
Distinguishes value-bearing signs, placeholders, modifiers and separators.
Sign roles
Records each recognised sign's contribution and its dependence on surrounding signs.
- Which signs carry values, mark empty positions, modify other signs or separate numeral components? definition
- Which visually similar or alternate signs may be treated as equivalent under this profile? boundary
Assembly rules
Records how order, repetition, grouping and spatial arrangement construct a numeral.
Numeral formation
Makes permitted combinations and structurally meaningful layout explicit.
- What ordering, repetition, grouping and spatial rules determine whether signs form an admissible numeral? definition
- Which changes to spacing, orientation or layout preserve a numeral, and which change or destroy its interpretation? boundary
Value and expressibility Connects admissible numerals to numbers and establishes representation limits.
An agent must know both how a numeral obtains its value and whether a requested value is expressible.
Evaluation rules
Records how signs, positions and modifiers jointly determine numerical value.
Value assignment
Specifies the applicable interpretation mechanism without assuming a positional system.
- How do sign values combine through positional weights, addition, subtraction, multiplication or other documented rules? definition
- If position contributes value, what radix or weight sequence applies and how is the reference position established? definition
Representation domain
Records which numbers can be expressed and which extensions are required.
Expressibility limits
Distinguishes core capabilities from optional conventions for zero, signs, fractions and extended forms.
- Can the selected profile express zero, negative numbers and non-integers, and which rules or extensions enable each? boundary
- Which values have exact finite representations, and what bounds or non-finite representation conventions apply? boundary
Validity and equivalence Supports judgments about admissibility, ambiguity and alternative forms of the same number.
Readable, valid and preferred numerals need not be the same set.
Interpretation status
Separates rule compliance from an agent's ability to infer an intended value.
Validity and ambiguity
Records how to assess malformed, incomplete or multiply interpretable representations.
- Which rule violations make a numeral invalid even when its intended value appears recoverable? boundary
- When a representation is incomplete or admits multiple readings, what evidence must an agent obtain before assigning a value? action
Equivalent forms
Records value equivalence and any documented preference among alternate numerals.
Normalisation permission
Determines when a numeral can be rewritten while preserving its value and relevant source information.
- Which distinct admissible numerals represent the same number, and is a preferred form documented? definition
- When may an agent normalise leading placeholders, repeated signs or alternate constructions, and what original-form information must it retain? action
Reading, writing and conversion Defines conditions for applying the system to concrete representations.
Successful numerical interpretation does not by itself justify rewriting a source or converting it to another system.
Contextual use
Determines whether a sign sequence is being used numerically and whether an output is suitable for its destination.
Use-sensitive actions
Records prerequisites for reading and generating numerals in their actual context.
- What evidence distinguishes a numeral expressing a number from the same signs used as a code, label or identifier? boundary
- Before generating a numeral, which convention profile, representable domain and required layout must the agent establish? action
Cross-system conversion
Records whether another numeral system can preserve the interpreted value and relevant representation properties.
Conversion fidelity
Distinguishes exact value conversion from approximation and loss of written-form information.
- Can the target system express the interpreted number exactly under its permitted representation conventions? boundary
- If conversion requires approximation or discards meaningful source-form information, what authorisation and accompanying record are required? action
Evidence and external alignment What the world already says about this thing, gathered so the model can be checked against it.
A model that cannot be lined up against existing standards, identifiers and practice cannot be adopted by anyone who already uses them.
Reported evidence
Findings from the breadth pass, kept separate from the structural claims.
Kinds and varieties
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- positional (place-value) systems
- additive (sign-value) systems
- hybrid / multiplicative-additive systems
- unary / tally systems
- bijective numeration
- signed-digit / balanced systems
- factoradic and mixed-radix systems
- spoken / linguistic numeral systems
- Which of these kinds and varieties hold for the sense of numeral system this model covers, and on what evidence? provenance
Sources
- Numeral system - Core definition distinguishing numeral systems from numbers, and the main practical kinds (positional, additive, hybrid, unary).
What the second pass must settle
- With no registry definition recorded, does this entry include spoken, signed and tactile numeral systems, or only written numerical notation?
- Which authoritative sources should establish the representation mechanisms and convention profiles covered by the model?
- Where should the boundary fall between a numeral system's core rules and extensions for fractions, exponents, repeating forms or special values?
- How should the model distinguish a variant of one numeral system from a separate system while respecting registry granularity?
- Which documented communities of use recognise preferred forms, tolerate irregular numerals or require preservation of original notation?