{ "model": { "rank": 1306, "code": "thing-q11567", "model_id": "vr.tr.complex-number", "name": "complex number", "purpose": "Let an agent handle complex numbers by form, operations, representation and applications.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q11567/", "tier": 2, "score": 157, "payload": { "layer": "wikidata", "aliases": [ "decimal fraction", "electric flux density phasor", "electric field strength phasor", "magnetic flux density phasor", "magnetic field strength phasor", "electric current phasor", "complex frequency", "conic parameter", "Madelung constant", "complex unit circle", "conic constant", "Hermite constant", "Chvátal–Sankoff constant", "Favard constant", "Feigenbaum constants", "Seshadri constant", "space vector representation", "complex unit", "rotating phasor", "real number", "definable complex number", "constant of integration", "Kleinian integer", "complex logarithm", "Eisenstein integer", "Complex random vector", "Lehmer number", "probability amplitude", "Gaussian rational", "quadratic integer", "phasor", "imaginary number", "harmonic number", "Chebyshev node", "Airy function zeros", "rational number", "constructible number", "non-negative real number", "nonzero real number", "open interval from −π/2 to +π/2" ], "aliasCount": 72, "merged": 72, "knownIn": 157, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q11567", "url": "https://www.wikidata.org/wiki/Q11567", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T07:52:46Z", "providers": [ "Claude" ], "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify", "pass": 2, "wave": 1, "engine": "claude" }, "metaModel": { "id": "THING-Q11567", "registryId": "vr.tr.complex-number", "name": "complex number", "version": "0.2.0-wave.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q11567/", "model": { "registry_id": "vr.tr.complex-number", "name": "complex number", "purpose": "Let an agent handle complex numbers by form, operations, representation and applications.", "definition": "A number of the form a + bi, where a and b are real numbers and i is the imaginary unit with i squared equal to -1; complex numbers extend the reals, and phasors in electrical engineering use them.", "what_it_is_for": "Solving polynomial equations, analysing waves and circuits, and complex analysis.", "affordances": [ "add, multiply and divide them", "convert between rectangular and polar forms", "use phasors in AC circuits", "plot on the complex plane" ], "distinguishing_features": [ "Real and imaginary parts", "Algebraically closed field", "Polar form with modulus and argument", "No natural ordering" ], "appearance": "Expressions a + bi or r(cos t + i sin t); points on the complex plane.", "visual_identification": [ "Imaginary unit i (or j in engineering)", "Argand diagrams", "Phasors for AC quantities" ], "physical_properties": [], "families_and_kinds": [ "rectangular form", "polar and exponential form", "phasors", "Gaussian integers (subset)", "extensions such as quaternions" ], "related_models": [ { "relation": "is a kind of", "target": "number", "why": "category" }, { "relation": "extends", "target": "real numbers", "why": "field" }, { "relation": "is generalised by", "target": "quaternion", "why": "extension" }, { "relation": "is used in", "target": "electrical engineering and quantum mechanics", "why": "applications" } ], "identifiers": [], "standards_and_regulation": [ "ISO 80000-2 notation" ], "failure_modes_and_hazards": [ "Branch cut errors with roots and logarithms", "Confusing i and j conventions" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "form", "name": "Form", "description": "Representation.", "rationale": "Forms convert.", "layers": [ { "id": "rectangular", "name": "Rectangular", "description": "a + bi.", "findings": [ { "id": "rectangular-finding", "name": "Rectangular", "description": "Rectangular form.", "questions": [ { "text": "What are its real and imaginary parts?", "kind": "measurement" }, { "text": "Is it written with i or j?", "kind": "definition" } ] } ] }, { "id": "polar", "name": "Polar", "description": "Modulus and argument.", "findings": [ { "id": "polar-finding", "name": "Polar", "description": "Polar form.", "questions": [ { "text": "What are its modulus and argument?", "kind": "measurement" }, { "text": "In which range is the argument?", "kind": "boundary" } ] } ] } ] }, { "id": "operations", "name": "Operations", "description": "Arithmetic.", "rationale": "Operations follow rules.", "layers": [ { "id": "arithmetic", "name": "Arithmetic", "description": "Basic operations.", "findings": [ { "id": "arithmetic-finding", "name": "Arithmetic", "description": "Arithmetic.", "questions": [ { "text": "What is the result of this operation?", "kind": "measurement" }, { "text": "Which form is easiest to use?", "kind": "action" } ] } ] }, { "id": "functions", "name": "Functions", "description": "Roots and logs.", "findings": [ { "id": "functions-finding", "name": "Functions", "description": "Complex functions.", "questions": [ { "text": "Which branch is used for roots or logarithms?", "kind": "boundary" }, { "text": "Is the result unique?", "kind": "boundary" } ] } ] } ] }, { "id": "applications", "name": "Applications", "description": "Uses.", "rationale": "Complex numbers are practical.", "layers": [ { "id": "circuits", "name": "Circuits", "description": "Phasors.", "findings": [ { "id": "circuits-finding", "name": "Circuits", "description": "AC circuits.", "questions": [ { "text": "How are phasors used to analyse AC circuits?", "kind": "definition" }, { "text": "What does impedance represent?", "kind": "definition" } ] } ] }, { "id": "signals", "name": "Signals", "description": "Fourier analysis.", "findings": [ { "id": "signals-finding", "name": "Signals", "description": "Signal analysis.", "questions": [ { "text": "How do complex exponentials describe signals?", "kind": "definition" }, { "text": "Why are they convenient?", "kind": "definition" } ] } ] } ] }, { "id": "learning", "name": "Learning", "description": "Teaching.", "rationale": "Intuition helps.", "layers": [ { "id": "visual", "name": "Visual", "description": "Complex plane.", "findings": [ { "id": "visual-finding", "name": "Visual", "description": "Visualisation.", "questions": [ { "text": "How can multiplication be seen as rotation and scaling?", "kind": "definition" }, { "text": "How is it plotted?", "kind": "action" } ] } ] }, { "id": "misconceptions", "name": "Misconceptions", "description": "Errors.", "findings": [ { "id": "misconceptions-finding", "name": "Misconceptions", "description": "Misconceptions.", "questions": [ { "text": "What mistakes are common?", "kind": "boundary" }, { "text": "How are they avoided?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should forms be separate entries?", "How should phasors be linked?", "Registry aliases include decimal fraction; should it be moved?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "abstract concept", "method": "Written from the archetype playbook - what this kind of thing needs beyond identity and provenance - and from the structure that recurred across 6,333 models already researched by two engines. Applied to this entry by rule. No source was read for this thing and no claim here is researched. This entry carries no facets of its own, so they were inferred from its domain - a guess about a whole domain applied to one thing.", "facetsInferred": true, "nextPass": "A researcher replaces this draft with a sourced specification. Treat every sentence below as a proposal to argue with.", "purpose": "Give an agent a durable, checkable way to recognise a complex number, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent handle complex numbers by form, operations, representation and applications.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: decimal fraction, electric flux density phasor, electric field strength phasor, magnetic flux density phasor, magnetic field strength phasor, electric current phasor, complex frequency, conic parameter, Madelung constant, complex unit circle, conic constant, Hermite constant.", "72 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 157 Wikipedia languages, which is a measure of how widely the thing is known, not of how important it is." ], "openQuestionsForResearch": [ "Which of the bundles below does a real task actually need, and which are ceremony?", "What does this thing have that the facets do not capture at all?", "Which neighbouring kind is most often confused with a complex number, and on what evidence are they told apart?" ], "whatItIsMadeOf": "an abstraction with no physical instance", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "observe it, measure it, record its state" ], "howItIsRecognised": [ "Nothing to see. What is recognised is an instance of it, and which instances count is exactly what is argued about." ], "relatedModels": [ { "relation": "covers", "note": "Finer kinds folded into this entry because telling them apart needs a task that asks for it. Each is a model waiting to be split out when one does.", "targets": [ "decimal fraction", "electric flux density phasor", "electric field strength phasor", "magnetic flux density phasor", "magnetic field strength phasor", "electric current phasor", "complex frequency", "conic parameter", "Madelung constant", "complex unit circle", "conic constant", "Hermite constant" ] } ], "standing": "Described in 157 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 72 finer distinctions are held inside this entry as names rather than as separate models.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one complex number from another, and a complex number from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a complex number, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a complex number, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a complex number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a complex number, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a complex number, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a complex number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a complex number sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a complex number, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a complex number, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a complex number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a complex number from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a complex number, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a complex number, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a complex number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a complex number passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a complex number between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a complex number, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a complex number, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a complex number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a complex number, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a complex number is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a complex number, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a complex number, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a complex number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definitions-in-contest", "name": "Definitions and who holds them", "description": "What complex number is taken to mean, and by whom.", "rationale": "When a field disagrees about a concept, the disagreement is the content. A model that picks one definition silently destroys the information.", "layers": [ { "id": "competing-definitions", "name": "Competing definitions", "description": "The main readings and the traditions behind them.", "findings": [ { "id": "definition-map", "name": "Definitions and their holders", "description": "Each definition with the school or body that holds it.", "questions": [ { "id": "definition-map-q01", "text": "Which definitions of complex number are in use, and which tradition or body holds each?", "kind": "definition" }, { "id": "definition-map-q02", "text": "What turns on the difference between them in practice?", "kind": "boundary" } ] } ] }, { "id": "operationalisation", "name": "Operationalisation", "description": "How it is measured or applied when it has to be.", "findings": [ { "id": "operational-record", "name": "Measures and proxies", "description": "Instruments and indicators used to stand in for it.", "questions": [ { "id": "operational-record-q01", "text": "How is complex number operationalised or measured in practice, and by what instrument?", "kind": "measurement" }, { "id": "operational-record-q02", "text": "What does that operationalisation leave out, and when does that matter?", "kind": "boundary" } ] } ] } ] }, { "id": "instances-and-use", "name": "Instances, use and consequence", "description": "What counts as an instance of complex number and what follows from calling something that.", "rationale": "Applying a concept is an act with consequences, so a model must say what the label licenses and what it does not.", "layers": [ { "id": "instances", "name": "What counts as an instance", "description": "Clear cases, borderline cases and non-cases.", "findings": [ { "id": "instance-tests", "name": "Tests for an instance", "description": "What would settle whether something falls under it.", "questions": [ { "id": "instance-tests-q01", "text": "What would settle whether something is an instance of complex number?", "kind": "boundary" }, { "id": "instance-tests-q02", "text": "Which borderline cases are argued about, and on what grounds?", "kind": "definition" } ] } ] }, { "id": "consequence", "name": "Consequence of application", "description": "Rights, duties or decisions that follow from the label.", "findings": [ { "id": "consequence-record", "name": "What the label licenses", "description": "What an agent may do once something is classified this way.", "questions": [ { "id": "consequence-record-q01", "text": "What follows practically once something is treated as complex number?", "kind": "action" }, { "id": "consequence-record-q02", "text": "What must an agent not infer from the label alone?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a complex number came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a complex number and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a complex number, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a complex number, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a complex number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a complex number is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a complex number, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a complex number, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a complex number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 5, "layers": 10, "findings": 12, "questions": 32 } } } }