{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.complex-number","code":"thing-q11567","url":"https://ver.cy/models/thing/q11567/","name":"complex number","alternateNames":["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"],"kind":"thing","status":"published","version":"0.2.0-wave.1","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QLT"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"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.","purpose":"Let an agent handle complex numbers by form, operations, representation and applications.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Real and imaginary parts","Algebraically closed field","Polar form with modulus and argument","No natural ordering"],"structure":{"bundles":[{"id":"form","name":"Form","description":"Representation.","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.","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.","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.","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"}]}]}]}]},"agentConduct":{"may":["Compute with complex numbers, stating the conventions used for branch cuts and the imaginary unit.","Explain complex arithmetic and its use in engineering and physics."],"mustNot":["Present a result that depends on a branch choice without saying which branch was used.","Mix the i and j conventions in one calculation without noting it.","Report a computed value with more precision than the method supports.","Call complex numbers unreal or meaningless in a way that misleads learners."],"requiresHuman":["Relying on complex-valued calculations for safety-critical design, such as power systems."]},"ethics":{"considerations":["Calculation errors in engineering models can lead to unsafe systems.","Mathematics teaching should be honest about where intuition fails."],"affectedParties":["Engineers and the users of what they build","Students"]},"owners":{"steward":"Nobody: a mathematical concept held in common.","roles":[],"masterSystems":[]},"relations":[{"target":"Q11563","type":"parent","note":"registry parent class"},{"target":"Q173853","type":"parent","note":"registry parent class"},{"target":"Q191290","type":"parent","note":"registry parent class"},{"target":"Q837414","type":"parent","note":"registry parent class"},{"target":"number","type":"related","note":"category"},{"target":"real numbers","type":"extends","note":"field"},{"target":"quaternion","type":"related","note":"extension"},{"target":"electrical engineering and quantum mechanics","type":"related","note":"applications"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.complex-number","Wikidata: Q11567 (https://www.wikidata.org/wiki/Q11567)"]},"properties":{"applicability":"not-applicable","items":[]},"recognition":{"applicability":"optional","items":["Imaginary unit i (or j in engineering)","Argand diagrams","Phasors for AC quantities","Expressions a + bi or r(cos t + i sin t); points on the complex plane."]},"capabilities":{"applicability":"required","items":["add, multiply and divide them","convert between rectangular and polar forms","use phasors in AC circuits","plot on the complex plane"]},"hazards":{"applicability":"required","items":["Branch cut errors with roots and logarithms","Confusing i and j conventions"]},"interfaces":{"applicability":"required","items":["ISO 80000-2 notation"]},"context":{"applicability":"required","items":["Solving polynomial equations, analysing waves and circuits, and complex analysis.","rectangular form","polar and exponential form","phasors","Gaussian integers (subset)","extensions such as quaternions"]}},"sources":[{"title":"Wikidata item Q11567: complex number","url":"https://www.wikidata.org/wiki/Q11567","note":"identity and sense of the item"},{"title":"Wikipedia: Complex number","url":"https://en.wikipedia.org/wiki/Complex_number","note":"general description of the item"}],"openQuestions":["Should forms be separate entries?","How should phasors be linked?","Registry aliases include decimal fraction; should it be moved?"],"resources":{"spec":"/models/things/publications/thing-q11567/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q11567/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-11T07:52:46Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-06T19:51:53+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties."},"score":1.0}}