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complex number

vr.tr.complex-number · thing-q11567

Let an agent handle complex numbers by form, operations, representation and applications.

Thing Registry Cross-cutting context XCT.QLT
Bundle → Layer → Finding → QuestionsClassifiersWhat it isWhy it existsDistinguishing featuresWhat robots and AI may and may not doMoral aspectsOwnersLinks to other meta-modelsIdentity and identifiersDirect propertiesRecognitionCapabilities and actionsHazards and failure modesStandards and interfacesContext of useSources

Bundle → Layer → Finding → Questions Filled

4 bundles · 8 layers · 8 findings · 16 questions

Form Representation.

Rectangular

a + bi.

Rectangular

Rectangular form.

  1. What are its real and imaginary parts? measurement
  2. Is it written with i or j? definition

Polar

Modulus and argument.

Polar

Polar form.

  1. What are its modulus and argument? measurement
  2. In which range is the argument? boundary
Operations Arithmetic.

Arithmetic

Basic operations.

Arithmetic

Arithmetic.

  1. What is the result of this operation? measurement
  2. Which form is easiest to use? action

Functions

Roots and logs.

Functions

Complex functions.

  1. Which branch is used for roots or logarithms? boundary
  2. Is the result unique? boundary
Applications Uses.

Circuits

Phasors.

Circuits

AC circuits.

  1. How are phasors used to analyse AC circuits? definition
  2. What does impedance represent? definition

Signals

Fourier analysis.

Signals

Signal analysis.

  1. How do complex exponentials describe signals? definition
  2. Why are they convenient? definition
Learning Teaching.

Visual

Complex plane.

Visual

Visualisation.

  1. How can multiplication be seen as rotation and scaling? definition
  2. How is it plotted? action

Misconceptions

Errors.

Misconceptions

Misconceptions.

  1. What mistakes are common? boundary
  2. How are they avoided? action

Classifiers Filled

Family
Thing Registry
Category
Cross-cutting context
Entry kind
thing
Plane
XCT
Domain
XCT.QLT
Other names and narrower kinds
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

What it is Filled

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.

Why it exists Filled

Let an agent handle complex numbers by form, operations, representation and applications.

Distinguishing features Filled

  • Real and imaginary parts
  • Algebraically closed field
  • Polar form with modulus and argument
  • No natural ordering

What robots and AI may and may not do Filled

Must not

  • 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.

Only with a human decision

  • Relying on complex-valued calculations for safety-critical design, such as power systems.

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.

Moral aspects Filled

  • Calculation errors in engineering models can lead to unsafe systems.
  • Mathematics teaching should be honest about where intuition fails.

Who is affected

  • Engineers and the users of what they build
  • Students

Owners Filled

Steward

Nobody: a mathematical concept held in common.

Links to other meta-models Filled

parent

  • Q11563 - registry parent class
  • Q173853 - registry parent class
  • Q191290 - registry parent class
  • Q837414 - registry parent class

related

  • number - category
  • quaternion - extension
  • electrical engineering and quantum mechanics - applications

extends

  • real numbers - field

What else AI and robots need to interact with it Filled

Identity and identifiers required Filled

  • Vercy registry: vr.tr.complex-number
  • Wikidata: Q11567 (https://www.wikidata.org/wiki/Q11567)

Direct properties not applicable Not applicable

Not applicable

Plane XCT: no invented physical properties.

Recognition optional Filled

  • 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 and actions required Filled

  • add, multiply and divide them
  • convert between rectangular and polar forms
  • use phasors in AC circuits
  • plot on the complex plane

Hazards and failure modes required Filled

  • Branch cut errors with roots and logarithms
  • Confusing i and j conventions

Standards and interfaces required Filled

  • ISO 80000-2 notation

Context of use required Filled

  • 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 Filled

  1. Wikidata item Q11567: complex number - identity and sense of the item
  2. Wikipedia: Complex number - general description of the item

Open questions

  • Should forms be separate entries?
  • How should phasors be linked?
  • Registry aliases include decimal fraction; should it be moved?

Machine files

Card as JSON Specification Card backlog vercy.card/1

Provenance

thing registry research (pass 2) · unreviewed

Built from: models/things/publications/thing-q11567/spec.json