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Research draft

complex number

vr.tr.complex-number · XCT.QLT

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

Thing Registry Cross-cutting context

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.

written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify

Researched by: Claude

Purpose and description

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

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.

It can be 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

What it looks like

Expressions a + bi or r(cos t + i sin t); points on the complex plane.

How it is recognised

Imaginary unit i (or j in engineering)

Argand diagrams

Phasors for AC quantities

Related models

is a kind of - category

number

extends - field

real numbers

is generalised by - extension

quaternion

is used in - applications

electrical engineering and quantum mechanics

In practice

Families and kinds

rectangular form

polar and exponential form

phasors

Gaussian integers (subset)

extensions such as quaternions

Standards and regulation

ISO 80000-2 notation

Failure modes and hazards

Branch cut errors with roots and logarithms

Confusing i and j conventions

Also called

decimal fractionelectric flux density phasorelectric field strength phasormagnetic flux density phasormagnetic field strength phasorelectric current phasorcomplex frequencyconic parameterMadelung constantcomplex unit circleconic constantHermite constantChvátal–Sankoff constantFavard constantFeigenbaum constantsSeshadri constantspace vector representationcomplex unitrotating phasorreal numberdefinable complex numberconstant of integrationKleinian integercomplex logarithmEisenstein integerComplex random vectorLehmer numberprobability amplitudeGaussian rationalquadratic integerphasorimaginary numberharmonic numberChebyshev nodeAiry function zerosrational numberconstructible numbernon-negative real numbernonzero real numberopen interval from −π/2 to +π/2

+32

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Drafted structure

Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.

Form Representation.

Forms convert.

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.

Operations follow rules.

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.

Complex numbers are practical.

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.

Intuition helps.

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

What the second pass must settle

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