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Canonical static concept record

Complex numbers $\mathbb C$

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Summary

The algebraically closed topological field obtained from $\mathbb R$ by adjoining a square root $i$ of $-1$ .

Record metadata

Canonical name: Complex numbers $\mathbb C$
Concept ID: complex_numbers
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Analysis
Domains: Analysis, Algebra, Topology, Number theory
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Number system

Carrier(s)

set $\mathbb C$

Data

field operations
Euclidean topology
complex conjugation

Axioms / constraints

$i^2=-1$

Canonically induces

two-dimensional real vector-space structure

Notes

$\mathbb C$ cannot be ordered as an ordered field. The pair and polynomial-quotient constructions are equivalent presentations, not simultaneous literal definitions.

Concept sources

Wikipedia — Complex number — Complex number · encyclopedic reference · source ID wp-complex-number
nLab — complex number — complex number · mathematical reference · source ID nlab-complex-number

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Ordered-pair complex field Complex numbers $\mathbb C$

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: complex_pairs_to_complex_numbers
Relation type: Representation / equivalence representation-equivalence
Direction: source → target
Endpoint roles: source: Representation / equivalence with; target: Representation / equivalence with
Authored annotation: ordered-pair realization

Authored explanation

Identify $(a,b)$ with $a+bi$ to obtain the standard complex field.

How to interpret this relation type

Reinterpret an object, pass to an equivalent presentation, or relate canonically corresponding structures; the carrier may change.

Relation sources

Wikipedia — Complex number — Complex number · encyclopedic reference · source ID wp-complex-number
nLab — complex number — complex number · mathematical reference · source ID nlab-complex-number

Polynomial-quotient complex field Complex numbers $\mathbb C$

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: complex_quotient_to_complex_numbers
Relation type: Representation / equivalence representation-equivalence
Direction: source → target
Endpoint roles: source: Representation / equivalence with; target: Representation / equivalence with
Authored annotation: polynomial-quotient realization

Authored explanation

Identify the residue class of $x$ with $i$ ; the quotient is isomorphic to the ordered-pair complex field.

How to interpret this relation type

Reinterpret an object, pass to an equivalent presentation, or relate canonically corresponding structures; the carrier may change.

Relation sources

Wikipedia — Complex number — Complex number · encyclopedic reference · source ID wp-complex-number
nLab — complex number — complex number · mathematical reference · source ID nlab-complex-number

Real numbers $\mathbb R$ Complex numbers $\mathbb C$

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: real_numbers_into_complex_numbers
Relation type: Canonical embedding canonical-embedding
Direction: source → target
Endpoint roles: source: Embeds canonically into; target: Contains a canonical copy of
Authored annotation: $a\mapsto a+0i$

Authored explanation

The map $a\mapsto(a,0)$ embeds $\mathbb R$ canonically as the real axis in $\mathbb C$ .

How to interpret this relation type

Map one structure injectively into another by the standard structure-preserving inclusion or representation. This records a canonical copy, not necessarily literal set containment.

Relation sources

Wikipedia — Complex number — Complex number · encyclopedic reference · source ID wp-complex-number
nLab — complex number — complex number · mathematical reference · source ID nlab-complex-number

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.