Select any concept, construction junction, or annotated relation by pointer, touch, search, or keyboard.

Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Ordered-pair complex field

Open this concept in the interactive graph Read the Markdown equivalent

Summary

The field on $\mathbb R^2$ with $(a,b)(c,d)=(ac-bd,ad+bc)$ .

Record metadata

Canonical name: Ordered-pair complex field
Concept ID: complex_pair_realization
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Analysis
Domains: Analysis, Algebra, Set theory, Number theory
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Set-theoretic realization

Carrier(s)

set $\mathbb R^2$

Data

componentwise addition
complex multiplication

Canonically induces

element $i=(0,1)$ with $i^2=-1$

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.

Real numbers $\mathbb R$ Ordered-pair complex field

Permalink to relation

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

Relation ID: real_numbers_to_complex_pairs
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: form $\mathbb R^2$ with complex multiplication

Authored explanation

Equip ordered pairs of reals with the standard complex field operations.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

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.