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

Real numbers $\mathbb R$

Open this concept in the interactive graph Read the Markdown equivalent

Summary

The complete Archimedean ordered field, unique up to unique order-preserving field isomorphism.

Record metadata

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

Carrier(s)

set $\mathbb R$

Data

field operations
total order
order topology and metric

Axioms / constraints

ordered-field laws
least-upper-bound property

Canonically induces

real line and continuum

Notes

Dedekind cuts and Cauchy sequences give different set-theoretic realizations of the same abstract ordered field.

Concept sources

Wikipedia — Construction of the real numbers — Construction of the real numbers · encyclopedic reference · source ID wp-construction-reals
nLab — real number — real number · mathematical reference · source ID nlab-real-number

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Dedekind-cut real line Real numbers $\mathbb R$

Permalink to relation

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

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

Authored explanation

The Dedekind-cut field is canonically order-isomorphic to the abstract complete Archimedean ordered field $\mathbb R$ .

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 — Construction of the real numbers — Construction of the real numbers · encyclopedic reference · source ID wp-construction-reals
nLab — real number — real number · mathematical reference · source ID nlab-real-number

Cauchy sequences of rationals Real numbers $\mathbb R$

Permalink to relation

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

Relation ID: cauchy_sequences_to_reals
Relation type: Quotient construction quotient-construction
Direction: source → target
Endpoint roles: source: Quotients to; target: Obtained as quotient of
Authored annotation: identify null differences

Authored explanation

Quotient Cauchy sequences by equality of limiting difference zero to obtain a complete ordered field representing $\mathbb R$ .

How to interpret this relation type

Identify elements by a stated equivalence relation and equip the quotient with the induced structure. The edge detail must state the representatives and equivalence relation.

Relation sources

Wikipedia — Construction of the real numbers — Construction of the real numbers · encyclopedic reference · source ID wp-construction-reals
nLab — real number — real number · mathematical reference · source ID nlab-real-number

Rational numbers $\mathbb Q$ Real numbers $\mathbb R$

Permalink to relation

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

Relation ID: rational_numbers_into_reals
Relation type: Canonical embedding canonical-embedding
Direction: source → target
Endpoint roles: source: Embeds canonically into; target: Contains a canonical copy of
Authored annotation: constant cut / constant sequence

Authored explanation

Each rational has a canonical image in either completion, giving the standard embedding $\mathbb Q\hookrightarrow\mathbb R$ .

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 — Construction of the real numbers — Construction of the real numbers · encyclopedic reference · source ID wp-construction-reals
nLab — real number — real number · mathematical reference · source ID nlab-real-number

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

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

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

Real numbers $\mathbb R$ Polynomial-quotient 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_quotient
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: adjoin a root of $x^2+1$

Authored explanation

Form $\mathbb R[x]/(x^2+1)$ , in which the class of $x$ squares to $-1$ .

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

Real numbers $\mathbb R$ Cardinality of the continuum $\mathfrak c$

Permalink to relation

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

Relation ID: real_numbers_to_continuum
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: cardinality $\mathfrak c$

Authored explanation

The real line has cardinality $2^{\aleph_0}$ , denoted $\mathfrak c$ .

How to interpret this relation type

Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.

Relation sources

Wikipedia — Cardinality of the continuum — Cardinality of the continuum · encyclopedic reference · source ID wp-cardinality-continuum
Wikipedia — Construction of the real numbers — Construction of the real numbers · encyclopedic reference · source ID wp-construction-reals

Real numbers $\mathbb R$ Ordered field

Permalink to relation

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

Relation ID: real_numbers_to_ordered_field
Relation type: Induced / forgotten induced-forgotten
Direction: source → target
Endpoint roles: source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation: forget completeness

Authored explanation

Forgetting completeness and the distinguished real realization leaves an ordered field.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

nLab — real number — real number · mathematical reference · source ID nlab-real-number

Real numbers $\mathbb R$ Real closed field

Permalink to relation

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

Relation ID: real_numbers_to_real_closed_field
Relation type: Induced / forgotten induced-forgotten
Direction: source → target
Endpoint roles: source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation: forget completeness and topology

Authored explanation

The real numbers are real closed; forgetting their completeness and topology retains that algebraic ordered-field property.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

nLab — real number — real number · mathematical reference · source ID nlab-real-number

Real numbers $\mathbb R$ Topological field

Permalink to relation

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

Relation ID: real_numbers_to_topological_field
Relation type: Induced / forgotten induced-forgotten
Direction: source → target
Endpoint roles: source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation: forget order completeness

Authored explanation

The usual topology and field operations make $\mathbb R$ a topological field.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

nLab — real number — real number · mathematical reference · source ID nlab-real-number