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Canonical static concept record

Rational numbers $\mathbb Q$

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Summary

The ordered field of fractions of $\mathbb Z$ .

Record metadata

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

Carrier(s)

quotient of integer fraction pairs

Data

field operations
usual total order

Axioms / constraints

ordered-field laws

Canonically induces

countable dense linear order without endpoints

Notes

The map $z\mapsto[(z,1)]$ canonically embeds $\mathbb Z$ into $\mathbb Q$ .

Concept sources

Wikipedia — Rational number — Rational number · encyclopedic reference · source ID wp-rational-number
nLab — rational number — rational number · mathematical reference · source ID nlab-rational-number

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Field Rational numbers $\mathbb Q$

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

Relation ID: field_to_rational_numbers
Relation type: Classification classification
Direction: source → target
Endpoint roles: source: Has subtype; target: Classified as
Authored annotation: prime ordered field

Authored explanation

The rational numbers form the characteristic-zero prime field, equipped here with their canonical order.

How to interpret this relation type

The target is a member or subtype of the broader source class.

Relation sources

Wikipedia — Rational number — Rational number · encyclopedic reference · source ID wp-rational-number
nLab — rational number — rational number · mathematical reference · source ID nlab-rational-number

Integers $\mathbb Z$ Rational numbers $\mathbb Q$

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

Relation ID: integers_into_rationals
Relation type: Canonical embedding canonical-embedding
Direction: source → target
Endpoint roles: source: Embeds canonically into; target: Contains a canonical copy of
Authored annotation: $z\mapsto z/1$

Authored explanation

The map $z\mapsto[(z,1)]$ embeds $\mathbb Z$ canonically into its field of fractions.

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 — Rational number — Rational number · encyclopedic reference · source ID wp-rational-number
nLab — rational number — rational number · mathematical reference · source ID nlab-rational-number

Integer fraction pairs Rational numbers $\mathbb Q$

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

Relation ID: rational_pairs_to_rationals
Relation type: Quotient construction quotient-construction
Direction: source → target
Endpoint roles: source: Quotients to; target: Obtained as quotient of
Authored annotation: identify equal fractions

Authored explanation

Quotient by $(a,b)\sim(c,d)$ exactly when $ad=bc$ ; induced operations produce the fraction field $\mathbb Q$ .

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 — Rational number — Rational number · encyclopedic reference · source ID wp-rational-number
nLab — rational number — rational number · mathematical reference · source ID nlab-rational-number

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Rational numbers $\mathbb Q$ Countable infinity $\aleph_0$

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

Relation ID: rational_numbers_to_aleph_zero
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: countably infinite

Authored explanation

$\mathbb Q$ is countably infinite.

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

nLab — rational number — rational number · mathematical reference · source ID nlab-rational-number
Wikipedia — Aleph number — Aleph number · encyclopedic reference · source ID wp-aleph-number

Rational numbers $\mathbb Q$ Dedekind-cut real line

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

Relation ID: rationals_to_dedekind_cuts
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: form Dedekind cuts

Authored explanation

Construct an order-complete field from proper lower cuts of $\mathbb Q$ .

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 — 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 $\mathbb Q$ Ordered field

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

Relation ID: rational_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 the distinguished rational realization

Authored explanation

Forgetting how elements were constructed leaves the usual ordered-field structure.

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

Wikipedia — Rational number — Rational number · encyclopedic reference · source ID wp-rational-number
nLab — rational number — rational number · mathematical reference · source ID nlab-rational-number

Rational numbers $\mathbb Q$ Cauchy sequences of rationals

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

Relation ID: rationals_to_cauchy_sequences
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: form Cauchy sequences

Authored explanation

Take rational Cauchy sequences in the absolute-value metric.

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 — 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 $\mathbb R$

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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