Primeros años de universidadNumberPlanificado
Constructing ℤ, ℚ and ℝ
Building the number systems from the natural numbers: ℤ from pairs of naturals, ℚ from pairs of integers, and ℝ from Dedekind cuts or Cauchy sequences, together with the uniqueness of the complete ordered field.
math.number.number-system-constructions
La lección de este tema todavía no se ha escrito. El esquema siguiente forma parte del mapa del plan de estudios y no ha sido revisado.
Se basa en
Relations, equivalence relations and partitionsOrders, lattices and well-ordersReal numbers and irrationalityAlgebraic structures: monoids, groups and rings
Requisitos previos obligatorios, hasta el fondo: 17
Aprenderás a
- Construct ℤ as equivalence classes of pairs of natural numbers and check that the operations are well defined.
- Construct ℚ as the field of fractions of ℤ.
- Construct ℝ by Dedekind cuts or Cauchy sequences and prove the least-upper-bound property.
Definición
- On ℕ × ℕ, (a, b) ~ (c, d) if and only if a + d = b + c; the class of (a, b) represents a − b.
- A Dedekind cut is a non-empty proper subset of ℚ that is closed downwards and has no greatest element.
- A complete ordered field is an ordered field in which every non-empty subset that is bounded above has a supremum.
Teorema
- Operations on equivalence classes are well defined: they do not depend on the chosen representatives.
- Any two complete ordered fields are isomorphic by a unique order-preserving field isomorphism.
- The Dedekind and Cauchy constructions produce isomorphic ordered fields.
Errores frecuentes
- Thinking well-definedness is automatic when operations are defined on equivalence classes.
- Reading the constructions as saying what numbers 'really are', rather than as showing that structures with the required properties exist.
Descubrimientos opcionales
- Real analysis Completeness of ℝ, established here by construction, is the property on which rigorous analysis rests.
- Universal properties The field of fractions has a universal property: every injective ring homomorphism from an integral domain into a field extends uniquely to its field of fractions.
- ← Integers
Lecturas complementarias
- Edmund Landau, Foundations of Analysis