Early undergraduateNumberPlanned

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

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Builds on

Relations, equivalence relations and partitions Orders, lattices and well-orders Real numbers and irrationality Algebraic structures: monoids, groups and rings

Required prerequisites, all the way down: 17

You will learn to

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.

Definition

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.

Theorem

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.

Common misconceptions

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.

Optional discoveries

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