The rationals as equivalence classes of pairs of integers

Depends on

• The integers as equivalence classes of pairs of naturals
• The integers form a commutative ring

Used by

• A rational root of xᵏ = m is an integer: if k ≥ 1, m ∈ ℤ, x ∈ ℚ and xᵏ is the image of m, then x is the image of an integer Corollary
• {q ∈ ℚ : q ≥ 0, q² < 2} is closed and bounded in ℚ and is not compact Counterexample
• On a closed interval of ℚ there is a continuous unbounded function, a bounded one with no maximum, and one without the intermediate value property Counterexample
• Over ℚ there is a nonconstant differentiable function with identically zero derivative, so Rolle and the mean value theorem both fail Counterexample
• ℚ ∩ [0,2] is bounded and disconnected, so being an interval of ℚ is not enough Counterexample
• ℝ/ℚ carries the indiscrete topology, although ℝ is metrizable and the quotient has more than one point Counterexample
• The additive rationals do not decompose as a product of finite cyclic prime-power groups Counterexample
• The truncated decimal approximations of √2 form a Cauchy sequence of rationals with no rational limit Counterexample
• Arithmetic on the rationals Definition
• Cauchy sequence of rationals Definition
• Dedekind cut Definition
• Null sequence Definition
• Order on the rationals Definition
• Rational powers aʳ of a positive base Definition
• The real numbers ℝ as Dedekind cuts Definition
• No rational squares to 3 or to 6, and none cubes to 2: three instances of the rational-root corollary Example
• ℚ(√2) carries exactly two distinct field orders, exchanged by the conjugation √2 ↦ -√2 Example
• ℝ as a vector space over ℚ has a basis, and every such basis is infinite; the existence proof exhibits none Example
• The 2-adic absolute value gives an ultrametric on ℚ, in which every triangle is isosceles and every point of a ball is a centre Example
• The completion of ℚ under the usual metric is ℝ Example
• FALSE: a^m/n := (a^1/n)ᵐ extends to negative bases False statement
• FALSE: in every ordered field a closed bounded set is compact, so Heine-Borel needs no completeness False statement
• FALSE: some rational number squares to 2 False statement
• Every rational has a positive-denominator representative Lemma
• Laws of rational exponents Lemma
• Rational arithmetic is well defined Lemma
• Rational powers do not depend on the representative Lemma
• The integers embed in the rationals Lemma
• The p-adic valuation extends to the nonzero rationals by vₚ(a/b) := vₚ(a) - vₚ(b) ∈ ℤ, independently of the representation; it satisfies vₚ(xy) = vₚ(x) + vₚ(y), and vₚ(x+y) ≥ min{vₚ(x), vₚ(y)} whenever x, y and x+y are nonzero Lemma
• The rational-defining relation is an equivalence relation Lemma
• The reciprocal on the rationals is well-defined Lemma
• Gauss lemma: primitive factorisations over ℚ can be cleared to primitive factorisations over ℤ Theorem
• Minkowski's inequality for finite sums (rational exponent) Theorem
• ℚ is countably infinite Theorem
• The number e is irrational Theorem
• Weighted AM-GM inequality with rational weights Theorem
• Young's inequality for products (rational conjugate exponents) Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 24 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

• T. Tao, Analysis I, 3rd ed., §4.2 (standard reference, not scraped)
• E. Landau, Foundations of Analysis (standard reference, not scraped)
• Rational number — formal construction (Wikipedia) (standard reference, not scraped)