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
• Morse Euler characteristic identity Corollary
• Rings of integers are Dedekind domains 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
• Weighted AM-GM needs nonnegative weights summing to 1 Counterexample
• Algebraic Lefschetz number via rational homology traces Definition
• Arithmetic on the rationals Definition
• Cauchy sequence of rationals Definition
• Convergents of a regular continued fraction Definition
• Dedekind cut Definition
• Euler characteristic of a compact manifold Definition
• Finite and infinite regular continued fractions Definition
• Null sequence Definition
• Order on the rationals Definition
• Quadratic irrationals Definition
• Rational powers aʳ of a positive base Definition
• Simple continued fractions, convergents, and the integer-coordinate coding of ℕ^ℕ Definition
• The geometric representation on the simple-root basis over a common splitting field, and the root set Definition
• The real numbers ℝ as Dedekind cuts Definition
• An explicit enumeration of the rationals, with repetitions and then without them Example
• Frac(ℤ) is canonically isomorphic to ℚ Example
• 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
• Rational powers of rational bases computed from the definition Example
• Real algebraic numbers are countably infinite, without choice 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
• A countable union of compact content-zero subsets of the line is null in ZF Lemma
• An additive f : ℝ → ℝ satisfies f(0) = 0, f(-x) = -f(x) and f(qx) = q f(x) for every rational q and every real x; in particular f(q) = q f(1) at every rational q Lemma
• Every rational has a positive-denominator representative Lemma

…and 17 more results.

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on 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)