The integers embed in the rationals

Depends on

• Arithmetic on the rationals
• Order on the rationals
• Order on the integers
• The rationals as equivalence classes of pairs of integers
• The integers form a commutative ring
• The naturals embed in the integers

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
• Complex de Moivre formula for every integer exponent Corollary
• The rational algebraic integers are exactly the integers Corollary
• ℤ and {n + 1/n : n ≥ 2} are disjoint closed subsets of ℝ at distance 0, so the set-to-set distance is not a metric Counterexample
• ψ(1/x) has no limit at 0: two sequences tending to 0 give values constantly 0 and constantly 1/2 Counterexample
• The Catalan generating function C(x)=∑_n≥0Cₙxⁿ in ℚ⟦ x⟧ Definition
• The classical Weierstrass function Definition
• The Dirichlet function 1_ℚ, and Thomae's function t with t(x) = 1/q at a rational x = p/q in lowest terms with q ≥ 1 and t(x) = 0 at every irrational x Definition
• Ceiling ⊣ inclusion ⊣ floor: an adjoint triple between (ℝ,≤) and (ℤ,≤) Example
• No rational squares to 3 or to 6, and none cubes to 2: three instances of the rational-root corollary 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 distance ψ(x) = d(x, ℤ) from a real number to the integers is 1-Lipschitz, hence uniformly continuous, takes values in [0,1/2], and vanishes exactly on ℤ Example
• The Prüfer p-group is Artinian but not Noetherian Example
• The trigonometry-free oscillator ψ(x) = inf_n ∈ ℤ |x - n| is well defined and attained at a nearest integer, takes values in [0, 1/2], vanishes exactly on ℤ, equals 1/2 at half-integers, and is 1-periodic Example
• ℤ sits inside ℚ as a subring that is not a subfield, so the inverse-closure clause of the subfield definition is doing work Example
• 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
• Integer part: for every real x there is exactly one integer m with m ≤ x < m + 1 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 rationals are Archimedean Lemma
• Euler's product formula φ(n)=n∏_p∣ n(1-1/p)=∏_pᵏ∥ n(pᵏ-pᵏ⁻¹) for n≥1, stated through a finite injective list of its prime divisors Theorem
• Gauss lemma: primitive factorisations over ℚ can be cleared to primitive factorisations over ℤ Theorem
• ℚ is countably infinite Theorem
• Rational root theorem Theorem
• The inclusion ℤ↪ℚ is monic and epic but neither surjective nor an isomorphism in Ring Theorem
• The number e is irrational Theorem

Dependency tree · two levels

21 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)
• Rational number — formal construction (Wikipedia) (standard reference, not scraped)