How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced — the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted — a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated — a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The integers embed in the rationals
Statement
The map is injective and preserves addition, multiplication, and order. Composing with The naturals embed in the integers embeds in ; we write for throughout.
Facts & Assumptions
Given: The map , .
Ring arithmetic in (The integers form a commutative ring).
The order on (Order on the rationals), whose defining inequality is read in the order on (Order on the integers).
Equality in : iff (The rationals as equivalence classes of pairs of integers).
Proof
Injectivity: means , i.e. .
Addition: .
Multiplication: .
Order: denominators are , so reads , i.e. .
embeds into , preserving arithmetic and order.
Depends on
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
- 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 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
- 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 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 · next 3 levels
Direct dependencies and their dependencies through the next three levels: 35 results over 14 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)
- Rational number — formal construction (Wikipedia) (standard reference, not scraped)