The integers as equivalence classes of pairs of naturals

Depends on

• The natural numbers $\mathbb{N}$ (von Neumann)
• Addition of natural numbers

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
• Division with remainder for any nonzero divisor: for a ∈ ℤ and b ≠ 0 there are unique q, r ∈ ℤ with a = qb + r and 0 ≤ r < |b| Corollary
• Every nonzero integer n is u ∏_i<r pᵢ with u ∈ {1,-1} and every pᵢ prime; u and r are determined by n, and the list is determined up to a permutation Corollary
• For an integer p > 1: p is prime if and only if, for all integers a and b, p ∣ ab implies p ∣ a or p ∣ b Corollary
• If a prime p divides a finite product ∏_i<n aᵢ of integers then p ∣ aᵢ for some i < n; at n = 0 the product is 1 and the hypothesis cannot hold Corollary
• The extended Euclidean algorithm: the same descent produces integers x, y with ax + by = gcd(a,b), so Bézout coefficients are computed and not merely shown to exist Corollary
• The index of a cycle is locally constant off its trace and vanishes far from it Corollary
• The principal logarithm is the normalised holomorphic branch on the slit plane Corollary
• The winding number is the increment of a continuous argument divided by 2π Corollary
• 2ℤ is closed under addition, negation and multiplication and is not a subring of ℤ, because it does not contain 1 Counterexample
• 6 ∣ 4 · 9 while 6 ∤ 4 and 6 ∤ 9: dividing a product does not force dividing a factor, and the coprimality hypothesis is what fails Counterexample
• A continuous function on [0,1] can have unbounded variation Counterexample
• A nonempty subset of a group closed under the operation need not be a subgroup: the nonnegative integers inside (ℤ, +) Counterexample
• If 1 were admitted as a prime, uniqueness would fail: 6 = 2 · 3 = 1 · 2 · 3 = 1 · 1 · 2 · 3, lists of different lengths that no permutation matches Counterexample
• In the multiplicative monoid H = {1, 4, 7, 10, …} of positive integers one more than a multiple of 3, the element 100 has two genuinely different factorisations into irreducibles, 4 · 25 and 10 · 10 Counterexample
• The common divisors of (0,0) are all of ℤ and have no greatest element in the order of ℤ, so gcd(0,0) cannot be defined as a maximum and is fixed by convention Counterexample
• The map n ↦ (n,0) from ℤ to ℤ × ℤ preserves addition and multiplication and does not preserve 1, so the clause f(1) = 1 is not redundant Counterexample
• 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
• ℤ is closed and not compact, and (0,1) is bounded and not compact: neither hypothesis of Heine-Borel can be dropped Counterexample
• ψ(1/x) has no limit at 0: two sequences tending to 0 give values constantly 0 and constantly 1/2 Counterexample
• Arithmetic on the integers Definition
• Complex chains, their traces, and cycles Definition
• Cyclic shifts of an integer word and its periodic partial-sum function Definition
• Diagonal lattice paths with steps U=(1,1) and D=(1,-1), and the height function Definition
• Divisibility in ℤ: d ∣ a when a = dq for some integer q Definition
• Dyadic cubes of generation k in ℝⁿ Definition
• Fredholm operator cokernel and index Definition
• Integer powers aᵐ Definition
• Integer powers in the complex field Definition
• Lattice paths, step sets and step words Definition
• Order on the integers Definition
• Powers gⁿ: natural exponents in a monoid and integer exponents in a group, with g⁰ = e Definition
• Prime and composite integers: p is prime when p > 1 and its only positive divisors are 1 and p Definition
• Sum-free subsets of the integers Definition
• The absolute value |a| of an integer Definition
• The circle as S¹=ℝ/ℤ with basepoint [0] Definition
• The classical Weierstrass function Definition
• The formal Laurent series ℝ((t⁻¹)): support bounded below, valuation, leading coefficient Definition
• The one-dimensional torus and its normalized Haar integral Definition
• The p-adic valuation vₚ(a) of a nonzero integer: the greatest k ∈ ℕ with pᵏ ∣ a Definition

…and 84 more results.

Dependency tree · two levels

9 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., §4.1 (standard reference, not scraped)
• E. Landau, Foundations of Analysis (standard reference, not scraped)
• Integer — construction from pairs of naturals (Wikipedia) (standard reference, not scraped)