Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-07-24
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Arithmetic on the rationals
Definition
On (The rationals as equivalence classes of pairs of integers) define, on representatives,
and, for (equivalently ), the inverse .
Remarks
- The denominators stay legal: because has no zero divisors (The integers have no zero divisors; multiplicative cancellation).
- Independence of representatives: Rational arithmetic is well defined ↗ for sum, product, and negation; The reciprocal on the rationals is well-defined ↗ for the reciprocal.
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
- Convergents of a regular continued fraction Definition
- Finite and infinite regular continued fractions Definition
- Order on the rationals Definition
- Simple continued fractions, convergents, and the integer-coordinate coding of ℕ^ℕ Definition
- No rational squares to 3 or to 6, and none cubes to 2: three instances of the rational-root corollary Example
- Convergents are given by the standard recurrences and tail formula Lemma
- Laws of rational exponents Lemma
- Rational arithmetic is well defined 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 reciprocal on the rationals is well-defined Lemma
- Minkowski's inequality for finite sums (rational exponent) Theorem
- The rationals form a field Theorem
- The rationals form a totally ordered field Theorem
- Weighted AM-GM inequality with rational weights Theorem
- Young's inequality for products (rational conjugate exponents) Theorem
Dependency tree · two levels
6 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)