Representations of Rational Numbers

Representations of Rational Numbers

Rational numbers are quantities that can be expressed as a ratio of two integers, where the denominator is not zero. These numbers can be depicted in various forms. The most common is the fraction form, such as 8/3. When a rational number includes both a whole number and a fraction, it is termed a mixed number, like 2 2/3. Another representation is a repeating decimal, which can be denoted by a bar over the repeating digits, for example, 2.6̅, or in parentheses, such as 2.(6). Continued fractions provide a unique way to express rational numbers through a sequence of integers and fractions, written as 2 + 1/(1 + 1/2), or in a compact form, [2; 1, 2]. Egyptian fractions represent a number as a sum of distinct unit fractions, for instance, 2 + 1/2 + 1/6. The prime factorization of a rational number involves expressing it as a product of prime numbers to various powers, such as 2^3 × 3^-1. Quote notation, often used in measurements, is another method, exemplified by 3'6". These diverse forms all convey the same rational value in different ways.

Formal Construction of Rational Numbers

The set of rational numbers, symbolized by ℚ, is constructed from equivalence classes of ordered pairs (m, n), where m and n are integers and n ≠ 0. An equivalence relation is established by declaring (m1, n1) equivalent to (m2, n2) if m1n2 = m2n1. This relation is congruent with the operations of addition and multiplication defined for these pairs: (m1, n1) + (m2, n2) = (m1n2 + m2n1, n1n2) and (m1, n1) × (m2, n2) = (m1m2, n1n2). The rational numbers are then the set of equivalence classes under this relation, with the defined operations of addition and multiplication.

Canonical Representation and Equivalence Classes

In the formal structure of rational numbers, each number is represented by an equivalence class of the form (m, n), with m and n being integers and n positive. The equivalence class [m/n] contains all pairs equivalent to (m, n). Although many pairs can represent the same rational number, there is a unique simplest form for each equivalence class where the numerator and denominator are coprime and the denominator is positive. This simplest form is the canonical representative of the rational number, also known as the reduced fraction or lowest terms.

Integrating Integers and Ordering Rational Numbers

Integers are included in the rational numbers, represented as n/1. This representation allows for the integration of integers into the rational number system. Rational numbers are ordered by comparing the cross-products of their numerators and denominators. For two positive denominators, m1/n1 ≤ m2/n2 if and only if m1n2 ≤ m2n1. If the denominators are negative, the direction of the inequality is reversed. This ordering is consistent with the natural order of integers and extends it to the entire set of rational numbers, enabling comparison and organization within the system.