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.