Rational Number - Formal Construction

Formal Construction

Mathematically we may construct the rational numbers as equivalence classes of ordered pairs of integers (m,n), with n ≠ 0. This space of equivalence classes is the quotient space (Z × (Z \ {0})) / ~, where (m₁,n₁) ~ (m₂,n₂) if, and only if, m₁n₂ − m₂n₁ = 0. We can define addition and multiplication of these pairs with the following rules:

and, if m₂ ≠ 0, division by

The equivalence relation (m₁,n₁) ~ (m₂,n₂) if, and only if, m₁n₂ − m₂n₁ = 0 is a congruence relation, i.e. it is compatible with the addition and multiplication defined above, and we may define Q to be the quotient set (Z × (Z \ {0})) / ~, i.e. we identify two pairs (m₁,n₁) and (m₂,n₂) if they are equivalent in the above sense. (This construction can be carried out in any integral domain: see field of fractions.) We denote by the equivalence class containing (m₁,n₁). If (m₁,n₁) ~ (m₂,n₂) then, by definition, (m₁,n₁) belongs to and (m₂,n₂) belongs to ; in this case we can write = . Given any equivalence class there are a countably infinite number of representation, since

The canonical choice for is chosen so that gcd(m,n) = 1, i.e. m and n share no common factors, i.e. m and n are coprime. For example, we would write instead of or, even though = = .

We can also define a total order on Q. Let ∧ be the and-symbol and ∨ be the or-symbol. We say that ≤ if:

The integers may be considered to be rational numbers by the embedding that maps m to .