Irreducible fraction

An irreducible fraction (or fraction in lowest terms or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and -1, when negative numbers are considered). In other words, a fraction ^a⁄_b is irreducible if and only if a and b are coprime, that is, if a and b have a greatest common divisor of 1. In higher mathematics, "irreducible fraction" may also refer to rational fractions such that the numerator and the denominator are coprime polynomials. Every positive rational number can be represented as an irreducible fraction in exactly one way.

An equivalent definition is sometimes useful: if a, b are integers, then the fraction ^a⁄_b is irreducible if and only if there is no other equal fraction ^c⁄_d such that |c| < |a| or |d| < |b|, where |a| means the absolute value of a. (Two fractions ^a⁄_b and ^c⁄_d are equal or equivalent if and only if ad = bc.)

For example, ¹⁄₄, ⁵⁄₆, and ⁻¹⁰¹⁄₁₀₀ are all irreducible fractions. On the other hand, ²⁄₄ is reducible since it is equal in value to ¹⁄₂, and the numerator of ¹⁄₂ is less than the numerator of ²⁄₄.