An irreducible fraction (or fraction in lowest terms) is a vulgar fraction in which the numerator and denominator are smaller than those in any other equivalent fraction. In arithmetic, a vulgar fraction (or common fraction) consists of one integer divided by a non-zero integer. ... In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ... In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ...

Stating it more formally, a fraction ^a⁄_b is irreducible if there is no other equivalent fraction ^c⁄_d with c having an absolute value less than the absolute value of a (where a, b, c, and d are all integers). In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...

For example, ¹⁄₄, ⁵⁄₆, and ^-101⁄₁₀₀ are all irreducible fractions. On the other hand, ²⁄₄ is not irreducible since it is equal in value to ¹⁄₂, and the numerator of the latter (1) is less than the numerator of the former (2).

It can be shown that a fraction ^a⁄_b is irreducible if, and only if, a and b are coprime (relatively prime), that is, if a and b have a greatest common divisor of 1. Coprime - Wikipedia /**/ @import /skins-1. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ...

A fraction that is not irreducible can be reduced by using the Euclidean algorithm to find the greatest common divisor of the numerator and the denominator, and then dividing both the numerator and the denominator by it. In number theory, the Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (GCD) of two elements of any Euclidean domain (for example, the integers). ...