An irreducible fraction (or fraction in lowest terms or reduced form) is a vulgar fraction in which the numerator and denominator are smaller than those in any other vulgar fraction equal to it. It can be shown that 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.
More formally, if a, b, c, and d are all 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. This definition is more rigorous and expandable than a simpler one involving common divisors, and it is often necessary to use it to determine the rationality or reducibility of numbers that are expressed in terms of variables.
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).
A fraction that is reducible can be reduced by dividing both the numerator and denominator by a common factor. It can be fully reduced to lowest terms if both are divided by their greatest common divisor. In order to find the greatest common divisor, the Euclidean algorithm may be used. Using the Euclidean algorithm is a simple method that can even be performed without a calculator.
Read more about Irreducible Fraction: Examples, Uniqueness, Generalization
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