Irreducible fraction

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 vulgar fraction. In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object Numerator may refer to A numeral used to indicate a count particularly of the equal parts in a fraction For example in 3/4 3 is the numerator 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. In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than In Mathematics, the greatest common divisor (gcd, sometimes known as the greatest common factor (gcf or highest common factor (hcf, of two non-zero

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 equivalent fraction ^c⁄_d such that |c| < |a| or |d| < |b|. Note that |a| means the absolute value of a. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. 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. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions

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 not irreducible 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 Mathematics, the greatest common divisor (gcd, sometimes known as the greatest common factor (gcf or highest common factor (hcf, of two non-zero In order to find the greatest common divisor, the Euclidean algorithm may be used. In Number theory, the Euclidean algorithm (also called Euclid's algorithm) is an Algorithm to determine the Greatest common divisor (GCD Using the Euclidean algorithm is a simple method that can even be performed without a calculator.

Examples

$\frac{120}{90}=\frac{12}{9}=\frac{4}{3} \,.$

In the first step both numbers were divided by 10, which is a factor common to both 120 and 90. In the second step, they were divided by 3. The final result, ⁴/₃, is an irreducible fraction because 4 and 3 have no common factors.

The original fraction could have also been reduced in a single step by using the greatest common divisor of 90 and 120, which would be gcd(90,120)=30.

$\frac{120}{90}=\frac{4}{3} \,.$

Which method is faster "by hand" depends on the fraction.

Examples

See also