Anirreducible fraction (orfraction in lowest terms,simplest form orreduced fraction) is afraction in which the numerator and denominator areintegers that have no other commondivisors than 1 (and −1, when negative numbers are considered).[1] In other words, a fractiona/b is irreducible if and only ifa andb arecoprime, that is, ifa andb have agreatest common divisor of 1. In highermathematics, "irreducible fraction" may also refer torational fractions such that the numerator and the denominator are coprimepolynomials.[2] Everyrational number can be represented as an irreducible fraction with positive denominator in exactly one way.[3]
An equivalent definition is sometimes useful: ifa andb are integers, then the fractiona/b is irreducible if and only if there is no other equal fractionc/d such that|c| < |a| or|d| < |b|, where |a| means theabsolute value ofa.[4] (Two fractionsa/b andc/d areequal orequivalentif and only ifad = 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 to1/2, and the numerator of1/2 is less than the numerator of2/4.
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 theirgreatest common divisor.[5] In order to find the greatest common divisor, theEuclidean algorithm orprime factorization can be used. The Euclidean algorithm is commonly preferred because it allows one to reduce fractions with numerators and denominators too large to be easily factored.[6]
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 other than 1.
The original fraction could have also been reduced in a single step by using thegreatest common divisor of 90 and 120, which is 30. As120 ÷ 30 = 4, and90 ÷ 30 = 3, one gets
Which method is faster "by hand" depends on the fraction and the ease with which common factors are spotted. In case a denominator and numerator remain that are too large to ensure they are coprime by inspection, a greatest common divisor computation is needed anyway to ensure the fraction is actually irreducible.
Every rational number has aunique representation as an irreducible fraction with a positive denominator[3] (however2/3 =−2/−3 although both are irreducible). Uniqueness is a consequence of theunique prime factorization of integers, sincea/b =c/d impliesad = bc, and so both sides of the latter must share the same prime factorization, yeta andb share no prime factors so the set of prime factors ofa (with multiplicity) is a subset of those ofc and vice versa, meaninga = c and by the same argumentb = d.
The fact that any rational number has a unique representation as an irreducible fraction is utilized in variousproofs of the irrationality of the square root of 2 and of other irrational numbers. For example, one proof notes that if could be represented as a ratio of integers, then it would have in particular the fully reduced representationa/b wherea andb are the smallest possible; but given thata/b equals so does2b −a/a −b (since cross-multiplying this witha/b shows that they are equal). Sincea > b (because is greater than 1), the latter is a ratio of two smaller integers. This is acontradiction, so the premise that the square root of two has a representation as the ratio of two integers is false.
The notion of irreducible fraction generalizes to thefield of fractions of anyunique factorization domain: any element of such a field can be written as a fraction in which denominator and numerator are coprime, by dividing both by their greatest common divisor.[7] This applies notably torational expressions over a field. The irreducible fraction for a given element is unique up to multiplication of denominator and numerator by the same invertible element. In the case of the rational numbers this means that any number has two irreducible fractions, related by a change of sign of both numerator and denominator; this ambiguity can be removed by requiring the denominator to be positive. In the case of rational functions the denominator could similarly be required to be amonic polynomial.[8]
