Irreducible fraction
Encyclopedia
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 equivalent vulgar fraction. 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 equivalent fraction c⁄d such that |c| < |a| or |d| < |b|. Note that |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.
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.
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.
of any unique 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. This applies notably to rational 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 a monic polynomial.
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
, that is, if a and b have a greatest common divisor
Greatest common divisor
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
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 equivalent fraction c⁄d such that |c| < |a| or |d| < |b|. Note that |a| means the absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
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
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
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
Greatest common divisor
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
. In order to find the greatest common divisor, the Euclidean algorithm
Euclidean algorithm
In mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...
may be used. Using the Euclidean algorithm is a simple method that can even be performed without a calculator.
Examples
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.
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.
Generalization
The notion of irreducible fraction generalizes to the field of fractionsField of fractions
In abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0...
of any unique factorization domain
Unique factorization domain
In mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
: 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. This applies notably to rational 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 a monic polynomial.