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.
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.
See also
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