An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and −1, when negative numbers are considered).[1] In other words, 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 higher mathematics, "irreducible fraction" may also refer to rational fractions such that the numerator and the denominator are coprime polynomials.[2] Every rational number can be represented as an irreducible fraction with positive denominator in exactly one way.[3]
An equivalent definition is sometimes useful: if a and b are 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.[4] (Two fractions a/b and c/d are equal or equivalent if and only if ad = 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 to 1/2, and the numerator of 1/2 is less than the numerator of 2/4.
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 their greatest common divisor.[5] In order to find the greatest common divisor, the Euclidean algorithm or prime 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]
^Stepanov, S. A. (2001) [1994], "Fraction", Encyclopedia of Mathematics, EMS Press
^E.g., see Laudal, Olav Arnfinn; Piene, Ragni (2004), The Legacy of Niels Henrik Abel: The Abel Bicentennial, Oslo, June 3-8, 2002, Springer, p. 155, ISBN 9783540438267
^Scott, William (1844), Elements of Arithmetic and Algebra: For the Use of the Royal Military College, College text books, Sandhurst. Royal Military College, vol. 1, Longman, Brown, Green, and Longmans, p. 75.
^Scott (1844), p. 74.
^Sally, Judith D.; Sally, Paul J. Jr. (2012), "9.1. Reducing a fraction to lowest terms", Integers, Fractions, and Arithmetic: A Guide for Teachers, MSRI mathematical circles library, vol. 10, American Mathematical Society, pp. 131–134, ISBN 9780821887981.
^Cuoco, Al; Rotman, Joseph (2013), Learning Modern Algebra, Mathematical Association of America Textbooks, Mathematical Association of America, p. 33, ISBN 9781939512017.
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