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An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a
fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
in which the numerator and denominator are
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s that have no other common
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s than 1 (and −1, when negative numbers are considered). In other words, a fraction is irreducible if and only if ''a'' and ''b'' are
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
, that is, if ''a'' and ''b'' have a
greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
of 1. In higher mathematics, "irreducible fraction" may also refer to
rational fraction In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are \frac and \frac. Algebraic fractions are subject to the same laws as arithmetic fractions. A ration ...
s such that the numerator and the denominator are coprime
polynomials In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
. Every positive
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
can be represented as an irreducible fraction in exactly one way.. An equivalent definition is sometimes useful: if ''a'' and ''b'' are integers, then the fraction is irreducible if and only if there is no other equal fraction such that or , where means the absolute value of ''a''. (Two fractions and are ''equal'' or ''equivalent''
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicond ...
''ad'' = ''bc''.) For example, , , and are all irreducible fractions. On the other hand, is reducible since it is equal in value to , and the numerator of is less than the numerator of . 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 mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
. In order to find the greatest common divisor, the
Euclidean algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...
or
prime factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are s ...
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.


Examples

: \frac=\frac=\frac 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, , 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 the
greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
of 90 and 120, which is 30. As , and , one gets : \frac=\frac 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.


Uniqueness

Every rational number has a ''unique'' representation as an irreducible fraction with a positive denominator (however = although both are irreducible). Uniqueness is a consequence of the unique prime factorization of integers, since implies ''ad'' = ''bc'', and so both sides of the latter must share the same prime factorization, yet ''a'' and ''b'' share no prime factors so the set of prime factors of ''a'' (with multiplicity) is a subset of those of ''c'' and vice versa, meaning ''a'' = ''c'' and by the same argument ''b'' = ''d''.


Applications

The fact that any rational number has a unique representation as an irreducible fraction is utilized in various proofs 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 representation where ''a'' and ''b'' are the smallest possible; but given that equals , so does (since cross-multiplying this with shows that they are equal). Since ''a'' > ''b'' (because is greater than 1), the latter is a ratio of two smaller integers. This is a
contradiction In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle' ...
, so the premise that the square root of two has a representation as the ratio of two integers is false.


Generalization

The notion of irreducible fraction generalizes to the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of any
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
: 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 In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form: :x^n+c_x^+\cd ...
..


See also

* Anomalous cancellation, an erroneous arithmetic procedure that produces the correct irreducible fraction by cancelling digits of the original unreduced form. *
Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by ...
, the approximation of real numbers by rational numbers.


References


External links

* {{DEFAULTSORT:Irreducible Fraction Fractions (mathematics) Elementary arithmetic