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
integers that have no other common
divisors 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 equival ...
, 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. Every positive
rational number 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
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of ''a''. (Two fractions and are ''equal'' or ''equivalent''
if and only if ''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 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.
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, , 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 of 90 and 120, which is 30. As , and , 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.
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's ...
, 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 anomalous cancellation or accidental cancellation is a particular kind of arithmetic procedural error that gives a numerically correct answer. An attempt is made to reduce a fraction by cancelling individual digits in the numerator and denomin ...
, an erroneous arithmetic procedure that produces the correct irreducible fraction by cancelling digits of the original unreduced form.
*
Diophantine approximation, the approximation of real numbers by rational numbers.
References
External links
*
{{DEFAULTSORT:Irreducible Fraction
Fractions (mathematics)
Elementary arithmetic