View source for Irreducible fraction

{{short description|Fully simplified fraction}}
An '''irreducible fraction''' (or '''fraction in lowest terms''', '''simplest form''' or '''reduced fraction''') is a [[Fraction (mathematics)|fraction]] in which the numerator and denominator are [[Integer|integer]]s that have no other common [[Divisor|divisor]]s than 1 (and −1, when negative numbers are considered).<ref>{{SpringerEOM|title=Fraction|id=Fraction|first=S. A.|last=Stepanov}}</ref> In other words, a fraction {{sfrac|''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|mathematics]], "'''irreducible fraction'''" may also refer to rational fractions such that the numerator and the denominator are coprime polynomials.<ref>E.g., see {{citation|title=The Legacy of Niels Henrik Abel: The Abel Bicentennial, Oslo, June 3-8, 2002|first1=Olav Arnfinn|last1=Laudal|first2=Ragni|last2=Piene|publisher=Springer|year=2004|page=155|isbn=9783540438267 |url=https://books.google.com/books?id=HiXwhBm42hcC&pg=PA155}}</ref> Every [[Rational number|rational number]] can be represented as an irreducible fraction with positive denominator in exactly one way.<ref name="unique">{{citation|title=Elements of Arithmetic and Algebra: For the Use of the Royal Military College|volume=1|series=College text books, Sandhurst. Royal Military College|first=William|last=Scott|publisher=Longman, Brown, Green, and Longmans|year=1844|page=75}}.</ref>

An equivalent definition is sometimes useful:  if ''a'' and ''b'' are integers, then the fraction {{sfrac|''a''|''b''}} is irreducible if and only if there is no other equal fraction {{sfrac|''c''|''d''}} such that {{nowrap|1={{abs|''c''}} < {{abs|''a''}}}} or {{nowrap|1={{abs|''d''}} < {{abs|''b''}}}}, where {{abs|''a''}} means the [[Absolute value|absolute value]] of ''a''.{{sfnp|Scott|1844|page=74}} (Two fractions {{sfrac|''a''|''b''}} and {{sfrac|''c''|''d''}} are ''equal'' or ''equivalent'' [[If and only if|if and only if]] ''ad''&nbsp;=&nbsp;''bc''.)

For example, {{sfrac|1|4}}, {{sfrac|5|6}}, and {{sfrac|−101|100}} are all irreducible fractions.  On the other hand, {{sfrac|2|4}} is reducible since it is equal in value to {{sfrac|1|2}}, and the numerator of {{sfrac|1|2}} is less than the numerator of {{sfrac|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.<ref>{{citation|title=Integers, Fractions, and Arithmetic: A Guide for Teachers|volume=10|series=MSRI mathematical circles library|first1=Judith D.|last1=Sally|first2=Paul J. Jr.|last2=Sally|publisher=American Mathematical Society|year=2012|isbn=9780821887981|contribution=9.1. Reducing a fraction to lowest terms|pages=131–134|url=https://books.google.com/books?id=Ntjq07-FA_IC&pg=PA131}}.</ref> 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.<ref>{{citation|title=Learning Modern Algebra|series=Mathematical Association of America Textbooks|first1=Al|last1=Cuoco|first2=Joseph|last2=Rotman|publisher=Mathematical Association of America|year=2013|isbn=9781939512017|page=33|url=https://books.google.com/books?id=LelYGuQHResC&pg=PA33}}.</ref>

==Examples==
:<math> \frac{120}{90}=\frac{12}{9}=\frac{4}{3}</math>

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, {{sfrac|4|3}}, 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 {{nowrap|1=120 ÷ 30 = 4}}, and  {{nowrap|1=90 ÷ 30 = 3}}, one gets
:<math> \frac{120}{90}=\frac{4}{3}</math>

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<ref name="unique"/> (however {{sfrac|2|3}} = {{sfrac|−2|−3}} although both are irreducible). Uniqueness is a consequence of the [[Fundamental theorem of arithmetic|unique prime factorization]] of integers, since {{nowrap|1={{sfrac|''a''|''b''}} = {{sfrac|''c''|''d''}}}} implies ''ad''&nbsp;=&nbsp;''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''&nbsp;=&nbsp;''c'' and by the same argument ''b''&nbsp;=&nbsp;''d''.

==Applications==

The fact that any rational number has a unique representation as an irreducible fraction is utilized in various [[Square root of 2#Proofs of irrationality|proofs of the irrationality of the square root of 2]] and of other irrational numbers. For example, one proof notes that if {{sqrt|2}} could be represented as a ratio of integers, then it would have in particular the fully reduced representation {{sfrac|''a''|''b''}} where ''a'' and ''b'' are the smallest possible; but given that {{sfrac|''a''|''b''}} equals {{sqrt|2}}, so does {{sfrac|2''b'' − ''a''|''a'' − ''b''}} (since cross-multiplying this with {{sfrac|''a''|''b''}} shows that they are equal). Since ''a''&nbsp;>&nbsp;''b'' (because {{sqrt|2}} is greater than 1), the latter is a ratio of two smaller integers. This is a [[Proof by contradiction|contradiction]], 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|field of fractions]] of any [[Unique factorization domain|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.<ref>{{citation|title=Abstract Algebra|first=Paul B.|last=Garrett|publisher=CRC Press|year=2007|isbn=9781584886907|page=183|url=https://books.google.com/books?id=CZzSBQAAQBAJ&pg=PA183}}.</ref> 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|monic polynomial]].<ref>{{citation|title=Abstract Algebra|volume=242|series=Graduate Texts in Mathematics|first=Pierre Antoine|last=Grillet|publisher=Springer|year=2007|isbn=9780387715681|at=Lemma 9.2, p.&nbsp;183|url=https://books.google.com/books?id=CZzSBQAAQBAJ&pg=PA183}}.</ref>

==See also==
*[[Anomalous cancellation]], 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==
{{reflist}}

==External links==
*{{MathWorld |title=Reduced Fraction |id=ReducedFraction}}

{{Fractions and ratios}}

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[[Category:Fractions (mathematics)]]
[[Category:Elementary arithmetic]]

{{Sourceattribution|Irreducible fraction}}