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| In [[algebra]], an '''algebraic fraction''' is a [[fraction (mathematics)|fraction]] whose numerator and denominator are [[algebraic expression]]s. Two examples of algebraic fractions are <math>\frac{3x}{x^2+2x-3}</math> and <math>\frac{\sqrt{x+2}}{x^2-3}</math>. Algebraic fractions are subject to the same laws as [[arithmetic fraction]]s.
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| A '''rational fraction''' is an algebraic fraction whose numerator and denominator are both [[polynomial]]s. Thus <math>\frac{3x}{x^2+2x-3}</math> is a rational fraction, but not <math>\frac{\sqrt{x+2}}{x^2-3},</math> because the numerator contains a square root function.
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| ==Terminology==
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| In the algebraic fraction <math>\tfrac{a}{b}</math>, the dividend ''a'' is called the ''numerator'' and the divisor ''b'' is called the ''denominator''. The numerator and denominator are called the ''[[Term (mathematics)|terms]]'' of the algebraic fraction.
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| A ''complex fraction'' is a fraction whose numerator or denominator, or both, contains a fraction. A ''simple fraction'' contains no fraction either in its numerator or its denominator. A fraction is in ''lowest terms'' if the only factor common to the numerator and the denominator is 1.
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| An expression which is not in fractional form is an ''integral expression''. An integral expression can always be written in fractional form by giving it the denominator 1. A ''mixed expression'' is the algebraic sum of one or more integral expressions and one or more fractional terms.
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| ==Rational fractions==
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| {{See also|Rational function}}
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| If the expressions ''a'' and ''b'' are [[polynomial]]s, the algebraic fraction is called a ''rational algebraic fraction''<ref>{{cite book|author=Bansi Lal|title=Topics in Integral Calculus|page=53|year=2006|url=http://books.google.com/books?id=RlQ-tHlWcxcC&pg=PA53&dq=%22rational+algebraic+fraction%22&hl=fr&ei=cyWcTqe1I5CPswaz5oTsAw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC0Q6AEwAA}}</ref> or simply ''rational fraction''.<ref>{{cite book|author=Ėrnest Borisovich Vinberg|title=A course in algebra|page=131|year=2003|url=http://books.google.com/books?id=rzNq39lvNt0C&pg=PA132&dq=%22rational+fraction%22&hl=fr&ei=JiucTp-qJIj0sgbY2PSfBA&sa=X&oi=book_result&ct=result&resnum=5&ved=0CEIQ6AEwBDgK}}</ref><ref>{{cite book|author=Parmanand Gupta|title=Comprehensive Mathematics XII|page=739|url=http://books.google.com/books?id=DoqIu7L2Yg8C&pg=PA739&dq=%22rational+fraction%22&hl=fr&ei=gi6cTom4Ic3FtAbQ58CbBA&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDUQ6AEwATgK}}</ref> Rational fractions are also known as rational expressions. A rational fraction <math>\tfrac{f(x)}{g(x)}</math> is called ''proper'' if <math>\deg f(x) < \deg g(x)</math>, and ''improper'' otherwise. For example, the rational fraction <math>\tfrac{2x}{x^2-1}</math> is proper, and the rational fractions <math>\tfrac{x^3+x^2+1}{x^2-5x+6}</math> and <math>\tfrac{x^2-x+1}{5x^2+3}</math> are improper. Any improper rational fraction can be expressed as the sum of a polynomial (possibly constant) and a proper rational fraction. In the first example of an improper fraction one has
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| :<math>\frac{x^3+x^2+1}{x^2-5x+6} = (x+6) + \frac{24x-35}{x^2-5x+6},</math> | |
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| where the second term is a proper rational fraction. The sum of two proper rational fractions is a proper rational fraction as well. The reverse process of expressing a proper rational fraction as the sum of two or more fractions is called resolving it into [[partial fraction]]s. For example,
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| :<math>\frac{2x}{x^2-1} = \frac{1}{x-1} + \frac{1}{x+1}.</math>
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| Here, the two terms on the right are called partial fractions.
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| ==Irrational fractions==
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| An ''irrational fraction'' is one that contains the variable under a fractional exponent.<ref>{{cite book|author=Washington McCartney|title=The principles of the differential and integral calculus; and their application to geometry|page=203|year=1844|url=http://books.google.com/books?id=o1dLAAAAMAAJ&pg=PA203&dq=%22irrational+fraction%22&hl=fr&ei=HRmcTuX6MoX0sgavm_DqAw&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDkQ6AEwAQ}}</ref> An example of an irrational fraction is
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| :<math>\frac{x^\tfrac12 - \tfrac13 a}{x^\tfrac13 - x^\tfrac12}.</math>
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| The process of transforming an irrational fraction to a rational fraction is known as [[Rationalisation (mathematics)|rationalization]]. Every irrational fraction in which the radicals are [[monomial]]s may be rationalized by finding the [[least common multiple]] of the indices of the roots, and substituting the variable for another variable with the least common multiple as exponent. In the example given, the least common multiple is 6, hence we can substitute <math>x = z^6</math> to obtain
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| :<math>\frac{z^3 - \tfrac13 a}{z^2 - z^3}.</math>
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| ==Notes==
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| {{reflist}}
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| ==References==
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| {{cite book|last=Brink|first=Raymond W.|title=College Algebra|year=1951|chapter=IV. Fractions|url=http://books.google.com/books?id=n-EjEuKqk1YC&pg=PR3&dq=%22complex+fraction%22+contains&hl=de&source=gbs_selected_pages&cad=3#v=onepage&q=%22complex%20fraction%22%20contains&f=false}}
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| [[Category:Elementary algebra]]
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| [[Category:Fractions]]
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