 Algebraic fraction

In elementary algebra, an algebraic fraction is the indicated quotient of two algebraic expressions.^{[1]} Two examples of algebraic fractions are and . Algebraic fractions are subject to the same laws as arithmetic fractions.
Contents
Terminology
In the algebraic fraction , the dividend a is called the numerator and the divisor b is called the denominator. The numerator and denominator are called the terms of the algebraic fraction.
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.
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.
Rational fractions
Main article: Rational functionIf the expressions a and b are polynomials, the algebraic fraction is called a rational algebraic fraction^{[2]} or simply rational fraction.^{[3]}^{[4]} Rational fractions are also known under the term rational function^{[3]}^{[4]} or rational expression. A rational fraction is called proper if deg f(x) < deg g(x), and improper otherwise. For example, the rational fraction is proper, and the rational fractions and 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
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 fractions. For example,
Here, the two terms on the right are called partial fractions.
Irrational fractions
An irrational fraction is one that contains the variable under a fractional exponent.^{[5]} An example of an irrational fraction is
The process of transforming an irrational fraction to a rational fraction is known as rationalization. Every irrational fraction in which the radicals are monomials 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 x = z^{6} to obtain
Notes
 ^ Slaught, H. E.; Lennes, N.J.. Intermediate algebra. p. 41. http://books.google.com/books?id=01l2b7SP9sIC&pg=PA41&dq=%22algebraic+fraction%22+%2Bdefinition&hl=fr&ei=LXi1TpzaENO5hAfD7dWXBA&sa=X&oi=book_result&ct=bookpreviewlink&resnum=9&ved=0CF0QuwUwCA#v=onepage&q=%22algebraic%20fraction%22%20%2Bdefinition&f=false.
 ^ Bansi Lal (2006). Topics in Integral Calculus. p. 53. http://books.google.com/books?id=RlQtHlWcxcC&pg=PA53&dq=%22rational+algebraic+fraction%22&hl=fr&ei=cyWcTqe1I5CPswaz5oTsAw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC0Q6AEwAA.
 ^ ^{a} ^{b} Ėrnest Borisovich Vinberg (2003). A course in algebra. p. 131. http://books.google.com/books?id=rzNq39lvNt0C&pg=PA132&dq=%22rational+fraction%22&hl=fr&ei=JiucTpqJIj0sgbY2PSfBA&sa=X&oi=book_result&ct=result&resnum=5&ved=0CEIQ6AEwBDgK.
 ^ ^{a} ^{b} Parmanand Gupta. Comprehensive Mathematics XII. p. 739. 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.
 ^ Washington McCartney (1844). The principles of the differential and integral calculus; and their application to geometry. p. 203. 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.
References
Brink, Raymond W. (1951). "IV. Fractions". College Algebra. http://books.google.com/books?id=nEjEuKqk1YC&pg=PR3&dq=%22complex+fraction%22+contains&hl=de&source=gbs_selected_pages&cad=3#v=onepage&q=%22complex%20fraction%22%20contains&f=false.
Categories: Elementary algebra
 Fractions
Wikimedia Foundation. 2010.
Look at other dictionaries:
Fraction (mathematics) — A cake with one quarter removed. The remaining three quarters are shown. Dotted lines indicate where the cake may be cut in order to divide it into equal parts. Each quarter of the cake is denoted by the fraction 1/4. A fraction (from Latin:… … Wikipedia
Algebraic torus — In mathematics, an algebraic torus is a type of commutative affine algebraic group. These groups were named by analogy with the theory of tori in Lie group theory (see maximal torus). The theory of tori is in some sense opposite to that of… … Wikipedia
fraction — /frak sheuhn/, n. 1. Math. a. a number usually expressed in the form a/b. b. a ratio of algebraic quantities similarly expressed. 2. Chem. (in a volatile mixture) a component whose range of boiling point temperatures allows it to be separated… … Universalium
fraction — /ˈfrækʃən / (say frakshuhn) noun 1. Mathematics a. one or more parts of a unit or whole number; the ratio between any two numbers. b. a ratio of algebraic quantities analogous to the arithmetical vulgar fraction and similarly expressed. 2. a part … Australian English dictionary
fraction — Synonyms and related words: Gaussian integer, adjunct, algebraic number, arithmetical proportion, cardinal, cardinal number, complex number, component, contingent, continued fraction, cross section, defective number, detachment, detail, division … Moby Thesaurus
fraction — frac•tion [[t]ˈfræk ʃən[/t]] n. 1) math. a) a number usu. expressed in the form a/b[/ex] b) a ratio of algebraic quantities similarly expressed 2) chem. a component in a volatile mixture whose range of boiling point temperatures allows it to be… … From formal English to slang
Partial fraction — In algebra, the partial fraction decomposition or partial fraction expansion is a procedure used to reduce the degree of either the numerator or the denominator of a rational function (also known as a rational algebraic fraction). In symbols, one … Wikipedia
Egyptian fraction — An Egyptian fraction is the sum of distinct unit fractions, such as frac{1}{2}+ frac{1}{3}+ frac{1}{16}. That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators… … Wikipedia
Continued fraction of Gauss — In complex analysis, the continued fraction of Gauss is a particular continued fraction derived from the hypergeometric functions. It was one of the first analytic continued fractions known to mathematics, and it can be used to represent several… … Wikipedia
Gauss's continued fraction — In complex analysis, Gauss s continued fraction is a particular class of continued fractions derived from hypergeometric functions. It was one of the first analytic continued fractions known to mathematics, and it can be used to represent several … Wikipedia