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In
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
, an algebraic fraction is a fraction whose numerator and denominator are
algebraic expression In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations ( addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). ...
s. Two examples of algebraic fractions are \frac and \frac. Algebraic fractions are subject to the same laws as
arithmetic 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 ...
s. A rational fraction is an algebraic fraction whose numerator and denominator are both
polynomial 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 ...
s. Thus \frac is a rational fraction, but not \frac, because the numerator contains a square root function.


Terminology

In the algebraic fraction \tfrac, 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

If the expressions ''a'' and ''b'' are
polynomial 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 ...
s, the algebraic fraction is called a ''rational algebraic fraction'' or simply ''rational fraction''. Rational fractions are also known as rational expressions. A rational fraction \tfrac is called ''proper'' if \deg f(x) < \deg g(x), and ''improper'' otherwise. For example, the rational fraction \tfrac is proper, and the rational fractions \tfrac and \tfrac 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 :\frac = (x+6) + \frac, 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 In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction a ...
s. For example, :\frac = \frac + \frac. 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. An example of an irrational fraction is :\frac. The process of transforming an irrational fraction to a rational fraction is known as rationalization. Every irrational fraction in which the radicals are
monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer expone ...
s may be rationalized by finding the
least common multiple In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by bo ...
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 :\frac.


See also

*
Partial fraction decomposition In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as ...


References

* {{Fractions and ratios Elementary algebra Fractions (mathematics)