quality:correction
Algebraic fractions

An algebraic fraction is the indicated quotient of two algebraic expressions. As with fractions of integers, the denominator of an algebraic fraction cannot be zero. Two examples of algebraic fractions are 3 x x 2 + 2 x − 3 {\displaystyle {\frac {3x}{x^{2}+2x-3}}} and x + 2 x 2 − 3 {\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}}}. Algebraic fractions are subject to the same field properties as arithmetic fractions.

If the numerator and the denominator are polynomials, as in 3 x x 2 + 2 x − 3 {\displaystyle {\frac {3x}{x^{2}+2x-3}}}, the algebraic fraction is called a rational fraction (or rational expression). An irrational fraction is one that is not rational, as, for example, one that contains the variable under a fractional exponent or root, as in x + 2 x 2 − 3 {\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}}}.

The terminology used to describe algebraic fractions is similar to that used for ordinary fractions. For example, an algebraic fraction is in lowest terms if the only factors common to the numerator and the denominator are 1 and −1. An algebraic fraction whose numerator or denominator, or both, contain a fraction, such as 1 + 1 x 1 − 1 x {\displaystyle {\frac {1+{\tfrac {1}{x}}}{1-{\tfrac {1}{x}}}}}, is called a complex fraction.

The field of rational numbers is the field of fractions of the integers, while the integers themselves are not a field but rather an integral domain. Similarly, the rational fractions with coefficients in a field form the field of fractions of polynomials with coefficient in that field. Considering the rational fractions with real coefficients, radical expressions representing numbers, such as 2 / 2 {\displaystyle \textstyle {\sqrt {2}}/2}, are also rational fractions, as are a transcendental numbers such as π / 2, {\textstyle \pi /2,} since all of 2, π, {\displaystyle {\sqrt {2}},\pi,} and 2 {\displaystyle 2} are real numbers, and thus considered as coefficients. These same numbers, however, are not rational fractions with integer coefficients.

The term partial fraction is used when decomposing rational fractions into sums of simpler fractions. For example, the rational fraction 2 x x 2 − 1 {\displaystyle {\frac {2x}{x^{2}-1}}} can be decomposed as the sum of two fractions: 1 x + 1 + 1 x − 1 {\displaystyle {\frac {1}{x+1}}+{\frac {1}{x-1}}}. This is useful for the computation of antiderivatives of rational functions (see partial fraction decomposition for more).