In algebra, an **algebraic fraction** is a fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are ${\frac {3x}{x^{2}+2x-3}}$ and ${\frac {\sqrt {x+2}}{x^{2}-3}}$. Algebraic fractions are subject to the same laws as arithmetic fractions.

A **rational fraction** is an algebraic fraction whose numerator and denominator are both polynomials. Thus ${\frac {3x}{x^{2}+2x-3}}$ is a rational fraction, but not ${\frac {\sqrt {x+2}}{x^{2}-3}},$ because the numerator contains a square root function.

## Terminology

In the algebraic fraction ${\tfrac {a}{b}}$, 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

See also:

polynomials. Thus

${\frac {3x}{x^{2}+2x-3}}$ is a rational fraction, but not

${\frac {\sqrt {x+2}}{x^{2}-3}},$ because the numerator contains a square root function.

In the algebraic fraction ${\tfrac {a}{b}}$, 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 polynomials, the algebraic fraction is called a *rational algebraic fraction*^{[1]} or simply *rational fraction*.^{polynomials, the algebraic fraction is called a rational algebraic fraction[1] or simply rational fraction.[2][3] Rational fractions are also known as rational expressions. A rational fraction ${\tfrac {f(x)}{g(x)}}$ is called proper if $\deg f(x)<\deg g(x)$, and improper otherwise. For example, the rational fraction ${\tfrac {2x}{x^{2}-1}}$ is proper, and the rational fractions ${\tfrac {x^{3}+x^{2}+1}{x^{2}-5x+6}}$ and ${\tfrac {x^{2}-x+1}{5x^{2}+3}}$ 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
}

- $\frac{{x}^{}}{}$
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,

- ${\frac {2x}{x^{2}-1}}={\frac {1}{x-1}}+{\frac {1}{x+1}}.$

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.^{[4]} An example of an irrational fraction is

- 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.^{[4]} An example of an irrational fraction is

- $\frac{{x}^{1/2}-{\textstyle \frac{1}{3}}a}{{x}^{1}}$
An *irrational fraction* is one that contains the variable under a fractional exponent.^{[4]} An example of an irrational fraction is

- $}The\; process\; of\; transforming\; an\; irrational\; fraction\; to\; a\; rational\; fraction\; is\; known\; asrationalization.\; Every\; irrational\; fraction\; in\; which\; the\; radicals\; aremonomialsmay\; be\; rationalized\; by\; finding\; theleast\; common\; multipleof\; 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
- ${\frac {z^{3}-{\tfrac {1}{3}}a}{z^{2}-z^{3}}}.$

## Notes