In mathematics a

**rational function**is a ratio of polynomials. For a single variable

*x*a typical rational function is therefore P(

*x*)/Q(

*x*) where P and Q are polynomials, and Q isn't the zero polynomial. Any non-zero polynomial Q is acceptable; but the possibility that a given value

*a*assigned to

*x*could make Q(

*a*) = 0 means that rational functions, unlike polynomials, do not always have an obvious function domain of definition. In fact if we take 1/(

*x*

^{2}+ 1), this function is everywhere defined for real numbers

*x*, but not for complex numbers where the denominator is 0 at

*x*= i and

*x*= -i.

From a mathematical point of view, a polynomial is firstly a formal expression, and only secondly a function (on some given domain). Despite the name, the same is equally true of rational functions. In abstract algebra a definition of **rational function** is given as *element of the fraction field of a polynomial ring*. For this definition to succeed, we must start with an integral domain R (for example, a field). Then R[X,Y,..., T], the ring of polynomials in some indeterminates X, ... , T, will also be an integral domain; and we can properly take a fraction field. (In greater generality for commutative rings the construction will be a localization of a polynomial ring.)