A real function $R(x)$ of a single variable $x$ is called rational if it can be written as a quotient$$ R(x) = \frac{P(x)}{Q(x)},$$ where $P(x)$ and $Q(x)$ are polynomials in $x$ with real coefficients. When one is only interested in algebraic properties of $R(x)$ or $P(x)$ and $Q(x)$ , it is convenient to forget that they define functions and simply treat them as algebraic expressions in $x$ . In this case $R(x)$ is referred to as a rational expression.

In general, a rational function (expression) $R(x_1,\ldots,x_n)$ has the form$$ R(x_1,\ldots,x_n) = \frac{P(x_1,\ldots,x_n)}{Q(x_1,\ldots,x_n)},$$ where $P(x_1,\ldots,x_n)$ and $Q(x_1,\ldots,x_n)$ are polynomials in the variables $(x_1,\ldots,x_n)$ with coefficients in some field or ring $S$ .

In this sense, $R(x_1,\ldots,x_n)$ can be regarded as an element of the fraction field $S(x_1,\ldots,x_n)$ of the polynomial ring $S[x_1,\ldots,x_n]$ .