# Definition:Rational Function

## Definition

Let $F$ be a field.

Let $P: F \to F$ and $Q: F \to F$ be polynomial functions on $F$.

Let $S$ be the set $F$ from which all the roots of $Q$ have been removed.

That is:

- $S = F \setminus \set {x \in F: \map Q x = 0}$

Then the equation $y = \dfrac {\map P x} {\map Q x}$ defines a mapping from $S$ to $F$.

Such a mapping is called a **rational function**.

The concept is usually encountered where the polynomial functions $P$ and $Q$ are either real or complex:

### Real Domain

Let $P: \R \to \R$ and $Q: \R \to \R$ be polynomial functions on the set of real numbers.

Let $S$ be the set $\R$ from which all the roots of $Q$ have been removed.

That is:

- $S = \R \setminus \left\{{x \in \R: Q \left({x}\right) = 0}\right\}$.

Then the equation $y = \dfrac {P \left({x}\right)} {Q \left({x}\right)}$ defines a function from $S$ to $\R$.

Such a function is a **rational function**.

### Complex Domain

Let $P: \C \to \C$ and $Q: \C \to \C$ be polynomial functions on the set of complex numbers.

Let $S$ be the set $\C$ from which all the roots of $Q$ have been removed.

That is:

- $S = \C \setminus \set {z \in \C: \map Q z = 0}$

Then the equation $y = \dfrac {\map P z} {\map Q z}$ defines a function from $S$ to $\C$.

Such a function is a **rational (algebraic) function**.

## Also see

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $4$: Fields: $\S 17$. The Characteristic of a Field: Example $26$