Definition:Rational Number

Definition
A number in the form $\dfrac p q$, where both $p$ and $q$ are integers ($q$ non-zero), is called a rational number.

The set of all rational numbers is usually denoted $\Q$.

Thus:
 * $\Q = \left\{{\dfrac p q: p \in \Z, q \in \Z_{\ne 0}}\right\}$

A rational number is positive iff $p \cdot q > 0$. It is negative iff $p \cdot q < 0$ and it is zero iff $p \cdot q = 0$. Every non-zero rational number is either positive or negative.

Linguistic Note
The name rational has two significances:


 * $(1): \quad$ The construct $\dfrac p q$ can be defined as the ratio between $p$ and $q$.
 * $(2): \quad$ In contrast to the concept irrational number, which can not be so defined. The ancient Greeks had such a term for an irrational number: alogon, which had a feeling of undesirably chaotic and unstructured, or, perhaps more literally: illogical. The proof that there exist such numbers was a shock to their collective national psyche.

For a rational number $\dfrac p q$, if $\left\vert{ p }\right\vert > \left\vert{ q }\right\vert$, it is sometimes referred to as an improper (or vulgar) fraction.

Also known as
A rational number such that $q \ne 1$ is colloquially and popularly referred to as a fraction. The similarity between that word and the word fracture is no accident.

Variants on $\Q$ are often seen, for example $\mathbf Q$ and $\mathcal Q$, or even just $Q$.

Also see

 * Definition:Canonical Form of Rational Number


 * Definition:Vulgar Fraction
 * Definition:Top-Heavy Fraction
 * Definition:Mixed Fraction