The integers allow subtraction without leaving the number system, but division can still produce values that are not integers. For example, there is no integer equal to $1\div2$.

To include such quotients, the integers are extended to the rational numbers, written $\mathbb Q$.

Writing rational numbers

A number is rational when it can be written as

$$\frac pq,\qquad p,q\in\mathbb Z,\quad q\ne0.$$

Different fractions can represent the same rational number: for example, $\frac12=\frac24$. Every integer is also rational, since an integer $n$ can be written as $n/1$.

Rational numbers on the number line

Rational numbers can be placed on the number line, and between any two distinct rational numbers there is always another rational number. In this sense, the rational numbers are densely distributed along the line.