Rational Numbers - Representation on the Number Line

A rational number is defined as a number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. On a number line, these numbers are represented as points at specific distances from a central point called the origin ($0$).

The number line extends infinitely in both directions. Positive rational numbers are always located to the right of $0$, while negative rational numbers are located to the left of $0$. The distance between any two consecutive integers (like $0$ and $1$) is called a 'unit length'.

The denominator ($q$) of a rational number $\frac{p}{q}$ determines how many equal parts each unit length on the number line must be divided into. For example, to represent $\frac{x}{3}$, you would visualize each integer interval divided into three equal segments by placing two equidistant marks between the integers.

The numerator ($p$) of a rational number $\frac{p}{q}$ indicates how many of these equal parts or divisions must be counted starting from zero. If the numerator is $2$, you move $2$ segments away from zero in the direction determined by the sign of the number.

Proper fractions, where the numerator is less than the denominator ($p < q$), always lie between $0$ and $1$ (if positive) or between $0$ and $-1$ (if negative). Visually, the point will be located within the very first unit interval from the origin.

Improper fractions, where the numerator is greater than the denominator ($p > q$), are best represented by first converting them into mixed numbers $a \frac{r}{q}$. The integer part '$a$' tells you the whole number of units to move, and the fraction part '$\frac{r}{q}$' shows how much further to go into the next unit interval. For instance, $2 \frac{1}{2}$ is located exactly halfway between $2$ and $3$.

Every rational number corresponds to a unique point on the number line. Conversely, every point on the number line that can be reached by dividing unit lengths into equal parts represents a rational number. This creates a dense set of points where multiple fractions can represent the same point if they are equivalent, such as $\frac{1}{2}$ and $\frac{2}{4}$.