Rational Numbers - Rational Numbers on a 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 $q \neq 0$. Visually, every rational number corresponds to a unique point on a horizontal line called the number line.

The number line is centered at a point called the origin, represented by the number $0$. All positive rational numbers are located to the right of $0$, and all negative rational numbers are located to the left of $0$. The distance between consecutive integers (like $0$ and $1$) is called a unit length.

To represent a positive proper fraction $\frac{a}{b}$ (where $a < b$), we divide the unit length between $0$ and $1$ into $b$ equal parts. The $a$-th point from $0$ towards the right represents the fraction $\frac{a}{b}$. For example, to show $\frac{1}{2}$, you mark the exact midpoint between $0$ and $1$.

To represent a negative rational number $-\frac{a}{b}$, we move to the left of $0$. If it is a proper fraction, the unit length between $0$ and $-1$ is divided into $b$ equal parts, and we count $a$ parts to the left. The markings on the left are a mirror image of the markings on the right.

Improper fractions (where the numerator is greater than the denominator) are best represented by first converting them into mixed fractions, such as $2 \frac{1}{3}$. This visually indicates that the number lies between the integers $2$ and $3$. On the number line, you locate the integer $2$ and then divide the space between $2$ and $3$ into $3$ equal sections, picking the first mark.

Every rational number has a standard form where the denominator $q$ is a positive integer and $p$ and $q$ have no common factors other than $1$. When plotting, it is helpful to simplify the fraction to its standard form first to determine the correct number of divisions needed between integers.

The concept of 'Density' states that between any two rational numbers on a number line, there are infinitely many other rational numbers. This means you can always zoom in on a segment of the number line and divide it into even smaller equal parts.