Rational Numbers - Definition and Need for Rational Numbers

Definition of Rational Numbers: A rational number is any number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. The word 'rational' arises from the term 'ratio'.

Numerator and Denominator: In the rational number $\frac{p}{q}$, the integer $p$ is the numerator and $q$ is the denominator. Visually, these are separated by a horizontal division bar.

The Need for Rational Numbers: While integers can represent whole values and their opposites, they cannot represent parts of a whole. Rational numbers are needed to represent measurements like $\frac{3}{4}$ km or positions between integers on a scale.

Integers as Rational Numbers: Every integer is a rational number because any integer $a$ can be written as $\frac{a}{1}$. For example, $7 = \frac{7}{1}$ and $-5 = \frac{-5}{1}$.

Positive and Negative Rational Numbers: A rational number is positive if both the numerator and denominator have the same sign (e.g., $\frac{2}{3}$ or $\frac{-2}{-3}$). It is negative if the numerator and denominator have opposite signs (e.g., $\frac{-4}{5}$ or $\frac{4}{-5}$).

Standard Form: A rational number $\frac{p}{q}$ is in standard form if the denominator $q$ is a positive integer and the numerator $p$ and denominator $q$ have no common factor other than $1$.

Number Line Representation: To visualize rational numbers, we use a number line where the distance between two integers is divided into equal parts. For example, to plot $\frac{3}{5}$, the space between $0$ and $1$ is divided into $5$ equal segments, and the $3$rd point from $0$ is marked.

Zero as a Rational Number: The number $0$ is a rational number because it can be expressed as $\frac{0}{1}, \frac{0}{2}, \frac{0}{-5}$, etc., satisfying the condition $q \neq 0$.