What is Rational Numbers Class 8: Definition & Key Concepts

Definition of Rational Numbers for Class 8 Students

Rational numbers are numbers that can be written in the form $\frac{p}{q}$, where:

• $p$ and $q$ are integers
• $q \neq 0$

For example, $\frac{3}{4}$, $-\frac{5}{2}$, and $0$ (which can be written as $\frac{0}{1}$) are rational numbers.

Key points:

• Rational numbers include positive and negative fractions.
• All integers are rational numbers because any integer $a$ can be written as $\frac{a}{1}$.

This definition is fundamental in Class 8 NCERT Mathematics and helps you understand more complex number systems.

Properties of Rational Numbers Explained

Rational numbers have several important properties that make calculations easier:

1. Closure Property: The sum, difference, and product of two rational numbers is always a rational number. 2. Commutative Property: $a + b = b + a$ and $a \times b = b \times a$ for rational numbers. 3. Associative Property: $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$. 4. Existence of Additive Inverse: For every rational number $a$, there exists $-a$ such that $a + (-a) = 0$. 5. Existence of Multiplicative Inverse: For every rational number $a \neq 0$, there exists $\frac{1}{a}$ such that $a \times \frac{1}{a} = 1$.

These properties are essential for performing arithmetic operations confidently.

How to Represent Rational Numbers on the Number Line

Representing rational numbers on the number line helps visualize their value:

• To plot $\frac{p}{q}$, divide the segment between 0 and 1 into $q$ equal parts.
• Move $p$ parts to the right if $p$ is positive, or to the left if $p$ is negative.

Example:

To plot $-\frac{3}{4}$:

• Divide the segment between 0 and -1 into 4 equal parts.
• Move 3 parts to the left of 0.

This method helps in understanding the size and position of rational numbers relative to integers.

Types of Rational Numbers: Positive, Negative, and Zero

Rational numbers can be classified based on their sign:

• Positive Rational Numbers: Numbers greater than zero, e.g., $\frac{2}{3}$, $5$.
• Negative Rational Numbers: Numbers less than zero, e.g., $-\frac{7}{4}$, $-3$.
• Zero: Zero is a rational number since it can be written as $\frac{0}{1}$.

Understanding these types helps in comparing and ordering rational numbers during problem-solving.

Decimal Representation of Rational Numbers

Every rational number can be expressed as a decimal. The decimal form is either:

• Terminating Decimal: The decimal ends after some digits. Example: $\frac{1}{4} = 0.25$.
• Repeating Decimal: A digit or group of digits repeats infinitely. Example: $\frac{1}{3} = 0.333...$

Worked Example:

Convert $\frac{5}{8}$ to decimal:

Divide 5 by 8:

$$5 \div 8 = 0.625$$

Since the division ends, $0.625$ is a terminating decimal.

This understanding is important for converting and comparing rational numbers.

Operations on Rational Numbers with Examples

You can perform addition, subtraction, multiplication, and division on rational numbers using these rules:

• Addition: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$
• Subtraction: $\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$
• Multiplication: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$
• Division: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$ (where $c \neq 0$)

Example: Add $\frac{2}{3}$ and $\frac{3}{4}$:

$$\frac{2}{3} + \frac{3}{4} = \frac{2 \times 4 + 3 \times 3}{3 \times 4} = \frac{8 + 9}{12} = \frac{17}{12}$$

This result is an improper fraction but still a rational number.