What Is Rational Numbers Class 8: Definition and Examples Explained

Definition of Rational Numbers for Class 8 Students

In Class 8 Mathematics, rational numbers are defined as any number that can be written in the form $$\frac{p}{q}$$ where $p$ and $q$ are integers and $q \neq 0$. Here:

• $p$ is called the numerator
• $q$ is called the denominator

For example, $$\frac{3}{4}$$, $$\frac{-5}{2}$$, and $$\frac{0}{7}$$ are rational numbers.

Important: The denominator cannot be zero because division by zero is undefined.

Rational numbers include:

• Positive fractions like $$\frac{2}{3}$$
• Negative fractions like $$\frac{-7}{5}$$
• Whole numbers (since they can be written as $$\frac{n}{1}$$)
• Zero (which is $$\frac{0}{1}$$)

This definition forms the foundation for understanding rational numbers in the NCERT Class 8 syllabus.

Properties of Rational Numbers in Class 8 NCERT

Rational numbers have several important properties that help in performing arithmetic operations:

1. Closure Property: The sum, difference, or product of two rational numbers is always a rational number. 2. Commutative Property:

• Addition: $$a + b = b + a$$
• Multiplication: $$a \times b = b \times a$$

3. Associative Property:

• Addition: $$(a + b) + c = a + (b + c)$$
• Multiplication: $$(a \times b) \times c = a \times (b \times c)$$

4. Distributive Property:

• $$a \times (b + c) = a \times b + a \times c$$

5. Existence of Additive Inverse: For every rational number $a$, there exists $-a$ such that $a + (-a) = 0$. 6. 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 solving problems involving rational numbers in Class 8.

Types of Rational Numbers with Examples

Rational numbers can be classified into different types based on their values:

• Positive Rational Numbers: Numbers greater than zero, e.g., $$\frac{3}{5}$$, 2, 7.8 (as $$\frac{39}{5}$$)
• Negative Rational Numbers: Numbers less than zero, e.g., $$\frac{-4}{9}$$, -3
• Zero: Zero itself is a rational number, expressed as $$\frac{0}{1}$$

Examples:

Understanding these types helps in identifying and working with rational numbers in various problems.

Operations on Rational Numbers: Addition, Subtraction, Multiplication, and Division

Class 8 students must be comfortable performing operations on rational numbers. Here's a quick guide:

Addition and Subtraction

• To add or subtract rational numbers, make their denominators the same.
• Example:

$$\frac{2}{3} + \frac{1}{6} = \frac{4}{6} + \frac{1}{6} = \frac{5}{6}$$

Multiplication

• Multiply the numerators and denominators directly.

$$\frac{3}{4} \times \frac{2}{5} = \frac{3 \times 2}{4 \times 5} = \frac{6}{20} = \frac{3}{10}$$

Division

• Multiply the first rational number by the reciprocal of the second.

$$\frac{3}{7} \div \frac{2}{5} = \frac{3}{7} \times \frac{5}{2} = \frac{15}{14}$$

Worked Example:

Calculate $$\frac{5}{8} - \frac{1}{4}$$

Solution:

• Convert $$\frac{1}{4}$$ to $$\frac{2}{8}$$
• Subtract: $$\frac{5}{8} - \frac{2}{8} = \frac{3}{8}$$

These operations are crucial for solving rational number problems in exams.

Rational Numbers on the Number Line

Class 8 NCERT explains how to represent rational numbers on the number line:

• The number line is a straight line where each point corresponds to a real number.
• Rational numbers can be positive, negative, or zero, so they appear to the right, left, or at the origin respectively.

Steps to plot a rational number $$\frac{p}{q}$$:

1. Divide the segment between 0 and 1 into $q$ equal parts. 2. Count $p$ parts from zero towards the right if $p$ is positive, or towards the left if $p$ is negative. 3. Mark the point.

For example, to plot $$\frac{-3}{4}$$:

• Divide the segment between 0 and -1 into 4 equal parts.
• Count 3 parts from 0 towards the left.
• Mark the point.

This visual understanding helps students grasp the density property: between any two rational numbers, there is always another rational number.

Comparison of Rational Numbers: Which Is Greater?

Comparing rational numbers is important for ordering and problem-solving.

Methods to compare:

| Cross Multiplication | For $$\frac{a}{b}$$ and $$\frac{c}{d}$$, compare $a \times d$ and $b \times c$.

Example:

Compare $$\frac{3}{7}$$ and $$\frac{2}{5}$$ using cross multiplication:

• $3 \times 5 = 15$
• $7 \times 2 = 14$

Since 15 > 14, $$\frac{3}{7} > \frac{2}{5}$$.

This method is quick and effective for Class 8 students.