What Is Rational Number Class 8 Answer: Definition & Examples

Definition of Rational Numbers for Class 8

A rational number is defined as any number that can be written in the form of a fraction $$\frac{p}{q}$$, where:

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

This means every rational number can be expressed as a ratio of two integers. For example, $\frac{3}{4}$, $-\frac{5}{2}$, and $0$ (which is $\frac{0}{1}$) are all rational numbers. The Class 8 NCERT textbook introduces this concept to help students understand different types of numbers beyond just whole numbers and integers.

Properties of Rational Numbers Explained

Rational numbers have several important properties that make them easy to work with:

• Closure Property: The sum, difference, and product of any two rational numbers is always a rational number.
• Commutative Property: Addition and multiplication of rational numbers are commutative, meaning $a + b = b + a$ and $a \times b = b \times a$.
• Associative Property: Addition and multiplication are associative: $(a + b) + c = a + (b + c)$.
• Existence of Additive Inverse: For every rational number $\frac{p}{q}$, there exists $-\frac{p}{q}$ such that their sum is zero.
• Existence of Multiplicative Inverse: For every rational number except zero, there exists its reciprocal $\frac{q}{p}$ such that their product is 1.

Understanding these properties helps in solving problems efficiently.

Types of Rational Numbers with Examples

Rational numbers can be:

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

Decimal Representation

Rational numbers can also be expressed as decimals which are either:

• Terminating Decimals: Decimals that end after a finite number of digits, e.g., $0.75 = \frac{3}{4}$.
• Repeating Decimals: Decimals that have a repeating pattern, e.g., $0.333... = \frac{1}{3}$.

How to Identify Rational Numbers: Quick Tips

To identify if a number is rational, check the following:

• Can it be written as a fraction $\frac{p}{q}$ with integers $p$, $q$ and $q \neq 0$?
• Is its decimal form terminating or repeating?
• Is it an integer (all integers are rational since $n = \frac{n}{1}$)?

Examples:

• $\sqrt{2}$ is not rational because it cannot be expressed as a fraction of integers.
• $0.125$ is rational because it equals $\frac{1}{8}$.

Worked Example

Is $-\frac{7}{3}$ a rational number?

• Yes, because it is expressed as a fraction of two integers with denominator not zero.
• Therefore, $-\frac{7}{3}$ is rational.

Operations on Rational Numbers: Addition and Multiplication

Class 8 NCERT Maths teaches how to perform operations on rational numbers:

Addition of Rational Numbers

To add two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$:

$$ \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} $$

Example:

$$ \frac{2}{5} + \frac{3}{10} = \frac{2 \times 10 + 3 \times 5}{5 \times 10} = \frac{20 + 15}{50} = \frac{35}{50} = \frac{7}{10} $$

Multiplication of Rational Numbers

Multiply numerator with numerator and denominator with denominator:

$$ \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} $$

Example:

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

These operations follow the properties of rational numbers and are essential for solving algebraic problems.

Why Rational Numbers Matter in Class 8 Mathematics

Understanding rational numbers is crucial for Class 8 students because:

• They form the basis for advanced topics like algebra, geometry, and number theory.
• Many real-life quantities such as measurements, probabilities, and financial calculations use rational numbers.
• NCERT exercises on rational numbers develop problem-solving skills and logical thinking.
• Mastery of rational numbers helps in scoring well in CBSE exams.

By focusing on the definition, properties, and operations of rational numbers, students build a strong foundation for higher classes.