Rational Numbers for Class 8 Mathematics

Rational numbers are numbers expressible as fractions p/q, with integers p and q (q ≠ 0). This topic expands the number system beyond whole numbers.

Core Concept

Understanding rational numbers is essential for arithmetic operations and algebraic thinking in higher grades.

Key Aspects

• Definition and examples of rational numbers.
• Representation on the number line.
• Basic properties and forms.

Here we have provided NCERT notes for Class 8 Mathematics in english Language, Just select the chapters below to get notes of the same:

Rational Numbers

Linear Equations in One Variable

Understanding Quadrilaterals

Practical Geometry

Data Handling

Understanding Rational Numbers

Rational numbers form a vital part of the number system in Class 8 Maths. They include integers, fractions, and terminating or repeating decimals. A rational number is defined as a number that can be written as p/q, where p and q are integers and q is not zero.

Properties of Rational Numbers

Rational numbers exhibit several fundamental properties that simplify calculations.

• Closure Property: The sum, difference, or product of two rational numbers is always a rational number.
• Commutative Property: Addition and multiplication are commutative for rational numbers, meaning a+b = b+a and a×b = b×a.
• Associative Property: Addition and multiplication are associative, so (a+b)+c = a+(b+c) and (a×b)×c = a×(b×c).
• Distributive Property: Multiplication distributes over addition, i.e., a×(b+c) = a×b + a×c.
• Identity Elements: 0 is the additive identity (a+0=a), and 1 is the multiplicative identity (a×1=a).
• Inverse Elements: Every rational number a has an additive inverse -a (a+(-a)=0), and every non-zero rational number a has a multiplicative inverse 1/a (a×(1/a)=1).

Operations on Rational Numbers

Addition and Subtraction

To add or subtract rational numbers, ensure they have a common denominator. If not, find the least common multiple (LCM) of denominators and convert them.

Example: Subtract 3/4 from 5/6. LCM of 4 and 6 is 12. So, 5/6 = 10/12 and 3/4 = 9/12. Therefore, 10/12 - 9/12 = 1/12.

Multiplication and Division

Multiply rational numbers by multiplying numerators and denominators separately. For division, multiply by the reciprocal of the divisor.

Example: Multiply 2/5 by -3/7. (2/5) × (-3/7) = (2×-3)/(5×7) = -6/35.

Example: Divide 4/9 by 2/3. (4/9) ÷ (2/3) = (4/9) × (3/2) = 12/18 = 2/3 after simplification.

Representation on Number Line

Rational numbers can be plotted on a number line by dividing segments into equal parts. For instance, to represent 2/3, divide the unit segment between 0 and 1 into three equal parts and mark the second point.

Equivalent Rational Numbers

Rational numbers are equivalent if they reduce to the same simplest form. For example, 2/4, 3/6, and 4/8 are all equivalent to 1/2. Simplify by dividing numerator and denominator by their greatest common divisor (GCD).

Standard Form of Rational Numbers

A rational number is in standard form when its denominator is a positive integer, and the numerator and denominator have no common factors other than 1. For example, -5/7 is in standard form, but 6/8 simplifies to 3/4.

Comparing Rational Numbers

To compare two rational numbers, convert them to have the same denominator or use cross-multiplication. For example, compare 5/8 and 3/5. Cross-multiply: 5×5=25 and 8×3=24. Since 25>24, 5/8 > 3/5.

Applications and Practice

Rational numbers are used in real-life scenarios like sharing quantities, calculating speeds, or managing finances. Solving NCERT exercises helps reinforce these concepts through problems on operations, comparison, and simplification.

Mastery of rational numbers prepares students for advanced topics like linear equations and data handling in subsequent chapters.