CBSE Class 8 Maths Chapter 1 – Fractions in Disguise Notes
Class 8 Maths Chapter 1 – Fractions in Disguise introduces students to rational numbers through an engaging and conceptual approach. The chapter builds on students' prior knowledge of fractions and integers to develop a complete understanding of rational numbers — numbers that can be expressed as a ratio of two integers. Rather than presenting rational numbers as a dry topic, this chapter uses the idea of "fractions in disguise" to show students that many numbers they already know — integers, decimals, percentages — are actually rational numbers in different forms. The chapter covers the properties of rational numbers, their representation on the number line, and operations on rational numbers.
This chapter forms the foundation of the entire Class 8 Maths course and is essential for all subsequent chapters involving algebraic operations and number comparisons. A thorough understanding of rational numbers and their properties prepares students for Class 9 topics like the real number system and algebraic identities.
Key Concepts and Notes
1. What is a Rational Number?
A rational number is any number that can be written in the form p/q where: p and q are integers q ≠ 0
Examples of rational numbers: 3/4, -5/7, 0, -3, 8, 0.5, -1.25
Every integer is a rational number because any integer n can be written as n/1. Every fraction is a rational number. Decimals that terminate or repeat are rational numbers.
2. Fractions in Disguise
Many numbers that do not look like fractions are actually rational numbers in disguise:
Integers as rational numbers: 5 = 5/1, -7 = -7/1, 0 = 0/1
Decimals as rational numbers: 0.25 = 1/4, 0.333... = 1/3, 1.5 = 3/2
Percentages as rational numbers: 50% = 1/2, 25% = 1/4, 75% = 3/4
This is why the chapter is called "Fractions in Disguise" — all these different forms are rational numbers wearing different costumes.
3. Properties of Rational Numbers
(a) Closure Property Rational numbers are closed under addition, subtraction, and multiplication. For any two rational numbers a and b: a + b is a rational number ✓ a – b is a rational number ✓ a × b is a rational number ✓ a ÷ b is a rational number only when b ≠ 0 ✓
(b) Commutativity Addition: a + b = b + a ✓ Multiplication: a × b = b × a ✓ Subtraction and division are NOT commutative.
(c) Associativity Addition: (a + b) + c = a + (b + c) ✓ Multiplication: (a × b) × c = a × (b × c) ✓ Subtraction and division are NOT associative.
(d) Distributive Property a × (b + c) = a × b + a × c ✓ This property connects multiplication with addition.
(e) Identity Elements Additive identity: 0 — Adding 0 to any rational number gives the same number. a + 0 = a Multiplicative identity: 1 — Multiplying any rational number by 1 gives the same number. a × 1 = a
(f) Inverse Elements Additive inverse: The additive inverse of a/b is -a/b. Their sum = 0. Multiplicative inverse: The multiplicative inverse (reciprocal) of a/b is b/a. Their product = 1. (Valid only when a ≠ 0)
4. Representation on the Number Line
Rational numbers can be plotted on the number line.
Positive rational numbers lie to the right of 0. Negative rational numbers lie to the left of 0. 0 is neither positive nor negative.
Steps to plot p/q on the number line: Step 1 — Divide the unit segment between consecutive integers into q equal parts. Step 2 — Count p parts from 0 in the appropriate direction.
Example: To plot 3/4 on the number line, divide the segment between 0 and 1 into 4 equal parts and mark the 3rd part.
Example: To plot -5/3, divide the segment between -1 and -2 into 3 equal parts and mark the 2nd part from -1 moving left.
5. Rational Numbers Between Two Rational Numbers
Between any two rational numbers, there are infinitely many rational numbers.
Method 1 — Using equivalent fractions: Convert both fractions to the same denominator and list numbers between them. Example: Between 1/3 and 1/2 — convert to 2/6 and 3/6 — rational number between them: 5/12, 4/12, etc.
Method 2 — Using the mean: The average of two rational numbers always lies between them. Mean of a/b and c/d = (a/b + c/d) / 2 This can be repeated to find infinitely many rational numbers between any two.
6. Comparison of Rational Numbers
To compare two rational numbers: Step 1 — Convert both to the same denominator. Step 2 — Compare the numerators.
Example: Compare -3/4 and -5/8 -3/4 = -6/8 -6/8 < -5/8 (since -6 < -5) So -3/4 < -5/8
Important rule: Among negative rational numbers, the one with the larger absolute value is the smaller number.
7. Operations on Rational Numbers
(a) Addition: Same denominator: a/c + b/c = (a+b)/c Different denominators: Find LCM of denominators, convert, then add.
(b) Subtraction: Same denominator: a/c – b/c = (a–b)/c Different denominators: Find LCM, convert, then subtract.
(c) Multiplication: (a/b) × (c/d) = (a×c)/(b×d) Always simplify the result to lowest terms.
(d) Division: (a/b) ÷ (c/d) = (a/b) × (d/c) Dividing by a fraction is the same as multiplying by its reciprocal.
8. Standard Form of a Rational Number
A rational number is in standard form (lowest terms) when: The denominator is positive. The numerator and denominator have no common factor other than 1 (HCF = 1).
Example: -6/(-9) — make denominator positive: 6/9 — divide by HCF 3: 2/3 (standard form)
Important Formulas and Rules
Additive inverse of p/q = –p/q
Multiplicative inverse of p/q = q/p (where p ≠ 0)
Between any two rational numbers a and b, their mean = (a+b)/2 always lies between them
Number of rational numbers between any two rational numbers = infinite
For negative rational numbers: –p/q = p/–q = –(p/q)
Common Mistakes to Avoid
Do not add or subtract fractions with different denominators directly — always find the LCM first.
When comparing negative rational numbers, remember that –3/4 is greater than –5/4 because –3 > –5.
Zero has no multiplicative inverse. Dividing by zero is undefined.
The additive inverse of 0 is 0 itself.
When multiplying a rational number by its reciprocal, the result is always 1, not 0.
Quick Revision Summary
Rational numbers are of the form p/q where q ≠ 0. All integers, fractions, terminating decimals, and repeating decimals are rational numbers. Rational numbers are closed under addition, subtraction, multiplication, and division (except by zero). Between any two rational numbers there are infinitely many rational numbers. The additive identity is 0 and the multiplicative identity is 1. Every rational number has an additive inverse. Every non-zero rational number has a multiplicative inverse.
FAQs – Chapter 1 Fractions in Disguise
Q1. Why is Chapter 1 called "Fractions in Disguise"? The chapter is called Fractions in Disguise because it reveals that many numbers we encounter in everyday life — integers, decimals, percentages — are actually rational numbers written in different forms. Just as a person in disguise is still the same person underneath, all these different-looking numbers are fundamentally fractions (rational numbers) in disguise. This approach makes the introduction to rational numbers more intuitive and engaging.
Q2. Is every integer a rational number? Yes, every integer is a rational number. Any integer n can be written as n/1, which is in the form p/q with q ≠ 0. For example, 5 = 5/1, –7 = –7/1, and 0 = 0/1. This means the set of integers is entirely contained within the set of rational numbers.
Q3. Are there infinitely many rational numbers between any two rational numbers? Yes, there are infinitely many rational numbers between any two rational numbers, no matter how close they are. This is called the density property of rational numbers. For example, between 1/3 and 1/2 there are infinitely many rational numbers such as 5/12, 7/18, 2/5, and so on. You can always find more by taking the average repeatedly.
Q4. What is the difference between additive inverse and multiplicative inverse? The additive inverse of a rational number a/b is –a/b, and their sum is always 0. For example, the additive inverse of 3/4 is –3/4 because 3/4 + (–3/4) = 0. The multiplicative inverse (reciprocal) of a/b is b/a, and their product is always 1. For example, the multiplicative inverse of 3/4 is 4/3 because 3/4 × 4/3 = 1. Zero has an additive inverse (0 itself) but no multiplicative inverse.
Q5. How are rational numbers used in real life? Rational numbers appear everywhere in real life. Fractions are used in cooking and measurements. Decimals are used in money and prices. Percentages are used in interest rates, discounts, and statistics. Negative rational numbers represent temperatures below zero, debts, and altitudes below sea level. Understanding rational numbers is therefore not just an academic exercise but a practical life skill.