Chapter 3 of 14

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CBSE Class 8 Mathematics★ 10-12 marks · CBSE Class 8 Maths (Ganita Prakash Chapter 3)

A Story of Numbers

'A Story of Numbers' traces humanity's journey through natural numbers, integers, rational numbers, and irrational numbers — exploring properties, the number line, and the magnificent Indian contribution to global mathematics (zero, decimal system, Aryabhata, Brahmagupta, Bhaskara). Master rational number operations, decimal expansions, and the place value of India in number-history. CBSE Class 8 Maths (Ganita Prakash) Chapter 3.

⏱ 25 min readEasy difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Classify numbers as natural, whole, integer, rational, irrational, or real
2Express rationals in standard form
3Apply properties (closure, commutativity, associativity, distributivity)
4Perform all four operations on rationals fluently
5Identify terminating vs non-terminating decimal expansions
6Appreciate India's historical role in number systems

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Why this chapter matters

Foundation for all higher mathematics. Master rational numbers, their properties, operations, and the Indian historical contribution to numbers (zero, decimal system, Aryabhata, Brahmagupta, Bhaskara, Madhava, Ramanujan).

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

Class 7 Rational Numbers

Foundation extended here

A Story of Numbers — Class 8 Mathematics (Ganita Prakash)

"Numbers are the children of the human mind. They were not discovered like rivers; they were invented like languages. India gave them their alphabet — the zero, the decimal place, and the symbols 0-9."

1. About the Chapter

'A Story of Numbers' is one of the most beautiful and important chapters in Ganita Prakash. It teaches you:

The expanding world of numbers — from natural numbers to real numbers
Operations on rational numbers — addition, subtraction, multiplication, division
Properties — closure, commutativity, associativity, distributivity
The Indian story — how numerals, zero, and the decimal system were born in India

Key Idea

Numbers are not random. They form a family tree — each new kind of number extends the previous, allowing more operations to be performed.

2. The Family of Numbers

Natural Numbers (N)

Counting numbers: 1, 2, 3, 4, 5, ...
Used to count physical objects
No zero, no negatives

Whole Numbers (W)

Natural numbers + 0
W = {0, 1, 2, 3, 4, ...}
Indian gift: The zero was developed in India around the 5th-6th century CE

Integers (Z)

Whole numbers + negatives
Z = {... −3, −2, −1, 0, 1, 2, 3 ...}
Enables subtraction (5 − 8 = −3)
The Indian mathematician Brahmagupta (7th century CE) was the first to systematically treat negative numbers

Rational Numbers (Q)

Numbers of the form p/q where p, q are integers, q ≠ 0
Examples: 1/2, 3/4, −7/3, 5 (= 5/1), 0 (= 0/1)
Enables division (5 ÷ 8 = 5/8)
Q comes from 'quotient'

Irrational Numbers

Numbers that CANNOT be written as p/q
Examples: √2, √3, π, e
Their decimal expansions are non-terminating and non-repeating
Studied by Indian mathematicians in the Sulba Sutras (800 BCE)

Real Numbers (R)

All rational + all irrational numbers
Every point on the number line corresponds to a real number
Class 9 chapter: 'Number Systems' goes deeper

3. Rational Numbers — Detailed Study

Definition

A rational number is any number that can be expressed as p/q, where:

p (numerator) and q (denominator) are integers
q ≠ 0

Standard Form

A rational number p/q is in standard form if:

p and q have no common factor (other than 1)
q is positive

Examples:

6/8 is not standard. Reduce: 6/8 = 3/4 (standard form)
−6/8 → −3/4 (standard form)
5/−7 → −5/7 (standard form — denominator positive)

Equivalent Rational Numbers

Multiply both numerator and denominator by the same non-zero integer:

1/2 = 2/4 = 3/6 = −5/−10
These are all the same rational number

Rational Numbers on the Number Line

Every rational number has a unique position on the number line
1/2 is exactly halfway between 0 and 1
−3/4 is between −1 and 0, three-quarters of the way to −1
Between any two rational numbers, there are infinitely many other rational numbers — this is called the 'density property'

4. Properties of Rational Numbers

Property 1: Closure

A set is closed under an operation if the result of the operation on its elements stays in the set.

Operation	Natural	Whole	Integer	Rational
Addition (+)	Yes	Yes	Yes	Yes
Subtraction (−)	No	No	Yes	Yes
Multiplication (×)	Yes	Yes	Yes	Yes
Division (÷)	No	No	No	Yes (excl. ÷0)

Rational numbers are closed under +, −, ×, and ÷ (except by zero).

Property 2: Commutativity

For rationals a and b:

a + b = b + a (commutative under +)
a × b = b × a (commutative under ×)

But:

a − b ≠ b − a (NOT commutative under −)
a ÷ b ≠ b ÷ a (NOT commutative under ÷)

Property 3: Associativity

For rationals a, b, c:

(a + b) + c = a + (b + c) (associative under +)
(a × b) × c = a × (b × c) (associative under ×)

NOT associative under − or ÷.

Property 4: Distributivity

a × (b + c) = a × b + a × c

Example: 2/3 × (1/2 + 1/4) = 2/3 × 1/2 + 2/3 × 1/4 = 1/3 + 1/6 = 1/2

Property 5: Identity Elements

Additive identity: 0 (because a + 0 = a)
Multiplicative identity: 1 (because a × 1 = a)

Property 6: Inverse Elements

Additive inverse of a is −a (because a + (−a) = 0)
Multiplicative inverse of a is 1/a (because a × 1/a = 1), for a ≠ 0

5. Operations on Rational Numbers

Addition

Same denominator: Add numerators, keep denominator.

3/7 + 2/7 = 5/7

Different denominators: Find LCM, convert to same denominator, then add.

1/2 + 1/3 = 3/6 + 2/6 = 5/6

Subtraction

Same as addition with opposite sign.

5/8 − 3/8 = 2/8 = 1/4
3/4 − 1/3 = 9/12 − 4/12 = 5/12

Multiplication

Multiply numerators; multiply denominators.

2/3 × 4/5 = 8/15
3/4 × 5/6 = 15/24 = 5/8 (after simplification)

Division

Multiply by reciprocal.

2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
6/7 ÷ 3 = 6/7 × 1/3 = 6/21 = 2/7

6. Representation on the Number Line

Plotting Rational Numbers

Example: Plot 5/3 on the number line.

5/3 = 1 + 2/3 = 1 and 2/3
Lies between 1 and 2, two-thirds of the way to 2

Example: Plot −3/4 on the number line.

Lies between −1 and 0, three-quarters of the way to −1

Finding Rationals Between Two Numbers

Method: Take the average.

A rational between 1/4 and 1/2: average = (1/4 + 1/2) / 2 = (1/4 + 2/4) / 2 = 3/4 / 2 = 3/8

Between 3/8 and 1/2: average = (3/8 + 4/8) / 2 = 7/8 / 2 = 7/16

You can keep doing this — there are INFINITELY many rationals between any two.

7. Decimal Expansion of Rationals

Two Types

Every rational number's decimal expansion is either:

Terminating — ends in finite digits
Non-terminating, repeating — has a repeating block

Rule

A rational p/q (in standard form) terminates ↔ the denominator q has ONLY 2s and 5s as prime factors.

Examples:

1/2 = 0.5 (terminates; q = 2)
1/4 = 0.25 (terminates; q = 2²)
1/5 = 0.2 (terminates; q = 5)
1/8 = 0.125 (terminates; q = 2³)
3/20 = 0.15 (terminates; q = 2² × 5)
1/3 = 0.333... = 0.3̄ (non-terminating repeating; q = 3)
1/6 = 0.1666... = 0.16̄ (non-terminating repeating; q = 2 × 3 — has 3)
2/7 = 0.285714285714... = 0.285714̄ (block of 6 digits repeats)

8. Irrational Numbers — A Glimpse

What Are They?

Numbers that CANNOT be written as p/q. Their decimal expansion is non-terminating AND non-repeating.

Famous Examples

√2 ≈ 1.41421356237... (Pythagoras, 5th century BCE)
√3 ≈ 1.73205080757...
π (pi) ≈ 3.14159265358... (ratio of circumference to diameter)
e (Euler's number) ≈ 2.71828182845...
φ (golden ratio) ≈ 1.61803398875...

Why They Matter

π appears in every circular calculation
√2 appears in every right-angled isosceles triangle
These numbers are EVERYWHERE in nature and engineering
Yet you can never write them exactly as fractions!

9. The Indian Story — Why We Owe So Much

The Decimal Place-Value System

The system where the position of a digit determines its value (123 means 1×100 + 2×10 + 3)
Developed in India by ~500 CE
Travelled through Arabic mathematicians to Europe (~1000 CE)
This is why our numerals are called 'Hindu-Arabic numerals' worldwide

Zero — India's Gift to the World

Earliest documented use of zero as a number: Brahmagupta (628 CE) in 'Brahma-Sphuta-Siddhanta'
Brahmagupta defined arithmetic rules for zero and negative numbers
The symbol '0' evolved over centuries — possibly inspired by 'sunya' (Sanskrit for emptiness)

Aryabhata (476-550 CE)

Place-value notation
Methods for square roots, cube roots
Approximated π as 3.1416 (very close to true value)

Brahmagupta (598-668 CE)

First systematic treatment of zero and negative numbers
Rules for arithmetic with negatives (e.g., −2 × −3 = 6)
Quadratic equation solutions

Bhaskara II (1114-1185 CE)

Author of 'Lilavati' (named after his daughter)
'Lilavati' is a mathematics textbook in poetic form
Worked on calculus concepts centuries before Newton/Leibniz
Asserted: division by zero gives infinity

Madhava of Sangamagrama (1340-1425 CE)

Founder of the Kerala School of Mathematics
Discovered infinite series for π, sine, cosine
His work anticipated calculus by 200+ years

Modern Indian Mathematics

Srinivasa Ramanujan (1887-1920) — 'man who knew infinity'
Manjul Bhargava — Fields Medal 2014, of Indian-American origin
India has produced and continues to produce world-class mathematicians

10. Worked Examples

Example 1: Standard Form

Convert 14/−21 to standard form.

Make denominator positive: −14/21
HCF of 14 and 21 is 7
Divide: −14/7 = −2, 21/7 = 3
Standard form: −2/3

Example 2: Simplification

Simplify: (3/4) + (5/6) − (1/3)

LCM of 4, 6, 3 is 12
= 9/12 + 10/12 − 4/12 = 15/12 = 5/4

Example 3: Find Three Rationals Between 1/4 and 1/2

1/4 = 3/12, 1/2 = 6/12
Rationals between: 4/12, 5/12
For three: convert to /24. 1/4 = 6/24, 1/2 = 12/24
Rationals between: 7/24, 8/24, 9/24, 10/24, 11/24 (5 options!)
Pick any three: 7/24, 9/24, 11/24

Example 4: Multiplication

Multiply: (−3/4) × (8/9)

= (−3 × 8) / (4 × 9) = −24/36
Simplify: −2/3

Example 5: Division by a Rational

Divide: (5/12) ÷ (−10/3)

= 5/12 × 3/−10
= (5 × 3) / (12 × −10) = 15/−120 = −1/8

Example 6: Decimal Expansion

Express 5/16 as a decimal.

5/16 = 5/16. Long division: 5 ÷ 16 = 0.3125 (terminates)
(Check: 16 = 2⁴, only 2s. So terminates.)

Example 7: Distributivity

Verify: (2/3) × ((4/5) + (3/10)) = (2/3) × (4/5) + (2/3) × (3/10)

LHS: (2/3) × (8/10 + 3/10) = (2/3) × (11/10) = 22/30 = 11/15
RHS: 8/15 + 6/30 = 16/30 + 6/30 = 22/30 = 11/15 ✓

11. Common Mistakes

Forgetting to make denominator positive in standard form
- 5/−7 should be −5/7
Wrong sign rules (Brahmagupta's rules!):
- (−) × (−) = (+)
- (−) × (+) = (−)
- (+) ÷ (−) = (−)
Adding without common denominator
- 1/2 + 1/3 ≠ 2/5; it equals 5/6
Dividing by 0
- NEVER allowed; mathematically undefined
Confusing repeating decimals
- 0.3̄ = 0.333... ≠ 0.3
Order of operations in mixed expressions
- Always follow BODMAS / PEMDAS

12. Key Concepts for Future Chapters

Closure & Density

You'll use these in Class 9 and 10 when proving theorems about number systems.

Properties (commutative, associative, distributive)

Used everywhere in algebra.

Decimal Expansions

Critical for understanding real numbers in Class 9.

Negative & Reciprocal

Foundation of algebra equation-solving.

13. Tips for Mastery

For Operations

Always reduce to standard form at the end
LCM first for addition/subtraction
Multiply numerators, multiply denominators for multiplication
Reciprocal for division

For Properties

Memorise the property names — exam questions often quote properties directly
Verify with examples — pick 2-3 rationals, test the property

For Number-Line Work

Convert mixed numbers (5/3 = 1 + 2/3) for plotting
Negative numbers go LEFT of 0

14. Conclusion

'A Story of Numbers' is not just about computations — it is about the human journey to understand quantity. From counting sheep in 5000 BCE to algebra at 500 CE to abstract number theory today, this story spans the entire history of human thought.

India's role in this story is central:

We invented the zero
We invented the decimal place-value system
We treated negative numbers systematically first
Our mathematicians anticipated calculus by centuries

When you write a number — any number — you are using Indian mathematical heritage. When you solve an equation involving fractions or negatives, you are using ideas that started right here.

Master this chapter not just for exams — master it because numbers are the language of the universe, and India helped write that language.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Rational number

p/q where p, q ∈ Z, q ≠ 0

Standard form

p/q with HCF(p,q)=1 and q > 0

Addition (same denom)

a/b + c/b = (a+c)/b

Addition (different denom)

a/b + c/d = (ad+bc)/bd

Multiplication

a/b × c/d = (ac)/(bd)

Division

a/b ÷ c/d = a/b × d/c = (ad)/(bc)

Multiply by reciprocal

Additive identity

a + 0 = a

0 is identity for +

Multiplicative identity

a × 1 = a

1 is identity for ×

Additive inverse

a + (−a) = 0

Multiplicative inverse

a × (1/a) = 1, for a ≠ 0

Distributive

a × (b + c) = a × b + a × c

Terminating decimal rule

p/q (standard form) terminates iff q has only 2s and 5s as prime factors

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Standard form with negative denominator

✓ 5/(-7) is NOT standard. Make denominator positive: -5/7.

WATCH OUT

✗ Adding fractions without LCM

✓ 1/2 + 1/3 = 5/6 (after taking LCM = 6). NOT 2/5.

WATCH OUT

✗ Sign errors in operations

✓ Apply Brahmagupta's rules: (−)×(−) = +, (−)×(+) = −, etc.

WATCH OUT

✗ Dividing by zero

✓ NEVER. Division by zero is UNDEFINED in mathematics.

WATCH OUT

✗ Confusing rational with irrational

✓ Rational = p/q. Irrational CANNOT be written as p/q (e.g., √2, π).

WATCH OUT

✗ Wrong terminating-decimal test

✓ Check ONLY denominator (in standard form). If it has only 2s and 5s, it terminates.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Try These - Properties

Verifying closure, commutativity, associativity

Questions

Try These - Operations

Addition, subtraction, multiplication, division

Questions

Try These - Number Line

Plotting rationals; rationals between two numbers

Questions

Ex 3.1

Exercise 3.1

Operations and standard form

Questions

Ex 3.2

Exercise 3.2

Properties verification

Questions

Ex 3.3 (Application)

Exercise 3.3 (Application)

Real-world problems involving rationals

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Standard form

Express 24/(-36) in standard form.

▶ Show solution

✦ Answer: Make denominator positive: -24/36. HCF(24,36) = 12. Divide both: -24/12 = -2; 36/12 = 3. Standard form: -2/3.

Q2EASY· Operation

Compute: (3/5) + (2/5)

▶ Show solution

✦ Answer: Same denominator. Add numerators: (3+2)/5 = 5/5 = 1.

Q3MEDIUM· Multiplication

Verify the distributive property for: (2/3) × (1/2 − 3/4).

▶ Show solution

Step 1 — LHS computation.
  Bracket first: 1/2 − 3/4. LCM = 4. So 2/4 − 3/4 = −1/4.
  Now multiply: (2/3) × (−1/4) = −2/12 = −1/6.

Step 2 — RHS computation (distributing).
  (2/3) × (1/2) − (2/3) × (3/4)
  = 2/6 − 6/12
  = 1/3 − 1/2 (after simplification)
  LCM = 6: 2/6 − 3/6 = −1/6.

Step 3 — Compare.
  LHS = −1/6 ✓
  RHS = −1/6 ✓
  EQUAL — distributive property verified.

✦ Answer: Both LHS and RHS equal −1/6. The distributive property a × (b − c) = a × b − a × c holds for rationals.

Q4MEDIUM· Number line

Find three rational numbers between 1/3 and 1/2.

▶ Show solution

Step 1 — Convert to common denominator.
  1/3 = 4/12, 1/2 = 6/12.
  Only one number between (5/12), not three. Need larger denominator.

Step 2 — Multiply numerator and denominator by 4.
  1/3 = 16/48, 1/2 = 24/48.
  Now MANY rationals between.

Step 3 — Pick three.
  17/48, 19/48, 21/48 — any three work.

Step 4 — Verify each is between 1/3 and 1/2.
  16/48 = 1/3 ✓ (boundary)
  17/48: 17/48 > 16/48 ✓ and 17/48 < 24/48 ✓ → between.
  Similarly for 19/48 and 21/48.

✦ Answer: Three rationals between 1/3 and 1/2 are 17/48, 19/48, and 21/48. (Many other answers possible — between any two rationals there are infinitely many others.)

Q5HARD· Application

A train travels 3/4 of a journey in 2 1/2 hours. At the same speed, how long would the full journey take? Express the time as a rational number in standard form.

▶ Show solution

Step 1 — Set up the problem.
  Distance covered: (3/4) of total journey
  Time taken: 2 1/2 hours = 5/2 hours

Step 2 — Find time per unit fraction.
  Time for 3/4 of journey = 5/2 hours
  Time for 1/4 of journey = (5/2) ÷ 3 = 5/6 hours

Step 3 — Find time for full journey.
  Total = 4 × (5/6) = 20/6 hours
  Simplify: 20/6 = 10/3 hours

Step 4 — Verify.
  Check: 3/4 of 10/3 = (3 × 10)/(4 × 3) = 30/12 = 5/2 ✓

Step 5 — Express in mixed number form.
  10/3 = 3 1/3 hours = 3 hours 20 minutes

✦ Answer: The full journey takes 10/3 hours = 3 1/3 hours = 3 hours 20 minutes. In standard rational form: 10/3.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Number family: Natural ⊂ Whole ⊂ Integer ⊂ Rational ⊂ Real
•Rational = p/q, q ≠ 0 (p, q integers)
•Standard form: HCF(p,q) = 1, q > 0
•Closure: Rationals closed under +, −, ×, ÷ (not by 0)
•Commutativity: + and × only
•Associativity: + and × only
•Distributivity: a(b ± c) = ab ± ac
•Additive identity: 0; Multiplicative identity: 1
•Additive inverse of a is −a; Multiplicative inverse of a is 1/a
•Add/Sub: take LCM, add numerators
•Multiply: numerator × numerator, denominator × denominator
•Divide: multiply by reciprocal
•Decimal: terminates iff denominator (standard form) has only 2s and 5s
•Between two rationals: infinitely many others (density)
•Indian heritage: zero, decimal system, negative numbers, Aryabhata, Brahmagupta, Bhaskara
•Brahmagupta (628 CE): first systematic treatment of zero and negatives
•Madhava: Kerala School, infinite series for π — anticipated calculus

CBSE marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 10-12 marks per chapter

Question type	Marks each	Typical count	What it tests
MCQ / Very Short	1	2-3	Standard form; identifying rational/irrational; basic operations
Short Answer	2-3	2	Operations; finding numbers between; verifying properties
Long Answer	5	1	Word problems; distributive property; multi-step calculations

Prep strategy

Memorise rules: closure, commutativity, associativity, distributivity
Practise 20 rational number operations (mixed) per week
Master LCM and HCF for fractions
Know terminating decimal rule (only 2s and 5s in denominator)
Practice finding rationals between two numbers (use larger denominators)
Know Indian historical contributions for bonus marks

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Banking and finance

Interest rates, currency conversion, share prices — all use rationals. Negative numbers track debt; decimals track precise values.

Cooking and recipes

1/2 cup, 3/4 teaspoon — rational numbers used daily. Scaling recipes requires multiplying rationals.

Probability and statistics

Probabilities are rationals between 0 and 1 (e.g., 3/8 chance of heads-heads-heads in 3 tosses).

Engineering tolerances

Manufacturing requires precision like 5.327 mm ± 0.001 mm — only possible with rational/decimal numbers.

Music theory

Musical intervals are ratios — 2:1 (octave), 3:2 (fifth). Pythagorean tuning is built on rationals.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Write down rationals' 6 properties at start (memory dump)
Convert to standard form FIRST
Use LCM for adding/subtracting different denominators
Verify by working backwards (substitute answer into the equation)
For verification questions, compute BOTH sides carefully
Include India's historical context if asked about number system origins

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Stern-Brocot tree of rationals — beautiful structure
Continued fractions and their properties
Pythagorean triples and rational right triangles
Brahmagupta's identity and quadratic forms
Madhava's series for π: π/4 = 1 − 1/3 + 1/5 − 1/7 + ...
Read about Ramanujan's amazing infinite series for π

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

CBSE Class 8 School ExamVery High

Class 8 Maths Olympiad (IMO)High

NTSE Mental AbilityVery High

Class 9 Number SystemsVery High — direct prerequisite

NMTC (Maths Talent)High

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

If a/0 = b, then a = 0 × b = 0. So unless a = 0, no value of b works. For 0/0, ANY value of b works (e.g., 0×3=0, 0×5=0) — so it's INDETERMINATE. Mathematicians decided to forbid division by 0 to keep arithmetic consistent.

NO. Only TERMINATING or REPEATING decimals are rational. Non-terminating, non-repeating decimals (like √2, π) are IRRATIONAL. The set of decimals splits into rational and irrational — together forming the real numbers.

Several major contributions: (1) The SYMBOL 0 and its use as a number (Brahmagupta 628 CE). (2) The DECIMAL place-value system (~500 CE). (3) Systematic rules for NEGATIVE numbers and zero arithmetic. (4) Methods for square roots, cube roots (Aryabhata). (5) Quadratic equation solutions (Brahmagupta). (6) Infinite series for π and trig functions (Madhava, Kerala School, 14th c.) — 200 years before Newton/Leibniz!

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