Class 8 Maths Chapter 3: A Story of Numbers Early Number Systems Explained

 Class 8 Maths – Chapter 3: A Story of Numbers
(Ganita Prakash Part 1)

Introduction

Chapter 3 takes us on a fascinating journey through the history of numbers — from sticks and tally marks to Roman numerals, Egyptian numerals, and base‑n systems. Students learn how humans developed counting methods, how number systems evolved, and why the Hindu‑Arabic system became the most efficient.

Basic Knowledge Required

• Counting and grouping

• Understanding of symbols

• Place value basics

• Simple addition and subtraction

• Concept of one‑to‑one correspondence

Important Definitions

1. Number System

A standard sequence of symbols, sounds, or objects used for counting.

2. Numerals

Written symbols used to represent numbers (e.g., 0–9 in the Hindu system).

3. One‑to‑One Mapping

Associating each object with exactly one symbol or item.

4. Landmark Numbers

Special numbers used as building blocks in a number system (e.g., I, V, X, L in Roman numerals).

5. Base‑n System

A number system where each landmark number is a power of n.

Formulas / Concepts Used

• Grouping numbers using landmark values

• Roman numeral rules (additive & subtractive)

• Base‑n representation using powers of n

• One‑to‑one mapping for counting

• Tally marks for representing quantities

Solved Examples (Step‑by‑Step)

Example 1: Represent 27 in Roman numerals

27 = 10 + 10 + 5 + 1 + 1

→ XXVII

Example 2: Represent 324 in Egyptian numerals

324 = 100 + 100 + 100 + 10 + 10 + 4

→ 𓏺𓏺𓏺 𓎆𓎆 IIII

(Insert Image No. from textbook)

Example 3: Represent 143 in base‑5 system

Largest landmark < 143 is 5³ = 125

143 = 125 + 5 + 5 + 5 + 1 + 1 + 1

→ 1 symbol of 125, 3 symbols of 5, 3 symbols of 1

(Insert Image No.)

Example 4: Add Egyptian numerals

(Example from textbook)

Count total symbols → regroup using powers of 10

Final answer shown using Egyptian symbols

FIGURE IT OUT — COMPLETE SOLUTIONS

SECTION 3.1 — Mechanism of Counting

1. Using sticks for operations

• Addition: Combine both collections of sticks.

• Subtraction: Remove sticks equal to the second number.

• Multiplication: Repeat the first collection as many times as the second number.

• Division: Group sticks into equal sets.

2. Extending letter‑based system

Use combinations like:

a, b, c… z, aa, ab, ac… az, ba, bb…

This allows infinite numbers.

3. Create your own number system

Students may choose:

• Shapes

• Colors

• Patterns

• Sounds

• Body gestures

SECTION 3.2 — Early Number Systems

Roman Numerals — Figure It Out

1. Convert to Roman numerals

(i) 1222 → MCCXXII

(ii) 2999 → MMCMXCIX

(iii) 302 → CCCII

(iv) 715 → DCCXV

Addition in Roman numerals

(b) LXXXVII + LXXVIII

87 + 78 = 165

165 = CLXV

Why different number names for different objects?

Because:

• Different objects require different grouping

• Cultural practices vary

• Easier to count specific items

Gumulgal System — Arithmetic

Let ukasar = 2, urapon = 1

(i) (2+2+2+2+1) + (2+2+2+1)

= 9 → ras (any number > 6)

(ii) (9) – (6) = 3 → ukasar‑urapon

(iii) 9 × 4 = 36 → ras

(iv) 16 + 4 = 20 → ras

Features of Hindu number system

• Uses only 10 digits

• Place value system

• Zero as placeholder

• Efficient for large numbers

• Easy for arithmetic

SECTION 3.3 — Base Systems

Egyptian System — Figure It Out

1. Convert to Egyptian numerals

10458

1023

2660

784

1111

70707

2. What numbers do these represent?

(i) 𓎆𓎆𓎆𓎆𓏺𓏺𓏺𓏺𓏺𓏺𓏺𓏺𓏺 → 999

(ii) 𓎆𓎆𓎆 → 300

Base‑5 System — Figure It Out

1. Convert to base‑5

Use powers: 1, 5, 25, 125…

15 = 5 + 5 + 5

50 = 25 + 25

137 = 125 + 5 + 5 + 1 + 1

293 = 125 + 125 + 25 + 5 + 5 + 5 + 1 + 1 + 1

651 = 625 + 25 + 1

(Insert symbols from Table 2)

2. Can any number be represented?

Yes — because powers of 5 can represent all numbers.

3. Landmark numbers of base‑7

1, 7, 49, 343, 2401…

Advantages of Base‑n Systems

• Easy regrouping

• Simple addition

• Simple multiplication

• Fewer symbols needed

• Works for large numbers

Addition in Base‑5 — Figure It Out

Add symbols → regroup every 5 into next landmark

Conclusion

This chapter beautifully explains how number systems evolved across civilizations and why the Hindu‑Arabic system became the most efficient. Students gain a deep appreciation for counting, grouping, and base systems.

Visit: www.fuzymathacademy.com

FAQs – Chapter 3: A Story of Numbers

1. What is a number system?

A number system is a standard sequence of symbols, sounds, or objects used for counting.

2. What is one-to-one mapping?

It is pairing each object with exactly one symbol or item, used for counting.

3. What are numerals?

Numerals are written symbols representing numbers, such as 0–9.

4. Why did ancient people use sticks for counting?

Because sticks allowed easy one-to-one mapping with objects being counted.

5. What is the limitation of using letters for counting?

A single alphabet set cannot represent large numbers without combinations.

6. What are Roman numerals?

A number system using symbols like I, V, X, L, C, D, M.

7. How is 27 written in Roman numerals?

27 = XXVII.

8. What is a landmark number?

A special number used as a building block in a number system.

9. What is a base-n system?

A system where each landmark number is a power of n.

10. What are the landmark numbers of base-5?

1, 5, 25, 125, 625…

11. How do Egyptians write numbers?

Using symbols for powers of 10 (1, 10, 100, 1000…).

12. What is the advantage of base-n systems?

They make addition and multiplication easier through regrouping.

13. How do tally marks work?

Each mark represents one object; groups show totals.

14. Why is the Hindu number system efficient?

It uses place value, zero, and only 10 digits to represent all numbers.

15. How is 1222 written in Roman numerals?

1222 = MCCXXII.