Short Introduction

The chapter “A Story of Numbers” explains the fascinating history and evolution of number systems from ancient civilizations to the modern Hindu-Arabic number system. Students learn about Roman numerals, Egyptian numerals, Mesopotamian systems, place value systems, bases, and the importance of zero in mathematics.

Quick Information Box

Particular	Details
Chapter Name	A Story of Numbers
Class	Grade 8
Subject	Mathematics
Main Topics	Number Systems, Roman Numerals, Bases
Difficulty Level	Moderate
Important For	School Exams, Olympiads
Key Skills	Logical Thinking, Number Representation

Concepts Used (Topics Covered)

Evolution of Numbers
Hindu-Arabic Numerals
Roman Numerals
Egyptian Number System
Mesopotamian Number System
Base Number Systems
Decimal System
Place Value System
Tally Marks
Landmark Numbers
Powers of Numbers
Placeholder Symbol (Zero)

Important Formulas

Powers in Base System

$n^0=1$

$n^1=n$

$n^2=n\times n$

Decimal Place Value

$(a\times10^3)+(b\times10^2)+(c\times10^1)+(d\times10^0)$

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Base-n Landmark Numbers

$1,n,n^2,n^3,\cdots$

Questions & Step-by-Step Solutions

Figure It Out – Page 54

Q1. Give methods for adding, subtracting, multiplying and dividing using sticks.

Solution

Addition

Combine both groups of sticks together.

Subtraction

Remove sticks of the smaller group from the larger group.

Multiplication

Create repeated groups.

Example:

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3 × 4 means 4 sticks repeated 3 times.

Division

Distribute sticks equally into groups.

Final Answer

Arithmetic can be performed by grouping and separating sticks systematically.

Q2. Extend the letter-based number system.

Solution

After z:

aa = 27
ab = 28
ac = 29

and so on.

Final Answer

Using combinations of letters extends the counting system infinitely.

Q3. Make your own number system.

Solution

Example:

Number	Symbol
1	●
2	●●
3	▲
4	▲●

Final Answer

Students may create their own symbols and rules.

Figure It Out – Page 59

Q1. Represent the following in Roman numerals

(i) 1222

Solution

1222 = 1000 + 100 + 100 + 10 + 10 + 1 + 1

Roman numeral:

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MCCXXII

Final Answer

MCCXXII

(ii) 2999

Solution

2999 = 2000 + 900 + 90 + 9

Roman numeral:

MMCMXCIX

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Final Answer

MMCMXCIX

(iii) 302

Solution

302 = 300 + 2

Roman numeral:

CCCII

Final Answer

CCCII

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(iv) 715

Solution

715 = 700 + 10 + 5

Roman numeral:

DCCXV

Final Answer

DCCXV

Figure It Out – Page 60

Q1. Why do Pacific island people use different number sequences for different objects?

Solution

Different objects may be grouped differently for easier counting and cultural practices.

Final Answer

Different counting systems improve convenience for specific objects.

Q2. Evaluate in Gumulgal System

(i)

(ukasar-ukasar-ukasar-ukasar-urapon)
+
(ukasar-ukasar-ukasar-urapon)

Solution

9 + 7 = 16

Final Answer

Equivalent to 16.

(ii)

9 − 6

Final Answer

(iii)

9 × 4

Final Answer

(iv)

16 ÷ 4

Final Answer

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Q3. Features of Hindu Number System

Solution

Uses place value
Uses only 10 symbols
Includes zero
Easy arithmetic operations

Final Answer

The Hindu number system is highly efficient and flexible.

Figure It Out – Page 62

Q1. Represent in Egyptian System

(i) 10458

Solution

10458 = 10000 + 400 + 50 + 8

Represent using Egyptian symbols.

(ii) 1023

1023 = 1000 + 20 + 3

(iii) 2660

2660 = 2000 + 600 + 60

(iv) 784

784 = 700 + 80 + 4

(v) 1111

1111 = 1000 + 100 + 10 + 1

(vi) 70707

70707 = 70000 + 700 + 7

Figure It Out – Page 63

Q1. Write numbers in Base-5 System

(i) 15

Solution

$15=3\times5+0$

Final Answer

30₅

(ii) 50

$50=2\times25+0$

Final Answer

200₅

(iii) 137

$137=1\times125+0\times25+2\times5+2$

Final Answer

1022₅

Q2. Can every number be represented?

Solution

Yes. Any number can be expressed using powers of 5.

Final Answer

Yes, all numbers can be represented.

Q3. Landmark numbers in Base-7

Solution

$1,7,49,343,2401$

Final Answer

1, 7, 49, 343, 2401 …

Figure It Out – Page 65

Q1. Add Egyptian Numerals

Solution

Group similar symbols together.

Whenever 10 similar symbols occur, replace them with the next higher symbol.

Final Answer

Egyptian addition is based on regrouping symbols.

Q2. Add Base-5 Numerals

Solution

Whenever 5 similar symbols occur, regroup into next landmark number.

Final Answer

Addition works similarly to carrying in decimal system.

Figure It Out – Page 69

Q1. Can a symbol appear 10 or more times?

Solution

No.

10 repeated symbols are replaced by the next higher symbol.

Final Answer

No, regrouping prevents repetition beyond 9.

Q2. Create Base-4 System

Solution

Landmark numbers:

$1,4,16,64,256$

Representations:

Decimal	Base-4
1	1
2	2
3	3
4	10
5	11

Q3. Rule for multiplying by 5 in Base-5

Solution

Appending zero shifts to next power.

Example:

$23_5\times5=230_5$

Final Answer

Add one zero at the end.

Figure It Out – Page 73

Q1. Represent in Mesopotamian System

(i) 63

$63=1\times60+3$

Final Answer

1 sixty and 3 ones.

(ii) 132

$132=2\times60+12$

Final Answer

2 sixties and 12 ones.

(iii) 200

$200=3\times60+20$

Final Answer

3 sixties and 20 ones.

(iv) 60

Final Answer

1 sixty.

(v) 3605

$3605=1\times3600+5$

Final Answer

1 three-thousand-six-hundred and 5 ones.

Common Mistakes

Confusing Roman numerals with Hindu numerals
Forgetting regrouping rules in base systems
Wrong place value interpretation
Ignoring zero as placeholder
Incorrect base conversions

Exam Tips

Memorise Roman numeral symbols
Practice base conversions regularly
Understand landmark numbers clearly
Learn place value carefully
Revise ancient number systems visually

Practice MCQs

1. Which symbol represents 100 in Roman numerals?

A. X
B. L
C. C
D. D

Answer

C. C

2. Which civilization developed the decimal system?

A. Roman
B. Egyptian
C. Indian
D. Greek

Answer

C. Indian

3. Which number system uses place value?

A. Roman
B. Tally Marks
C. Hindu-Arabic
D. Stick System

Answer

C. Hindu-Arabic

FAQ Section

Q1. What is a positional number system?

A number system where digit position determines value.

Q2. Why is zero important?

It acts as a placeholder in place value systems.

Q3. What are Roman numerals?

Ancient numerals used in Rome using symbols like I, V, X, L, C, D, M.

Q4. What is a base-n system?

$1,n,n^2,n^3,\cdots$

A system based on powers of n.

Q5. What is the decimal system?

A base-10 number system.

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