NCERT Textbook: A Story of Numbers | Mathematics Class 8- New NCERT (Ganita Prakash) PDF Download

 Page 1


3
A STORY OF 
NUMBERS
3.1 Reema’s Curiosity
One lazy afternoon, Reema was flipping through an old book when — 
whoosh! — a piece of paper slipped out and floated to the floor. She 
picked it up and stared at the strange symbols all over it. “What is this?” 
she wondered.
She ran to her father, holding the paper as if it were a secret treasure. 
He looked at it and smiled. “Around 4000 years ago, there flourished a 
civilisation in a region called Mesopotamia, in the western part of Asia, 
containing a major part of the present-day Iraq and a few other 
neighbouring countries. This is one of the ways they wrote their 
numbers!”
Reema’s eyes lit up, “Seriously? These strange symbols were numbers?” 
Her curiosity was sparked, and questions started swirling in her head.
Since 
when have 
humans been 
counting?
What was their 
need for counting? 
What were they 
counting?
Since when 
have people 
been writing 
numbers in 
the modern 
form?
How would the 
Mesopotamians  
have written  
20? 50? 100?
Chapter 3.indd   48 Chapter 3.indd   48 10-07-2025   17:44:17 10-07-2025   17:44:17
Page 2


3
A STORY OF 
NUMBERS
3.1 Reema’s Curiosity
One lazy afternoon, Reema was flipping through an old book when — 
whoosh! — a piece of paper slipped out and floated to the floor. She 
picked it up and stared at the strange symbols all over it. “What is this?” 
she wondered.
She ran to her father, holding the paper as if it were a secret treasure. 
He looked at it and smiled. “Around 4000 years ago, there flourished a 
civilisation in a region called Mesopotamia, in the western part of Asia, 
containing a major part of the present-day Iraq and a few other 
neighbouring countries. This is one of the ways they wrote their 
numbers!”
Reema’s eyes lit up, “Seriously? These strange symbols were numbers?” 
Her curiosity was sparked, and questions started swirling in her head.
Since 
when have 
humans been 
counting?
What was their 
need for counting? 
What were they 
counting?
Since when 
have people 
been writing 
numbers in 
the modern 
form?
How would the 
Mesopotamians  
have written  
20? 50? 100?
Chapter 3.indd   48 Chapter 3.indd   48 10-07-2025   17:44:17 10-07-2025   17:44:17
A Story of Numbers
49
Sensing her curiosity, her father started telling her how the idea of 
number and number representation evolved over the course of time, 
across geographies, to finally reach its modern efficient form. 
Get ready to travel back in time with them!
Humans had the need to count even as early as the Stone 
Age. They were counting to determine the quantity of food 
they had, the number of animals in their livestock, details 
regarding trades of goods, the number of offerings given in 
rituals, etc. They also wanted to keep track of the passing 
days, e.g., to know and predict when important events such 
as the new moon, full moon, or onset of a season would occur. 
However, when they said or wrote down such numbers, they 
didn’t make use of the numbers that we use today.
The structure of the modern oral and written numbers that we use 
today had its origin thousands of years ago in India. Ancient Indian texts, 
such as the Yajurveda Samhita, mentioned names of numbers based on 
powers of 10, almost as we say them orally today. For example, they listed 
names for the numbers one (eka), ten (dasha), hundred (shata), thousand 
(sahasra), ten thousand (ayuta), etc., all the way up to 10
12
 and beyond.
The way we write our numbers today — using the digits 0 through 9 — 
also originated and were developed in India, around 2000 years ago. The 
first known instance of numbers being 
written using ten digits, including the digit 0 
(which was then notated as a dot), occurs in 
the Bakhshali manuscript (c. 3rd century 
CE). Aryabhata (c. 499 CE) was the first 
mathematician to fully explain, and do 
elaborate scientific computations with the 
Indian system of 10 symbols.
The Indian number system was transmitted to the Arab world 
by around 800 CE. It was popularised in the Arab world by the great 
Persian mathematician Al-Khwarizmi (after whom the word ‘algorithm’ 
is named) through his book On the Calculation with Hindu Numerals 
(c. 825) and by the noted philosopher Al-Kindi through his work On the 
Use of the Hindu Numerals (c. 830).
From the Arab world, the Hindu numerals were transmitted to Europe 
and to parts of Africa by around 1100 CE. Though Al-Khwarizmi’s work 
on calculation with Hindu numerals was translated into Latin, it was the 
Italian mathematician Fibonacci who around the year 1200 really made 
the case to Europe to adopt the Indian numerals. However, the Roman 
numerals were so ingrained in European thinking and writing at the 
time that the Indian numerals did not gain widespread use for several 
more centuries. But eventually, during the European Renaissance and 
by the 17th century, not adopting them became impossible or it would 
impede scientific progress. 
Zero in the Bakhshali manuscript
Chapter 3.indd   49 Chapter 3.indd   49 10-07-2025   17:44:18 10-07-2025   17:44:18
Page 3


3
A STORY OF 
NUMBERS
3.1 Reema’s Curiosity
One lazy afternoon, Reema was flipping through an old book when — 
whoosh! — a piece of paper slipped out and floated to the floor. She 
picked it up and stared at the strange symbols all over it. “What is this?” 
she wondered.
She ran to her father, holding the paper as if it were a secret treasure. 
He looked at it and smiled. “Around 4000 years ago, there flourished a 
civilisation in a region called Mesopotamia, in the western part of Asia, 
containing a major part of the present-day Iraq and a few other 
neighbouring countries. This is one of the ways they wrote their 
numbers!”
Reema’s eyes lit up, “Seriously? These strange symbols were numbers?” 
Her curiosity was sparked, and questions started swirling in her head.
Since 
when have 
humans been 
counting?
What was their 
need for counting? 
What were they 
counting?
Since when 
have people 
been writing 
numbers in 
the modern 
form?
How would the 
Mesopotamians  
have written  
20? 50? 100?
Chapter 3.indd   48 Chapter 3.indd   48 10-07-2025   17:44:17 10-07-2025   17:44:17
A Story of Numbers
49
Sensing her curiosity, her father started telling her how the idea of 
number and number representation evolved over the course of time, 
across geographies, to finally reach its modern efficient form. 
Get ready to travel back in time with them!
Humans had the need to count even as early as the Stone 
Age. They were counting to determine the quantity of food 
they had, the number of animals in their livestock, details 
regarding trades of goods, the number of offerings given in 
rituals, etc. They also wanted to keep track of the passing 
days, e.g., to know and predict when important events such 
as the new moon, full moon, or onset of a season would occur. 
However, when they said or wrote down such numbers, they 
didn’t make use of the numbers that we use today.
The structure of the modern oral and written numbers that we use 
today had its origin thousands of years ago in India. Ancient Indian texts, 
such as the Yajurveda Samhita, mentioned names of numbers based on 
powers of 10, almost as we say them orally today. For example, they listed 
names for the numbers one (eka), ten (dasha), hundred (shata), thousand 
(sahasra), ten thousand (ayuta), etc., all the way up to 10
12
 and beyond.
The way we write our numbers today — using the digits 0 through 9 — 
also originated and were developed in India, around 2000 years ago. The 
first known instance of numbers being 
written using ten digits, including the digit 0 
(which was then notated as a dot), occurs in 
the Bakhshali manuscript (c. 3rd century 
CE). Aryabhata (c. 499 CE) was the first 
mathematician to fully explain, and do 
elaborate scientific computations with the 
Indian system of 10 symbols.
The Indian number system was transmitted to the Arab world 
by around 800 CE. It was popularised in the Arab world by the great 
Persian mathematician Al-Khwarizmi (after whom the word ‘algorithm’ 
is named) through his book On the Calculation with Hindu Numerals 
(c. 825) and by the noted philosopher Al-Kindi through his work On the 
Use of the Hindu Numerals (c. 830).
From the Arab world, the Hindu numerals were transmitted to Europe 
and to parts of Africa by around 1100 CE. Though Al-Khwarizmi’s work 
on calculation with Hindu numerals was translated into Latin, it was the 
Italian mathematician Fibonacci who around the year 1200 really made 
the case to Europe to adopt the Indian numerals. However, the Roman 
numerals were so ingrained in European thinking and writing at the 
time that the Indian numerals did not gain widespread use for several 
more centuries. But eventually, during the European Renaissance and 
by the 17th century, not adopting them became impossible or it would 
impede scientific progress. 
Zero in the Bakhshali manuscript
Chapter 3.indd   49 Chapter 3.indd   49 10-07-2025   17:44:18 10-07-2025   17:44:18
Ganita Prakash | Grade 8 
50
“The ingenious method of expressing every possible number using a set of ten 
symbols (each symbol having a place value and an absolute value) emerged in 
India. The idea seems so simple nowadays that its significance and profound 
importance is no longer appreciated. Its simplicity lies in the way it facilitated 
calculations and placed arithmetic foremost among useful inventions.”
— Pierre-Simon Laplace (1749 – 1827)
Their use then spread to every continent, and are now used in every 
corner of the world.
 Because European scholars learned the Indian numerals from the 
Arab world, they called them ?Arabic numerals’ to reflect their European 
perspective. On the other hand, as noted above, Arab scholars, such as 
Al-Khwarizmi and Al-Kindi, called them ?Hindu numerals’. During the 
period of European colonisation, the European term Arabic numbers 
became widely used. However, in recent years, this mistake is being 
corrected in many textbooks and documents around the world, including 
in Europe.  The most commonly used terminologies for the numbers we 
use today are ?Hindu numerals’, ?Indian numerals’, and the transitional 
?Hindu-Arabic numerals’. It is worth noting that the word ?Hindu’ here 
does not refer to a religion, but rather a geography/people from whom 
these numbers came.
The shape of the digits 0, 1, 2, ..., 9 used to write numbers in the Indian 
number system today evolved over a period of time, as shown below:
Evolution of the digits used in the Indian number system
Prior to the global adoption of the Indian system of numerals, different 
groups of people used different methods of representing numbers. We 
Chapter 3.indd   50 Chapter 3.indd   50 10-07-2025   17:44:18 10-07-2025   17:44:18
Page 4


3
A STORY OF 
NUMBERS
3.1 Reema’s Curiosity
One lazy afternoon, Reema was flipping through an old book when — 
whoosh! — a piece of paper slipped out and floated to the floor. She 
picked it up and stared at the strange symbols all over it. “What is this?” 
she wondered.
She ran to her father, holding the paper as if it were a secret treasure. 
He looked at it and smiled. “Around 4000 years ago, there flourished a 
civilisation in a region called Mesopotamia, in the western part of Asia, 
containing a major part of the present-day Iraq and a few other 
neighbouring countries. This is one of the ways they wrote their 
numbers!”
Reema’s eyes lit up, “Seriously? These strange symbols were numbers?” 
Her curiosity was sparked, and questions started swirling in her head.
Since 
when have 
humans been 
counting?
What was their 
need for counting? 
What were they 
counting?
Since when 
have people 
been writing 
numbers in 
the modern 
form?
How would the 
Mesopotamians  
have written  
20? 50? 100?
Chapter 3.indd   48 Chapter 3.indd   48 10-07-2025   17:44:17 10-07-2025   17:44:17
A Story of Numbers
49
Sensing her curiosity, her father started telling her how the idea of 
number and number representation evolved over the course of time, 
across geographies, to finally reach its modern efficient form. 
Get ready to travel back in time with them!
Humans had the need to count even as early as the Stone 
Age. They were counting to determine the quantity of food 
they had, the number of animals in their livestock, details 
regarding trades of goods, the number of offerings given in 
rituals, etc. They also wanted to keep track of the passing 
days, e.g., to know and predict when important events such 
as the new moon, full moon, or onset of a season would occur. 
However, when they said or wrote down such numbers, they 
didn’t make use of the numbers that we use today.
The structure of the modern oral and written numbers that we use 
today had its origin thousands of years ago in India. Ancient Indian texts, 
such as the Yajurveda Samhita, mentioned names of numbers based on 
powers of 10, almost as we say them orally today. For example, they listed 
names for the numbers one (eka), ten (dasha), hundred (shata), thousand 
(sahasra), ten thousand (ayuta), etc., all the way up to 10
12
 and beyond.
The way we write our numbers today — using the digits 0 through 9 — 
also originated and were developed in India, around 2000 years ago. The 
first known instance of numbers being 
written using ten digits, including the digit 0 
(which was then notated as a dot), occurs in 
the Bakhshali manuscript (c. 3rd century 
CE). Aryabhata (c. 499 CE) was the first 
mathematician to fully explain, and do 
elaborate scientific computations with the 
Indian system of 10 symbols.
The Indian number system was transmitted to the Arab world 
by around 800 CE. It was popularised in the Arab world by the great 
Persian mathematician Al-Khwarizmi (after whom the word ‘algorithm’ 
is named) through his book On the Calculation with Hindu Numerals 
(c. 825) and by the noted philosopher Al-Kindi through his work On the 
Use of the Hindu Numerals (c. 830).
From the Arab world, the Hindu numerals were transmitted to Europe 
and to parts of Africa by around 1100 CE. Though Al-Khwarizmi’s work 
on calculation with Hindu numerals was translated into Latin, it was the 
Italian mathematician Fibonacci who around the year 1200 really made 
the case to Europe to adopt the Indian numerals. However, the Roman 
numerals were so ingrained in European thinking and writing at the 
time that the Indian numerals did not gain widespread use for several 
more centuries. But eventually, during the European Renaissance and 
by the 17th century, not adopting them became impossible or it would 
impede scientific progress. 
Zero in the Bakhshali manuscript
Chapter 3.indd   49 Chapter 3.indd   49 10-07-2025   17:44:18 10-07-2025   17:44:18
Ganita Prakash | Grade 8 
50
“The ingenious method of expressing every possible number using a set of ten 
symbols (each symbol having a place value and an absolute value) emerged in 
India. The idea seems so simple nowadays that its significance and profound 
importance is no longer appreciated. Its simplicity lies in the way it facilitated 
calculations and placed arithmetic foremost among useful inventions.”
— Pierre-Simon Laplace (1749 – 1827)
Their use then spread to every continent, and are now used in every 
corner of the world.
 Because European scholars learned the Indian numerals from the 
Arab world, they called them ?Arabic numerals’ to reflect their European 
perspective. On the other hand, as noted above, Arab scholars, such as 
Al-Khwarizmi and Al-Kindi, called them ?Hindu numerals’. During the 
period of European colonisation, the European term Arabic numbers 
became widely used. However, in recent years, this mistake is being 
corrected in many textbooks and documents around the world, including 
in Europe.  The most commonly used terminologies for the numbers we 
use today are ?Hindu numerals’, ?Indian numerals’, and the transitional 
?Hindu-Arabic numerals’. It is worth noting that the word ?Hindu’ here 
does not refer to a religion, but rather a geography/people from whom 
these numbers came.
The shape of the digits 0, 1, 2, ..., 9 used to write numbers in the Indian 
number system today evolved over a period of time, as shown below:
Evolution of the digits used in the Indian number system
Prior to the global adoption of the Indian system of numerals, different 
groups of people used different methods of representing numbers. We 
Chapter 3.indd   50 Chapter 3.indd   50 10-07-2025   17:44:18 10-07-2025   17:44:18
A Story of Numbers
51
shall take a glimpse of some of them. We will not be looking at different 
systems in a chronological order, but rather an order that shows us the 
main stages in the development of the idea of number representation. 
But first, let us explore some of the foundational ideas needed to count 
and to determine the number of objects in a given collection.
The Mechanism of Counting
Imagine that we are living in the Stone Age, say, around ten thousand 
years ago. Suppose we have a herd of cows. Here are some natural 
questions that we might ask about our herd —
Q1.  How do we ensure that all cows have returned safely after 
grazing?
Q2. Do we have fewer cows than our neighbour?
Q3.   If there are fewer, how many more cows would we need so 
that we have the same number of cows as our neighbour?
We need to tackle these questions without the use of the 
number names or written numbers of the Hindu number 
system. How do we do it?
Here are some possible methods.
Method 1: We could tackle the questions by using pebbles, sticks or 
any object that is available in abundance. Let us choose sticks. For every 
cow in the herd, we could keep a stick. The final collection of sticks tells 
us the number of cows, which can be used to check if any cows have 
gone missing.
Math 
Talk
Chapter 3.indd   51 Chapter 3.indd   51 11-07-2025   14:29:33 11-07-2025   14:29:33
Page 5


3
A STORY OF 
NUMBERS
3.1 Reema’s Curiosity
One lazy afternoon, Reema was flipping through an old book when — 
whoosh! — a piece of paper slipped out and floated to the floor. She 
picked it up and stared at the strange symbols all over it. “What is this?” 
she wondered.
She ran to her father, holding the paper as if it were a secret treasure. 
He looked at it and smiled. “Around 4000 years ago, there flourished a 
civilisation in a region called Mesopotamia, in the western part of Asia, 
containing a major part of the present-day Iraq and a few other 
neighbouring countries. This is one of the ways they wrote their 
numbers!”
Reema’s eyes lit up, “Seriously? These strange symbols were numbers?” 
Her curiosity was sparked, and questions started swirling in her head.
Since 
when have 
humans been 
counting?
What was their 
need for counting? 
What were they 
counting?
Since when 
have people 
been writing 
numbers in 
the modern 
form?
How would the 
Mesopotamians  
have written  
20? 50? 100?
Chapter 3.indd   48 Chapter 3.indd   48 10-07-2025   17:44:17 10-07-2025   17:44:17
A Story of Numbers
49
Sensing her curiosity, her father started telling her how the idea of 
number and number representation evolved over the course of time, 
across geographies, to finally reach its modern efficient form. 
Get ready to travel back in time with them!
Humans had the need to count even as early as the Stone 
Age. They were counting to determine the quantity of food 
they had, the number of animals in their livestock, details 
regarding trades of goods, the number of offerings given in 
rituals, etc. They also wanted to keep track of the passing 
days, e.g., to know and predict when important events such 
as the new moon, full moon, or onset of a season would occur. 
However, when they said or wrote down such numbers, they 
didn’t make use of the numbers that we use today.
The structure of the modern oral and written numbers that we use 
today had its origin thousands of years ago in India. Ancient Indian texts, 
such as the Yajurveda Samhita, mentioned names of numbers based on 
powers of 10, almost as we say them orally today. For example, they listed 
names for the numbers one (eka), ten (dasha), hundred (shata), thousand 
(sahasra), ten thousand (ayuta), etc., all the way up to 10
12
 and beyond.
The way we write our numbers today — using the digits 0 through 9 — 
also originated and were developed in India, around 2000 years ago. The 
first known instance of numbers being 
written using ten digits, including the digit 0 
(which was then notated as a dot), occurs in 
the Bakhshali manuscript (c. 3rd century 
CE). Aryabhata (c. 499 CE) was the first 
mathematician to fully explain, and do 
elaborate scientific computations with the 
Indian system of 10 symbols.
The Indian number system was transmitted to the Arab world 
by around 800 CE. It was popularised in the Arab world by the great 
Persian mathematician Al-Khwarizmi (after whom the word ‘algorithm’ 
is named) through his book On the Calculation with Hindu Numerals 
(c. 825) and by the noted philosopher Al-Kindi through his work On the 
Use of the Hindu Numerals (c. 830).
From the Arab world, the Hindu numerals were transmitted to Europe 
and to parts of Africa by around 1100 CE. Though Al-Khwarizmi’s work 
on calculation with Hindu numerals was translated into Latin, it was the 
Italian mathematician Fibonacci who around the year 1200 really made 
the case to Europe to adopt the Indian numerals. However, the Roman 
numerals were so ingrained in European thinking and writing at the 
time that the Indian numerals did not gain widespread use for several 
more centuries. But eventually, during the European Renaissance and 
by the 17th century, not adopting them became impossible or it would 
impede scientific progress. 
Zero in the Bakhshali manuscript
Chapter 3.indd   49 Chapter 3.indd   49 10-07-2025   17:44:18 10-07-2025   17:44:18
Ganita Prakash | Grade 8 
50
“The ingenious method of expressing every possible number using a set of ten 
symbols (each symbol having a place value and an absolute value) emerged in 
India. The idea seems so simple nowadays that its significance and profound 
importance is no longer appreciated. Its simplicity lies in the way it facilitated 
calculations and placed arithmetic foremost among useful inventions.”
— Pierre-Simon Laplace (1749 – 1827)
Their use then spread to every continent, and are now used in every 
corner of the world.
 Because European scholars learned the Indian numerals from the 
Arab world, they called them ?Arabic numerals’ to reflect their European 
perspective. On the other hand, as noted above, Arab scholars, such as 
Al-Khwarizmi and Al-Kindi, called them ?Hindu numerals’. During the 
period of European colonisation, the European term Arabic numbers 
became widely used. However, in recent years, this mistake is being 
corrected in many textbooks and documents around the world, including 
in Europe.  The most commonly used terminologies for the numbers we 
use today are ?Hindu numerals’, ?Indian numerals’, and the transitional 
?Hindu-Arabic numerals’. It is worth noting that the word ?Hindu’ here 
does not refer to a religion, but rather a geography/people from whom 
these numbers came.
The shape of the digits 0, 1, 2, ..., 9 used to write numbers in the Indian 
number system today evolved over a period of time, as shown below:
Evolution of the digits used in the Indian number system
Prior to the global adoption of the Indian system of numerals, different 
groups of people used different methods of representing numbers. We 
Chapter 3.indd   50 Chapter 3.indd   50 10-07-2025   17:44:18 10-07-2025   17:44:18
A Story of Numbers
51
shall take a glimpse of some of them. We will not be looking at different 
systems in a chronological order, but rather an order that shows us the 
main stages in the development of the idea of number representation. 
But first, let us explore some of the foundational ideas needed to count 
and to determine the number of objects in a given collection.
The Mechanism of Counting
Imagine that we are living in the Stone Age, say, around ten thousand 
years ago. Suppose we have a herd of cows. Here are some natural 
questions that we might ask about our herd —
Q1.  How do we ensure that all cows have returned safely after 
grazing?
Q2. Do we have fewer cows than our neighbour?
Q3.   If there are fewer, how many more cows would we need so 
that we have the same number of cows as our neighbour?
We need to tackle these questions without the use of the 
number names or written numbers of the Hindu number 
system. How do we do it?
Here are some possible methods.
Method 1: We could tackle the questions by using pebbles, sticks or 
any object that is available in abundance. Let us choose sticks. For every 
cow in the herd, we could keep a stick. The final collection of sticks tells 
us the number of cows, which can be used to check if any cows have 
gone missing.
Math 
Talk
Chapter 3.indd   51 Chapter 3.indd   51 11-07-2025   14:29:33 11-07-2025   14:29:33
Ganita Prakash | Grade 8 
52
This way of associating each cow with a stick, such that no two cows are 
associated or mapped to the same stick is called a one-to-one mapping. 
This mapping can then be used to come up with a way to represent 
numbers, as shown in the table.
Number Its representation (using sticks)
1
2
3
4
5
.
.
.
.
.
.
How will you use such sticks to answer the other two questions (Q2 and 
Q3)?
Method 2: Instead of objects, we could use a standard sequence of 
sounds or names. For example, we could use the sounds of the letters 
of any language. While counting, we could make a one-to-one mapping 
between the objects and the letters: that is, associate each object to be 
counted with a letter, following the letter-order. This mapping can then 
be used to come up with a way of verbally representing numbers.
For example, we get the following number representation if we use 
English letters ‘a’ to ‘z’.
Number Its representation (using 
sounds or names)
1
2
3
4
5
.
.
.
26
.
.
.
a
b
c
d
e
z
Chapter 3.indd   52 Chapter 3.indd   52 10-07-2025   17:44:18 10-07-2025   17:44:18