A Story of Numbers | IT

Question 1

Q. Reema's curiosity was sparked, and questions started swirling in her head:

(i) Since when have humans been counting?

(ii) What was their need for counting? What were they counting?

(iii) Since when have people been writing numbers in the modern form?

(iv) How would the Mesopotamians have written $20$ ? $50$ ? $100$ ?

Solution

Understand the Question

Counting originated tens of thousands of years ago from everyday practical needs such as managing livestock, trading, and tracking lunar or solar cycles.
Different civilizations developed different numeral systems over time:
- Mesopotamians (Babylonians): Used a base- $60$ (sexagesimal) positional system using two primary symbols: a vertical wedge ( $1$ ) and a chevron ( $10$ ).
- Modern System: The base- $10$ decimal place-value system with digits $0\text{--}9$ (Hindu-Arabic numerals) originated in India and later spread globally.

(i) Since when have humans been counting?

Step 1 · Origins of Counting

Archaeological evidence, such as tally marks notched on ancient bones, shows that humans have been counting for at least 50,000 years, dating back to prehistoric times.

Answer

(i) At least 50,000 years ago (prehistoric times).

(ii) What was their need for counting? What were they counting?

Step 1 · Reasons and Objects of Early Counting

Early humans needed to count to manage resources and organize community life:

Need for counting: Managing livestock, keeping track of group members, bartering/trading, and tracking time (days, moon phases, and seasons).
What they were counting: Animals, food supplies, people, days/seasons, and trade goods.

Answer

(ii) To manage resources, livestock, people, trade, and time (counting animals, supplies, days, and trade goods).

(iii) Since when have people been writing numbers in the modern form?

Step 1 · Origins of Modern Numerals

The modern Hindu-Arabic numeral system ( $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ ) was developed in India around the 6th or 7th century CE. It spread to the Arab world and was later introduced to Europe, where it became widely adopted between the 13th and 15th centuries.

Answer

(iii) Around the 6th or 7th century CE in India (widely adopted in Europe between the 13th and 15th centuries).

(iv) How would the Mesopotamians have written $20$ ? $50$ ? $100$ ?

Step 1 · Represent Numbers in the Mesopotamian Base-60 System

Diagram 1

The Mesopotamians used a base- $60$ (sexagesimal) positional system using two symbols:

A vertical wedge representing $1$
A chevron representing $10$

Representing each value:

$20$ : Two chevrons ( $\ll \ll$ )
$50$ : Five chevrons ( $\ll \ll \ll \ll \ll$ )
$100$ : In base- $60$ , $100 = (1 \times 60) + 40$ . This is represented as $1$ unit of $60$ (one vertical wedge) followed by $40$ units of $1$ (four chevrons).

Answer

(iv) * 20: $\ll \ll$ (two chevrons)

50: $\ll \ll \ll \ll \ll$ (five chevrons)
100: One vertical wedge for $60$ followed by four chevrons for $40$ ( $1 \times 60 + 40$ )

Common Mistakes

Base-10 vs Base-60 Confusion: Representing $100$ as $10$ chevrons instead of recognizing that Mesopotamian numerals were base- $60$ , where $100 = 1 \times 60 + 40$ .
Origins of Numerals: Confusing the origin of the Hindu-Arabic numeral system in India (6th–7th century CE) with its later transmission into Europe (13th–15th centuries).

More questions in IT

Q. Reema's curiosity was sparked, and questions started swirling in her head:

(i) Since when have humans been counting?

(ii) What was their need for counting? What were they counting?

(iii) Since when have people been writing numbers in the modern form?

(iv) How would the Mesopotamians have written $20$ ? $50$ ? $100$ ?

Context: Imagine that we are living in the Stone Age, say, around ten thousand years ago. Suppose we have a herd of cows.

Q. How do we ensure that all cows have returned safely after grazing?

Context: Imagine that we are living in the Stone Age, say, around ten thousand years ago. Suppose we have a herd of cows.

Q. Do we have fewer cows than our neighbour?

Context: Imagine that we are living in the Stone Age, say, around ten thousand years ago. Suppose we have a herd of cows.

Q. If there are fewer, how many more cows would we need so that we have the same number of cows as our neighbour?

Context: Suppose we have a herd of cows. We want to answer the following questions: 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?

Q. How will you use such sticks to answer the other two questions (Q2 and Q3)?

How many numbers can you represent in this way using the sounds of the letters of your language?

Do you see a way of extending this method to represent bigger numbers as well? How?

Context: A group of indigenous people in Australia called the Gumulgal had the following words for their numbers:

urapon
ukasar
ukasar-urapon
ukasar-ukasar
ukasar-ukasar-urapon
ukasar-ukasar-ukasar

Q. Can you see how their number names are formed?

Context: A group of indigenous people in Australia called the Gumulgal had the following words for their numbers:

urapon
ukasar
ukasar-urapon
ukasar-ukasar
ukasar-ukasar-urapon
ukasar-ukasar-ukasar

Q. Can you see how the names of the other numbers are formed?

Q10

What could be the difficulties with using a number system that counts only in groups of a single particular size? How would you represent a number like 1345 in a system that counts only by 5s?

Q11

Do it yourself now:

(b) $\text{LXXXVII} + \text{LXXVIII}$

Q12

How will you multiply two numbers given in Roman numerals, without converting them to Hindu numerals? Try to find the product of the following pairs of landmark numbers: $V \times L$ , $L \times D$ , $V \times D$ , $VII \times IX$ .

Q13

Instead of grouping together 10 collections of size equal to the previous landmark number (as in the case of the Egyptian system), can we get a number system by grouping together 5 collections of size equal to the previous landmark number? Can this 5 be replaced by any positive integer?

Q14

What is any landmark number multiplied by $\cap$ (that is $10$ )? Find the following products—

Each landmark number is a power of $10$ and so multiplying it with $10$ increases the power by $1$ , which is the next landmark number.

Q15

What is any landmark number multiplied by $?$ ( $10^2$ )? Find the following products—

Q16

Find the following products—

Thus, the product of any two landmark numbers is another landmark number!

Q17

Context: Thus, the product of any two landmark numbers is another landmark number!

Q. Does this property hold true in the base-5 system that we created? Does this hold for any number system with a base?

Q18

Now find the following products—

Q19

What would be a simple rule to multiply a number with $\cap$ ?

Q20

Context: To get an idea of how the abacus was used for calculations, let us consider a simple addition problem: $2907 + 43$ . The two numbers were taken on either side of the vertical partition.

Q. How would you use this to find the sum?

Q21

Context: To get an idea of how the abacus was used for calculations, let us consider a simple addition problem: $2907 + 43$ . The two numbers were taken on either side of the vertical partition. The counters along each line were brought together.

Q. What is to be done if the total in a line exceeded $10$ ?

Q22

Look at the representation of 60. What will be the representation for 3,600?

Q23

How is this to be read?

Q24

Represent the following numbers using the Mayan system:

(i) 77 (ii) 100 (iii) 361 (iv) 721

Q25

Where does the Hindu/Indian number system figure in the evolution of ideas of number representation? What are its landmark numbers? And does it use a place value system?