A Story of Numbers | IT

Question 24

Represent the following numbers using the Mayan system:

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

Solution

Understand the Question

The Mayan numeral system uses three primary symbols:
- Dot (•): Represents $1$
- Bar (— or =): Represents $5$
- Shell (O): Represents $0$
Numbers are written vertically with higher place values at the top and the ones place at the bottom.
Place values in the Mayan system are modified base-20:
- $1^{\text{st}}$ level (bottom): $1$
- $2^{\text{nd}}$ level: $20$
- $3^{\text{rd}}$ level: $360$ ( $20 \times 18$ )
- $4^{\text{th}}$ level: $7200$ ( $20^2 \times 18$ )
To convert a number, divide successively by the highest possible place value to find the digit at each level, using the remainder for the lower levels.

(i) Represent 77 using the Mayan system.

Step 1 · Convert 77 to Mayan Numerals

The largest place value $\le 77$ is $20$ .

Divide $77$ by $20$ to find the $20$ s place: $77 \div 20 = 3 \text{ with a remainder of } 17$

20s place: $3 = \text{•••}$
1s place: $17 = (3 \times 5) + (2 \times 1) = \text{===••}$

Answer

(i) $\begin{array}{c} \text{•••} \\ \text{===••} \end{array}$

(ii) Represent 100 using the Mayan system.

Step 1 · Convert 100 to Mayan Numerals

The largest place value $\le 100$ is $20$ .

Divide $100$ by $20$ to find the $20$ s place: $100 \div 20 = 5 \text{ with a remainder of } 0$

20s place: $5 = \text{—}$
1s place: $0 = \text{(O)}$

Answer

(ii) $\begin{array}{c} \text{—} \\ \text{(O)} \end{array}$

(iii) Represent 361 using the Mayan system.

Step 1 · Convert 361 to Mayan Numerals

The largest place value $\le 361$ is $360$ .

Divide $361$ by $360$ : $361 \div 360 = 1 \text{ with a remainder of } 1$

360s place: $1 = \text{•}$

Now divide the remainder $1$ by $20$ : $1 \div 20 = 0 \text{ with a remainder of } 1$

20s place: $0 = \text{(O)}$
1s place: $1 = \text{•}$

Answer

(iii) $\begin{array}{c} \text{•} \\ \text{(O)} \\ \text{•} \end{array}$

(iv) Represent 721 using the Mayan system.

Step 1 · Convert 721 to Mayan Numerals

The largest place value $\le 721$ is $360$ .

Divide $721$ by $360$ : $721 \div 360 = 2 \text{ with a remainder of } 1$

360s place: $2 = \text{••}$

Now divide the remainder $1$ by $20$ : $1 \div 20 = 0 \text{ with a remainder of } 1$

20s place: $0 = \text{(O)}$
1s place: $1 = \text{•}$

Answer

(iv) $\begin{array}{c} \text{••} \\ \text{(O)} \\ \text{•} \end{array}$

Common Mistakes

Omitting the Zero Placeholder: Forgetting to write the shell symbol (O) for empty positions (such as the 20s place in 361 and 721, or the 1s place in 100).
Incorrect Third Place Value: Assuming the 3rd level is $20 \times 20 = 400$ instead of $18 \times 20 = 360$ , which is specific to the Mayan calendar numeral system.
Writing Horizontally: Arranging place values from left to right instead of vertically from top (highest) to bottom (lowest).

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?