Context: To get an idea of how the abacus was used for calculations, let us consider a simple addition problem: . The two numbers were taken on either side of the vertical partition. Q. How would you use this to find the sum?

Understand the Question The abacus is divided by a vertical partition: represent the first number () on the left side and the second number () on the right side across place value lines (Thousands, Hundreds, Tens, and Units). Addition is done by combining beads column-by-column starting from the units () place moving leftward. Whenever a place value accumulates beads, they are cleared () and bead is carried over to the next higher place value on the left. Step 1 · Set Up the Numbers on the Abacus Represent on the left side and on the right side according to place values: Left side (): beads on , beads on , beads on , and beads on . Right side (): beads on , beads on , beads on , and beads on . Step 2 · Add the Units (1s) Place Combine the beads from both sides: Since units equal ten: Remove all beads from the line (leaving beads on the line). Carry over bead to the line on the left side. Step 3 · Add the Tens (10s) Place Combine the beads including the carry-over: The line on the left now has beads. Step 4 · Add the Hundreds (100s) Place Combine the beads (no carry-over): The line on the left remains beads. Step 5 · Add the Thousands (1000s) Place Combine the beads (no carry-over): The line on the left remains beads. Step 6 · Read the Final Sum Read the beads on the left side of the abacus: line: line: line: line: Answer > Common Mistakes Place Value Misalignment: Aligning incorrectly under hundreds and tens rather than tens and units. Forgetting the Carry-Over: Forgetting to convert units into ten bead on the tens line, which leaves the units place incorrect.

A Story of Numbers | IT

Question 20

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?

Solution

Understand the Question

The abacus is divided by a vertical partition: represent the first number ( $2907$ ) on the left side and the second number ( $43$ ) on the right side across place value lines (Thousands, Hundreds, Tens, and Units).
Addition is done by combining beads column-by-column starting from the units ( $1\text{s}$ ) place moving leftward.
Whenever a place value accumulates $10$ beads, they are cleared ( $0$ ) and $1$ bead is carried over to the next higher place value on the left.

Step 1 · Set Up the Numbers on the Abacus

Represent $2907$ on the left side and $43$ on the right side according to place values: Diagram 1

Left side ( $2907$ ): $2$ beads on $1000\text{s}$ , $9$ beads on $100\text{s}$ , $0$ beads on $10\text{s}$ , and $7$ beads on $1\text{s}$ .
Right side ( $43$ ): $0$ beads on $1000\text{s}$ , $0$ beads on $100\text{s}$ , $4$ beads on $10\text{s}$ , and $3$ beads on $1\text{s}$ .

Step 2 · Add the Units (1s) Place

Combine the $1\text{s}$ beads from both sides: $7 + 3 = 10 \text{ beads}$

Since $10$ units equal $1$ ten:

Remove all $10$ beads from the $1\text{s}$ line (leaving $0$ beads on the $1\text{s}$ line).
Carry over $1$ bead to the $10\text{s}$ line on the left side.

Step 3 · Add the Tens (10s) Place

Combine the $10\text{s}$ beads including the carry-over: $0 + 4 + 1 = 5 \text{ beads}$

The $10\text{s}$ line on the left now has $5$ beads.

Step 4 · Add the Hundreds (100s) Place

Combine the $100\text{s}$ beads (no carry-over): $9 + 0 = 9 \text{ beads}$

The $100\text{s}$ line on the left remains $9$ beads.

Step 5 · Add the Thousands (1000s) Place

Combine the $1000\text{s}$ beads (no carry-over): $2 + 0 = 2 \text{ beads}$

The $1000\text{s}$ line on the left remains $2$ beads.

Step 6 · Read the Final Sum

Read the beads on the left side of the abacus:

$1000\text{s}$ line: $2$
$100\text{s}$ line: $9$
$10\text{s}$ line: $5$
$1\text{s}$ line: $0$

$\text{Sum} = 2950$

Answer

$2950$

Common Mistakes

Place Value Misalignment: Aligning $43$ incorrectly under hundreds and tens rather than tens and units.
Forgetting the Carry-Over: Forgetting to convert $10$ units into $1$ ten bead on the tens line, which leaves the units place incorrect.

More questions in IT

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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
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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
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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!

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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.

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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?