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?
We use the abacus to add numbers by combining beads at each place value.
Step 1 — Initial setup of the numbers
First, we place the first number, 2907, on the left side. We put 2 beads on the 1000s line. We put 9 beads on the 100s line. We put 0 beads on the 10s line. We put 7 beads on the 1s line. Then, we place the second number, 43, on the right side. We put 0 beads on the 1000s line. We put 0 beads on the 100s line. We put 4 beads on the 10s line. We put 3 beads on the 1s line. The abacus now shows 2907 on the left and 43 on the right.
Step 2 — Adding the 1s (units) place
We start by adding the 1s beads. On the left, there are 7 beads. On the right, there are 3 beads. We combine these beads: Ten 1s beads make one 10s bead. So, we remove all 10 beads from the 1s line. We add 1 bead to the 10s line on the left side. The 1s line on the left now has 0 beads.
Step 3 — Adding the 10s place
Next, we add the 10s beads. Initially, there are 0 beads on the left. There are 4 beads on the right. We also have 1 bead carried over from the 1s place. We combine these beads: We move the 4 beads from the right 10s line to the left. We also add the 1 carried-over bead to the left. The 10s line on the left now has 5 beads.
Step 4 — Adding the 100s place
Now, we add the 100s beads. There are 9 beads on the left. There are 0 beads on the right. There is no carry-over from the 10s place. We combine these beads: The 100s line on the left remains with 9 beads.
Step 5 — Adding the 1000s place
Finally, we add the 1000s beads. There are 2 beads on the left. There are 0 beads on the right. There is no carry-over from the 100s place. We combine these beads: The 1000s line on the left remains with 2 beads.
Step 6 — Reading the final sum
After all additions, we read the number on the left side. The 1000s line has 2 beads. The 100s line has 9 beads. The 10s line has 5 beads. The 1s line has 0 beads. The final sum is 2950.
Answer
(i) We represent the first number (2907) on the left side of the abacus. (ii) We represent the second number (43) on the right side of the abacus. (iii) Starting from the 1s place, we combine the beads from both sides. (iv) If the combined beads for a place value are 10 or more, we remove 10 beads and add one bead to the next higher place value on the left side (this is called carrying over). (v) We repeat this process for the 10s, 100s, and 1000s places, moving from right to left. (vi) The beads remaining on the left side of the abacus show the final sum, which is 2950.
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?
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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?
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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?
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?
Do it yourself now:
(b) LXXXVII + LXXVIII
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 × L, L × D, V × D, VII × IX.
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?
What is any landmark number multiplied by (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.
What is any landmark number multiplied by ()? Find the following products—
Find the following products—
Thus, the product of any two landmark numbers is another landmark number!
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?
Now find the following products—
What would be a simple rule to multiply a number with ∩?
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?
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?
Look at the representation of 60. What will be the representation for 3,600?
How is this to be read?
Represent the following numbers using the Mayan system:
(i) 77 (ii) 100 (iii) 361 (iv) 721
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?