Discover the fascinating world of numbers, factors, and multiples in this chapter. The detailed Class 8 Maths Ganita Prakash NCERT Solutions help students understand divisibility, prime numbers, and other number concepts clearly.
Textbook Page 51 – 52
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.
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.
Q. How will you use such sticks to answer the other two questions (Q2 and Q3)?
Solution:
Q2. Do we have fewer cows than our neighbour?
Answer:
- Represent each cow in your herd with one stick, and ask your neighbor to do the same.
- Pair your sticks with your neighbor’s sticks one by one.
- Observe the results:
- If both piles run out at the same time, you have an equal number of cows.
- If your pile finishes first while your neighbor still has sticks, you have fewer cows.
- If your neighbor’s pile finishes first, you have more cows.
Q3. If there are fewer, how many more cows would we need so that we have the same number of cows as our neighbour?
Answer:
- Use the same one-to-one matching of sticks as before.
- Count any extra sticks left in your neighbor’s pile after matching.
- The number of extra sticks shows how many cows you are short of; you need that many more cows.
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’.
An obvious limitation of using only the letters of the English alphabet in this form is that it cannot be used to count collections having more than 26 objects.
Q. How many numbers can you represent in this way using the sounds of the letters of your language?
Solution: We can represent 52 numbers using the sounds of Hindi letters.
Method 3: We could use a sequence of written symbols as follows.
Q. Do you see a way of extending this method to represent bigger numbers as well? How?
Solution:
Yes, this method can be extended to represent much larger numbers by combining the basic symbols using additive and subtractive rules.
Rules to extend the system:
- Additive rule: If a smaller or equal symbol comes after a larger one, add their values.
- Example: XV = 10 + 5 = 15
- Example: MCC = 1000 + 100 + 100 = 1200
- Subtractive rule: If a smaller symbol comes before a larger one, subtract the smaller from the larger.
- Example: IV = 5 − 1 = 4
- Example: IX = 10 − 1 = 9
Textbook Page 54
Figure it out
Q1. Suppose you are using the number system that uses sticks to represent numbers, as in Method 1. Without using either the number names or the numerals of the Hindu number system, give a method for adding, subtracting, multiplying and dividing two numbers or two collections of sticks.
Solution:
1. Addition
- Take two groups of sticks (Group A and Group B).
- Combine both groups into a single pile.
- The total number of sticks in the pile shows the sum.
2. Subtraction
- Start with the larger group of sticks.
- Remove from it the number of sticks in the smaller group.
- The remaining sticks show the difference.
3. Multiplication
- Take one group of sticks (representing one set).
- Repeat this group as many times as indicated by the second number (e.g., 4 times).
- Combine all repeated groups into one pile. The total shows the product.
4. Division
- Start with a pile of sticks (the total quantity).
- Distribute the sticks into equal-sized groups.
- The number of sticks in each group is the quotient, and any leftover sticks are the remainder.
Q2. One way of extending the number system in Method 2 is by using strings with more than one letter — for example, we could use ‘aa’ for 27. How can you extend this system to represent all the numbers? There are many ways of doing it!
Solution:
One way to extend the number system in Method 2 is by using strings with more than one letter — for example, using “aa” for 27. You can extend this system to represent all numbers in many different ways!
Textbook Page 58
Q. 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?
Solution:
Difficulties with using a number system that counts only in groups of a single size:
- Numbers that are not multiples of 5 cannot be represented directly and require a way to account for the remainder.
- Representing large numbers may need many symbols or steps, making the system time-consuming and inefficient.
- Arithmetic operations like addition, subtraction, multiplication, and division become more complicated with fixed-size groupings.
Example:
1347÷5=269 remainder 2
So,
1347=5×269+2
In a system that only counts in groups of 5, you could show 269 groups of 5, but there would be 2 leftover, which the system cannot directly represent.
Textbook Page 59
Figure it Out
Q1. Represent the following numbers in the Roman system.
(i) 1222 (ii) 2999 (iii) 302 (iv) 715
Solution:
(i) 1222 = 1000 + 100 + 100 + 10 + 10 + 1 + 1 = MCCXXII.
(ii) 2999 = 1000 + 1000 + 900 + 90 + 9
= 1000 + 1000 + (1000 – 100) + (100 – 10) + (10 – 1)
= MMCMXCIX.
(iii) 302 = 100 + 100 + 100 + 1 + 1 = CCCII.
(iv) 715 = 500 + 100 + 100 + 10 + 5 = DCCXV.
Example: Try adding the following numbers without converting them to Hindu numerals:
(a) CCXXXII + CCCCXIII
Let us find the total number of Is, Xs, and Cs, and group them starting from the largest landmark number.
Apparently, it looks like the largest landmark number is C, but note that 5 Cs (100s) make a D (500). So the sum is
Q. Do it yourself now:
(b) LXXXVII + LXXVIII
Solution:
Textbook Page 60 – 61
Q. 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.
Solution:
Multiplying two numbers in Roman numerals without converting them to Hindu-Arabic numerals can be done using repeated addition or other techniques like doubling and halving.
Examples:
- V × L
- Add V (5) a total of L (50) times:L+L+L+L+L=50+50+50+50+50=100+100+50=CCL
- L × D
- Add D (500) a total of L (50) times:500+500+500+…(50 times)=25,000
- V × D
- Add D (500) a total of V (5) times:500+500+500+500+500=1000+1000+500=MMD
- VII × IX
- Add VII (7) a total of IX (9) times:7+7+7+7+7+7+7+7+7=63=50+10+3=LXIII
Figure it Out
Q1. A group of indigenous people in a Pacific island use different sequences of number names to count different objects. Why do you think they do this?
Solution:
Different number sequences are used because:
- Context and importance: Different objects (like coconuts or fish) may have special social, cultural, or economic significance. Using separate number names highlights their value.
- Oral tradition: These counting systems are often passed down orally, and object-specific sequences make it easier to remember quantities.
- Cultural development: Not all cultures developed a general-purpose number system like Hindu-Arabic numerals; instead, they created specific systems to count different objects.
Q2. Consider the extension of the Gumulgal number system beyond 6 in the same way of counting by 2s. Come up with ways of performing the different arithmetic operations (+, –, ×, ÷) for numbers occurring in this system, without using Hindu numerals. Use this to evaluate the following:
(i) (ukasar-ukasar-ukasar-ukasar-urapon) + (ukasar-ukasar- ukasar-urapon)
(ii) (ukasar-ukasar-ukasar-ukasar-urapon) – (ukasar-ukasar- ukasar)
(iii) (ukasar-ukasar-ukasar-ukasar-urapon) × (ukasar-ukasar)
(iv) (ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar) ÷ (ukasar-ukasar)
Solution:
(i) (ukasar-ukasar-ukasar-ukasar-urapon) + (ukasar-ukasar- ukasar-urapon)
= (2 + 2 + 2 + 2 + 1) + (2 + 2 + 2 + 1) = 9 + 7 = 16.
(ii) (ukasar-ukasar-ukasar-ukasar-urapon) – (ukasar-ukasar- ukasar)
= (2 + 2 + 2 + 2 + 1) – (2 + 2 + 2) = 9 – 6 = 3.
(iii) (ukasar-ukasar-ukasar-ukasar-urapon) × (ukasar-ukasar)
= (2 + 2 + 2 + 2 + 1) × (2 + 2) = 9 × 4 = 36.
(iv) (ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar) ÷ (ukasar-ukasar)
= (2 + 2 + 2 + 2 + 2 + 2 + 2 + 2) ÷ (2 + 2) = 16 ÷ 4 = 4.
Q3. Identify the features of the Hindu number system that make it efficient when compared to the Roman number system.
Solution:
The Hindu number system is far more efficient than the Roman number system because:
- Compact representation: Large numbers can be written concisely (e.g., 1,000,000), whereas Roman numerals become long and cumbersome.
- Place value: A digit’s position (units, tens, hundreds, etc.) determines its value, which Roman numerals do not have.
- Use of zero: Zero acts both as a number and a placeholder, a feature absent in Roman numerals.
- Fewer symbols: Only 10 digits (0–9) are needed to write any number, unlike Roman numerals, which require many symbols.
- Ease of arithmetic: Standard operations like addition, subtraction, multiplication, and division are straightforward, while calculations in Roman numerals are complex and often require conversion.
Textbook Page 62
Q1. Represent the following numbers in the Egyptian system: 10458, 1023, 2660, 784, 1111, 70707.
Solution:
(i) 10458
= 10000 + 100 + 100 + 100 + 100 + 10 + 10 + 10 + 10 + 10 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 =
(ii) 1023
= 1000 + 10 + 10 + 1 + 1 + 1 =
(iii) 2660
= 1000 + 1000 + 100 + 100 + 100 + 100 + 100 + 100 + 10 + 10 + 10 + 10 + 10 + 10 =
(iv) 784
= 100 + 100 + 100 + 100 + 100 + 100 + 100 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 1 + 1 + 1 + 1 =
(v) 1111
= 1000 + 100 + 10 + 1 =
(vi) 70707
= 10000 + 10000 + 10000 + 10000 + 10000 + 10000 + 10000 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 1 + 1 + 1 + 1 + 1 + 1 + 1 =
Q2. What numbers do these numerals stand for?
Solution:
(I) =(100+100) +(10+10+10+10+10+10+10) +(1+1+1) +(1+1+1)= 200 + 70 + 6 = 276.
(II) =(1000+1000+1000 + 1000) + (100 +100+100)+(10+10) + (1+1) = 4000+300 +20 +2 = 4322.
Textbook Page 63
Figure it Out
Q1. Write the following numbers in the above base-5 system using the symbols in Table 2: 15, 50, 137, 293, 651.
Solution:
(i) 15 = 5 + 5 + 5 =
(ii) 50 = 25 + 25 =
(iii) 137 = 125 + 5 + 5 + 1 + 1 =
(iv) 293 = 125 + 125 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 1 + 1 + 1 =
(v) 651 = 625 + 5 + 5 + 5 + 5 + 5 + 1 =
Q2. Is there a number that cannot be represented in our base-5 system above? Why or why not?
Solution:
There is no number that cannot be represented in base-5. Every whole number can be represented in base-5, because the system uses digits 0 to 4 combined with increasing powers of 5.
Q3. Compute the landmark numbers of a base-7 system. In general, what are the landmark numbers of a base-n system?
Solution:
In a base-7 system, starting with the first landmark number as 1, each subsequent landmark number is 7 times the previous one. They are the powers of 7:
70=1, 71, 72, 73, 74,…
More generally, in a base‑n system, the landmark numbers are the powers of n starting from n0=1:
1, n, n2, n3,…
Textbook Page 65
Figure it Out
Q1. Add the following Egyptian numerals:
Solution:
Q2. Add the following numerals that are in the base-5 system that we created:
Solution:
Textbook Page 69
Figure it Out
Q1. Can there be a number whose representation in Egyptian numerals has one of the symbols occurring 10 or more times? Why not?
Solution:
No number in the Egyptian numeral system can have any symbol appear 10 or more times. This is because:
- The Egyptian system is additive, not positional.
- Symbols are repeated to represent sums, but each symbol is used no more than 9 times.
- When a value would require repeating a symbol 10 times or more, Egyptians would instead use the next higher symbol, keeping their notation compact and systematic.
Q2. Create your own number system of base 4 and represent numbers from 1 to 16.
Solution:
I have created a number system called the Quad-Code System, which is based on base-4. Instead of regular digits, I use four special symbols:
- A = 0
- B = 1
- C = 2
- D = 3
In base-4, the place values increase as powers of 4: the rightmost place is 40=1, the next is 41=4, then 42=16, and so on. Using these symbols, any number can be written compactly.
Here’s how numbers from 1 to 16 are represented in the Quad-Code System:
- 1 → B
- 2 → C
- 3 → D
- 4 → BA
- 5 → BB
- 6 → BC
- 7 → BD
- 8 → CA
- 9 → CB
- 10 → CC
- 11 → CD
- 12 → DA
- 13 → DB
- 14 → DC
- 15 → DD
- 16 → ABA
Q3. Give a simple rule to multiply a given number by 5 in the base-5 system that we created.
Solution:
In my base-5 system using symbols (A = 0, B = 1, C = 2, D = 3, E = 4), multiplying a number by 5 is simple:
Rule: Add a zero (A) at the end of the number.
This works because in base-5, multiplying by 5 is equivalent to shifting all digits one place to the left, just like adding a zero in base-10 when multiplying by 10.
Example:
- Number: BC (which is 1×5+2=7 in decimal)
- Multiply by 5: add A at the end → BCA
- BCA in base-5 = 7×5=35 in decimal
So, adding A at the right end multiplies the number by the base itself (5).
Textbook Page 73
Figure it Out
Q1. Represent the following numbers in the Mesopotamian system —
(i) 63 (ii) 132 (iii) 200 (iv) 60 (v) 3605
Answer:
Textbook Page 80
Figure it Out
Q1. Why do you think the Chinese alternated between the Zong and Heng symbols? If only the Zong symbols were to be used, how would 41 be represented? Could this numeral be interpreted in any other way if there is no significant space between two successive positions?
Solution:
Using Zong and Heng Symbols in the Chinese Number System:
- The Chinese number system used Zong (vertical) and Heng (horizontal) symbols to clearly represent place values.
- They alternated the direction of the symbols for each place (units, tens, hundreds, etc.) to avoid confusion when reading numbers.
- This made it easy to identify which digit belonged to which place, even when the symbols were close together or spaces were small.
Example:
- Number: 41 = 4 tens + 1 unit
- Using only Zong symbols, it would be written as: ZZZZ Z (four Zong for tens, one Zong for units)
- Without alternating symbol direction or proper spacing, ZZZZ Z could be misread as 5 instead of 41, because there’s no visual clue separating tens from units.
By alternating between Zong and Heng, the Chinese system ensured place values were visually clear, making numbers easier to read and reducing the risk of mistakes in handwritten or closely packed texts.
Q2. Form a base-2 place value system using ‘ukasar’ and ‘urapon’ as the digits. Compare this system with that of the Gumulgal’s.
Solution:-
Base-2 System Using ‘ukasar’ and ‘urapon’:
We can create a base-2 place value system by assigning:
- ‘ukasar’ = 0
- ‘urapon’ = 1
This system works like the binary system, where each position from right to left represents increasing powers of 2:
- 1st place: 20=1
- 2nd place: 21=2
- 3rd place: 22=4, and so on
Examples:
- 1 → urapon
- 2 → urapon ukasar
- 3 → urapon urapon
- 4 → urapon ukasar ukasar
Each position indicates how many of that power of 2 are present, using ukasar for 0 and urapon for 1.
Comparison with the Gumulgal System:
- The Gumulgal system is additive and group-based. For example, 7 is represented as ukasar-ukasar-ukasar-urapon (2 + 2 + 2 + 1).
- The ukasar-urapon base-2 system is a place value system, making it much more efficient for representing large numbers.
- The Gumulgal method works fine for small numbers but becomes cumbersome as numbers grow.
Q3. Where in your daily lives and in which professions, do the Hindu numerals and 0, play an important role? How might our lives have been different if our number system and 0 hadn’t been invented or conceived of?
Solution:
Importance of Hindu Numerals and Zero
Hindu numerals, including the digit 0, are used every day—for telling time, counting money, reading prices, doing math in school, and writing phone numbers. Many professions, such as banking, teaching, engineering, and science, depend heavily on this number system.
Zero is especially important for place value and calculations, making it easy to write and understand large numbers.
Without zero and the Hindu number system, life would be much harder. Calculations, trade, and even writing dates would be complicated. Modern technology like computers and calculators might not exist, slowing progress in all areas of life.
Q4. The ancient Indians likely used base 10 for the Hindu number system because humans have 10 fingers and so we can use our fingers to count. But what if we had only 8 fingers? How would we be writing numbers then? What would the Hindu numerals look like if we were using base 8 instead? Base 5? Try writing the base-10 Hindu numeral 25 as base-8 and base-5 Hindu numerals, respectively. Can you write it in base-2?
Solution:
If Humans Had Only 8 Fingers
If humans had only 8 fingers, we would likely have developed a base-8 (octal) number system instead of base-10. Just as we now count from 0 to 9 in base-10, we would count from 0 to 7 in base-8. All numerals, calculations, and mathematics would be based on powers of 8.
Conversion of the decimal number 25:
- In base-8:
25 ÷ 8 = 3 remainder 1
3 ÷ 8 = 0 remainder 3
→ 25 in base-8 = 31 - In base-5:
25 ÷ 5 = 5 remainder 0
5 ÷ 5 = 1 remainder 0
1 ÷ 5 = 0 remainder 1
→ 25 in base-5 = 100 - In base-2 (binary):
25 ÷ 2 = 12 remainder 1
12 ÷ 2 = 6 remainder 0
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
→ 25 in base-2 = 11001