A Story of Numbers Class 8 Solutions Ganita Prakash Maths Chapter 3

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NCERT Class 8 Maths Chapter 3 A Story of Numbers Solutions Question Answer

Ganita Prakash Class 8 Chapter 3 Solutions A Story of Numbers

NCERT Class 8 Maths Ganita Prakash Chapter 3 A Story of Numbers Solutions Question Answer

3.1 Reema’s Curiosity

Figure It Out (Page 54)

Question 1.
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:
Suppose there are two groups of sheep.
The first group has 6 sheep and the second has 4 sheep.
We have counted using pebbles.
To add, put all pebbles in the same pouch.
To subtract, take out as many pebbles as in the pouch having fewer pebbles from the pouch having more pebbles.
Now, suppose we wish to know how many sheep will be twice the number in the first group.
For this, we count twice using pebbles and put all the pebbles in the same pouch.
Suppose we have 12 pebbles and want to divide them into three equal groups.
We put them one by one in three bowls till we exhaust all. The number of pebbles in each bowl gives the quotient.

Question 2.
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:
In method 2, numbers have been represented by English letters as a = 1, b = 2,…, z = 26.
On extending the number system in method 2, as given:
aa = 27, we can represent numbers as ab = 28, ac = 29, …, az = 52, like a base-26 system, and every number will be mapped uniquely to a letter string.
As ba = 53, bb = 54, …, bz = 78; ca = 79, … cz = 104; and so on.

Question 3.
Try making your own number system.
Solution:
Let base be 3
Then 3⁰ = 1 = A, 3¹ = 3 = B, 3² = 9 = C….,

3.2 Some Early Number Systems

Figure It Out (Pages 59)

Question 1.
Represent the following numbers in the Roman system.
(i) 1222
(ii) 2999
(iii) 302
(iv) 715
Solution:
(i) 1222 = 1000 + 100 + 100 + 10 + 10 + 2
= M + CC + XX + II
= MCCXXII

(ii) 2999 = 1000 + 1000 + 900 + 90 + 9
= MM + CM + XC + IX
= MMCMXCIX

(iii) 302 = 100 + 100 + 100 + 2
= CCC + II
= CCCII

(iv) 715 = 500 + 100 + 100 + 10 + 5
= D + CC + X + V
= DCCXV

Figure It Out (Pages 60-61)

Question 1.
A group of indigenous people on a Pacific island uses different sequences of number names to count different objects. Why do you think they do this?
Solution:
They probably use different number sequences to make counting more meaningful, accurate, and culturally appropriate, depending on the type of object.

Question 2.
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:

Question 3.
Identify the features of the Hindu number system that make it efficient when compared to the Roman number system.
Solution:
Hindu numbers have ‘0’ and a place value system, which Roman numerals do not have.
The Hindu number system is a positional system, whereas the Roman system is not.

Question 4.
Using the ideas discussed in this section, try refining the number system you might have made earlier.
Solution:
Try it yourself.

3.3 The Idea of a Base

Figure It Out (Page 62)

Question 1.
Represent the following numbers in the Egyptian system:
10458, 1023, 2660, 784, 1111, 70707
Solution:

Question 2.
What numbers do these numerals stand for?

Solution:
(i) 100 + 100 + 10 + 10 + 10 + 10 + 10 + 10 + 6 + 10
= 200 + 70 + 6
= 276

(ii) 1000 + 1000 + 1000 + 1000 + 100 + 100 + 100 + 1 + 1 + 10 + 10
= 4000 + 300 + 20 + 2
= 4322

Figure It Out (Page 63)

Question 1.
Write the following numbers in the above base-5 system using the symbols in Table 2: 15, 50, 137, 293, 651.
Solution:

Question 2.
Is there a number that cannot be represented in our base-5 system above? Why or why not?
Solution:
No, every number can be represented in our base-5 system. This is because any positive integer can be uniquely expressed as a sum of powers of 5 multiplied by coefficients or digits from 0 to 4. This is the significance of the positional number system.

Question 3.
Compute the landmark numbers of a base-7 system. In general, what are the landmark numbers of a base-n system?
Solution:
7⁰ = 1, 7¹ = 7, 7² = 49, 7³ = 343, 7⁴ = 2401
Hence, 1, 7, 49, 343, 2401 are landmark numbers of base 7.
The landmark numbers of a base-n number system are the powers of n starting from n⁰ = 1, n, n², n³,…

Figure It Out (Page 65)

Question 1.
Add the following Egyptian numerals:

Solution:

Question 2.
Add the following numerals that are in the base-5 system that we created:

Remember that in this system, 5 times a landmark number gives the next one!
Solution:

Figure It Out (Pages 69-70)

Question 1.
Can there be a number whose representation in Egyptian numerals has one of the symbols occurring 10 or more times? Why not?
Solution:
No, in the Egyptian numeral system, if a symbol occurs 10 times, it is replaced by the next landmark symbol. So, no symbol is used 10 or more times in a valid representation.

Question 2.
Create your own number system of base 4, and represent numbers from 1 to 16.
Solution:

Question 3.
Give a simple rule to multiply a given number by 5 in the base-5 system that we created.
Solution:
Rule: The Product of a landmark number with another landmark number gives a landmark number.

3.4 Place Value Representation

Figure It Out (Page 73)

Question 1.
Represent the following numbers in the Mesopotamian system:
(i) 63
(ii) 132
(iii) 200
(iv) 60
(v) 3605
Solution:

Figure It Out (Page 80)

Question 1.
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:
The ancient Chinese number system used Zong (vertical) and Heng (horizontal) rod symbols to represent numbers in a place-value format.
These symbols were alternated at each place to make it easier to identify the position of each digit.
This alternation helped prevent confusion, especially when numbers were written closely together or lacked proper spacing.
Now, if only the Zong symbol were used, how would 41 be written?
In the Chinese system,
41 = 4 tens + 1 unit.
Using only Zong symbols, it would be written as:
4 tens → | | | |
1 unit → |
So, 41 written as: | | | | |
Without alternating directions or leaving sufficient space, it could be misunderstood. A reader might interpret it as five ones (5) or as a different value like 14.
A visual clue for place value is missing. So, the alternation of Zong and Heng symbols in the Chinese numeral system was an effective method to ensure that each digit’s place value was clearly identified. Without this alternation or proper spacing, a number like 41 could easily be misread.

Question 2.
Form a base-2 place value system using ‘ukasar’ and ‘urapon’ as the digits. Compare this system with that of the Gumulgal’s.
Solution:
Let 2⁰ = 1 = A, 2¹ = 2 = B, 2² = 4 = C, 2⁴ = 16 = D,….

Both have the same base 2, but the base 2 system has many landmark numbers, whereas the Gumulgal system has only two landmark numbers.

Question 3.
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:
They are useful wherever we have to read unit numbers or do any calculation.
In case ‘0’ was not there, all the above would have become much more tedious and cumbersome.
Also, there would have been no computers.

Question 4.
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:
For base 8, the numerals would be: 0, 1, 2, 3, 4, 5, 6, 7
For base 5, the numerals would have been: 0, 1, 2, 3, 4

25₁₀ = 31₈
25 can be written as 31 in base 8.

25₁₀ = 50₅
25 can be written as 50 in base 5.

25₁₀ = 11001₂
25 can be written as 1101 in base 2.

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