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

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Class 8 Maths Chapter 3 A Story of Numbers Solutions

Ganita Prakash Class 8 Chapter 3 Solutions

Class 8 Maths Ganita Prakash Chapter 3 Solutions Power Play

Figure it Out (NCERT 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:
In method 1, we represent numbers using individual sticks
(i) Addition We can combine the two collections of sticks into one group and count the total sticks.
(ii) Subtraction We can remove sticks of one collection from another and count the remaining sticks.
(iii) Multiplication Make multiple groups of sticks of same size and then count all the sticks together.
(iv) Division We can split the total collection of sticks equally into the required number of parts and count how many sticks are there in each part.

Question 2.
One way of extending the number system in Method 2 is by using strings with more than one letter—for examples, 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:
Yes, we can extend the system by using two letter combinations like aa, ab, ac, ad, ……………, az, ba, bb, bc, ………….. and so on. This creates new unique symbols for numbers beyond the first 26 letters.

Question 3.
Try making your own number system.
Solution:
Do yourself.

(NCERT Page 57-58)

Question 1.
Quickly count the number of objects in each of the following boxes.

Up to what group size couid you immediately see the number of objects without counting?
Solution:
We can usually instantly recognize up to 4 or 5 objects without counting.

Question 2.
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:
If a number system counts only in fixed groups then it becomes very difficult to represent numbers that are not exact multiples of that group size.
To write 1345, in groups of 5, we divide it by 5 as
1345 ÷ 5 = 269

Figure It Out (NCERT Page 59)

Question 1.
Represent the following numbers in the Roman System.
(i) 1222
(ii) 2999
(iii) 302
(iv) 715
Solution:
(i) We have, 1222
First, we write 1222 as
1222 = 1000 + 100 + 100+10+10+1+1
Then, in Roman numerals,
1222 =MCCXXII

(ii) Do it same as part (i)
Ans. MMCMXCIX

(iii) Do it same as part (i)
Ans. CCCII

(iv) Do it samp as part (i)
Ans. DCCXV

(NCERT Page 60)

Question 2.
Do it yourself now LXXXVII + LXXVIII
Solution:
We have, LXXXVII + LXXVIII

So, LXXXVII + LXXVIII = CLXV

Question 3.
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:
It is not possible to multiply Roman numerals without converting them to Hindu numerals.
Here, we have V × L
As, V = 5 and L = 50
So, V × L = 5 × 50 = 250 = CCL
Next, L × D = 50 × 500 = 25000, which is not possible to write in standard Roman numerals.
Note that a bar over a Roman numeral indicates multiplication by 1000.
25000 = XXV
Next, V × D = 5 × 500 = 2500 = MMD
Also, VII × IX = 7 × 9 = 63 = LXIII
We have, CCXXXIX MDCCCLII = 231 × 1852 =427812,
which is not possible to write in standard Roman numerals.

Figure it Out (NCERT Page 60-61)

Question 1.
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:
They use different sequences of number names to count different objects because their systems are designed to specific cultural, practical or linguistic needs. These systems may reflect the way they group, use or value those objects and help to simplify counting.

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-ukasarr-irRSsar-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:
We know that urapon corresponds to 1 and ukasar corresponds to 2 and numbers in Gumulgal number system are counted in 2s. Here are the following ways of performing different arithmetic operations.

• Addition can be done by joining the number names together.
  e.g. (ukasar) + (ukasar) = ukasar-ukasar, (ukasar-ukasar) + urapon = ukasar-ukasar-urapon
• Subtraction can be done by removing some number names from a longer sequence.
  e.g. (ukasar-ukasar-urapon) – (urapon)
  = ukasar-ukasar, (ukasar-ukasar-ukasar) – (ukasar-ukasar) = ukasar
• Multiplication can be done by repeating the same word.
  e.g. ukasar × ukasar = ukasar – ukasar, (ukasar-ukasar) × ukasar
  = ukasar-ukasar-ukasar- ukasar
• Division can be done by splitting the word into equal groups.
  e.g. (ukasar-ukasar-ukasar-ukasar) + (ukasar)
  = ukasar – ukasar, (ukasar-ukasar-ukasar-ukasar) ÷ (ukasar-ukasar)
  = ukasar

Now, we have,
(i) (ukasar- ukasar-ukasar-ukasar-urapon) + (ukasar-ukasar-ukasar-urapon)
= ukasar-ukasar-ukasar-ukasar-ukasar-ukasar – ukasar-ukasar [∵ urapon + urapon = ukasar]

(ii) (ukasar-ukasar-ukasar-ukasar-urapon) – (ukasar-ukasar-ukasar) = ukasar-urapon

(iii) (ukasar-ukasar-ukasar-ukasar-urapon) × (ukasar-ukasar)
= ukasar-ukasar-ukasar-ukasar-ukasar-ukasar – ukasar – ukasar-ukasar-ukasar-ukasar-ukasar – ukasar-ukasar-ukasar-ukasar-ukasar-ukasar

(iv) (ukasar-ukasar-ukasar-ukasar-ukasar- ukasar -ukasar-ukasar) + (ukasar-ukasar) = ukasar – ukasar

Question 3.
Identify the features of the Hindu number system that make it efficient when compared to the Roman number system.
Solution:
(i) The Hindu number system, is a place value system, where Roman number system is not.
(ii) The Hindu number system has symbol for nothing that is 0, but Roman number system does not have.
(iii) The Hindu number system allows writing and reading of large numbers, while Roman number system does not allow.
(iv) The Hindu number system allows to perform addition, subtraction, multiplication and division, while Roman number system allows only basic addition.

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

Figure it Out (NCERT Page 62)

Question 1.
Represent the following numbers in the Egyptain system.
(i) 10458
Solution:
We have, 10458
First we write it as
10458 = 10000 +100 +100 + 100 + 100 + 10 + 10 + 10 + 10 + 10 + 1 + 1 + 1 + 1 + 1
If 1 + 1 = 10⁴ + 4 × 10² + 5 × 10¹ + 8 × 10°
So, in Egyption system,

(ii) 1023
Solution:
Do, it same as part (i)

(iii) 2660
Solution:
Do it same as part (i)

(iv) 784
Solution:
Do it same as part (i)

(v) 1111
Solution:
Do it same as part (i)

(vi) 70707.
Solution:
Do it same as part (i)

Question 2.
What numbers do these numerals stands for ?

Solution:
Here, we have,

So, 2 × 10² + 7 × 10 + 6 × 1 = 276

Solution:
Here, we have

So, 4 × 10³ + 3 × 10² + 2 x 10 + 2 × 1 = 4322

Figure It Out (NCERT Page 63)

Question 1.
Write the following numbers in the base-5 system using the symbols in Table 2.
(i) 15
Solution:
We have, 15 = 3 × 5¹ + 0 × 5⁰
So, base-5 representation of 15 is

(ii) 50
Solution:
We have, 50 = 2 × 5² + 0 × 5¹ + 0 × 5⁰
So, base-5 representation of 50 is

(iii) 137
Solution:
We have, 137 = 1 × 5³ + 0 × 5² + 2 × 5¹ + 2 × 5°
So, base-5 representation of 137 is

(iv) 293
Solution:
We have, 293 = 2 × 5³ + 1 × 5² + 3 × 5¹ + 3 × 5°
So, base-5 representation of 293 is

(v) 651
Solution:
We have, 651 = 1 × 5⁴ + 0 × 5³ + 1 × 5² + 0 × 5¹ + 1 × 5°
So, base-5 representation of 651 is

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 coefficient or digit from 0 to 4. This is the significance of 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:
In a base-7 system, the landmark numbers are the powers of 7,
i.e. 7° =1, 7¹ = 7, 7² = 49, 7³ = 343, 7⁴ = 2401,….
In general, the landmark numbers of a base-u systems are the powers of n,
i.e. n° = 1, n, n², n³, …………………

Figure it Out (NCERT Page 65)

Question 1.
Add the following Egyptian numerals.

Answer:
We have,

Answer:
We have,

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:
We know that
5° = 1, 5¹ = 5, 5² = 25, 5³ =125, 5⁴ = 625, 5⁶ = 3125

= 4 × 5³ + 3 × 5² + 3 × 5¹ + 4 × 5°
= 500 + 75 + 15 + 4
= 594

(NCERT Page 66-68)

Question 1.
What is any landmark number multiplied by (that is 10)? Find the following products.

Solution:

So, we observe that if any landmark number is multiplied by (10¹) then we get the successive landmark number of that given landmark number.

Question 2.
What is any landmark number multiplied by (10²)? Find the following products.

Solution:

So, we observe that when a landmark is multiplied by (10²), then the result is the second successive landmark number of that given landmark number.

Question 3.
Find the following products.

Solution:

10⁴ × 10⁶ = 10¹⁰,
which has no symbol in the Egyption number system.

Question 4.
Does this property hold true in the base-5 system that we created? Does this hold for any number system with a base?
Solution:
Yes, this property (i.e. the product of any two landmark numbers is another landmark number) holds true in the base- 5 system. Infact, it holds true in any number system with a base.

Question 5.
Now find the following products

Solution:
We have,

Solution:
We have,

Figure it Out (NCERT Page 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, there can not be any number whose representation in Egyptian numerals has one of the symbols occurring 10 or more times. It is because the Egyptian system has landmark numbers as the powers of 10. It means, when 10 symbols of the same type occur, they are replaced by the next higher symbol.

Question 2.
Create your own number system of base 4, and represent numbers from 1 to 16.
Solution:
In a base -4 system, 4 symbols would be used that are 0, 1, 2 and 3 and the landmark numbers would be the powers of 4 that are 4° = 1, 4¹ = 4, 4² = 16 and so on.
Now, representation ofbase-10 numbers 1-16 into base-4 numbers.

Base-10 system	Numbers of base-10 system as a sum of landmark numbers of base-4 system	Base-4 system
1	1 = 1 × 4° Base-4 system	1
2	2 = 2 × 4°	2
3	3 = 3 × 4° 3	3
4	4 = 1 × 41 + 0 × 4°	10
5	5 = 1 × 41 + 1 × 4°	11
6	6 = 1 × 41 + 2 × 4°	12
7	7 = 1 × 41 + 3 × 4°	13
8	8 = 2 × 41 + 0 × 4°	20
9	9 = 2 × 41 + 1 × 4°	21
10	10 = 2  × 41 + 2  ×  4°	22
11	11 = 2 × 41 + 3 × 4°	23
12	12 = 3  × 41 + 0  ×  4°	30
13	13 = 3 ×  41 + 1 ×  4°	31
14	14 = 3  ×  41 + 2  ×  4°	32
15	15 = 3 ×  41 + 3 × 4°	33
16	16 = 1 ×  42 + 0 × 41 + 0 × 4°	100

Question 3.
Given a simple rule to multiply a given number by 5 in the base-5 system that we created.
Solution:
In base-5 system, the symbols for landmark numbers we use, are

So, in order to multiply any given number by 5 in the base-5 system, we only need to add a ‘0’ at the end of the given number.
Symbolically, we can use the distributivity property.

Figure it Out (NCERT Page 73)

Question 1.
Represent the following numbers in the Mesopotamian system.
(i) 63
Solution:
We have, 63
First, we break it as
6³ = 1 × 60 + 3
Now, we write the symbols for each digit and corresponding landmark numbers together to get the required Mesopotamian numeral.

(ii) 132
Solution:
Do it same as part (i)

(iii) 200
Solution:
Do it same as part (i)

(iv) 60
Solution:
Do it same as part (i)

(v) 3605
Solution:
Do it same as part (i)

(NCERT Page 76)

Question 1.
Represent the following numbers using the Mayan system.
(i) 77
Solution:
We have, 77
First, we write it as
77 = 3 × 20 + 17
Now, we write the landmark numbers vertically from top to bottom, along with their count of occurrence and respective symbols as.

(ii) 100
Solution:
Do it same as part (i)

(iii) 361
Solution:
Do it same as part (i)

(iv) 721
Solution:
Do it same as part (i)

Figure it Out (NCERT Page 80)

Question 1.
Why do you think the Chinese alternated between the Zong and Heng symbols? If only the Zong A 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 Chinese alternated between the Zong and Heng symbols to clearly distinguish place values in their decimal system.
If only Zong (vertical rods) symbols were used then 41 would be represented as four vertical rods followed by one vertical rod, i.e.

This could be easily misread as 5, if there is no significant space between two successive positions.

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:
For a base-2 system, we use two symbols 0 and 1. Here, we use urapon and ukasar to denote 0 to 1, respectively. Then, the new base-2 system in Gumulgal’s style is given below.

Decimal system	New Base-2 system	Gumulgal’s style Base-2 system
0	–	urapon
1	–	ukasar
2	2 = 1 × 21 + 0 × 2° = 10	ukasar- urapon
3	3 = 1 × 21 + 1 × 2° = 11	ukasar- ukasar
4	4 = 1 × 22 + 0 × 2‘ + 0 × 2° = 100	ukasar-urapon-urapon
5	5 = 1 × 22 + 0 × 21 + 1 × 2° = 101	ukasar- urapon- ukasar
6	6 = 1 × 22 + 1 × 21 + 0 × 2° = 110	ukasar- ukasar- urapon

Comparison between original Gumulgal system and our New Base-2 system.
(i) We can count unlimited number using our base- 2 system, but not in Gumulgal’s. Gumulgals used ‘ras’ after 6.
(ii) Our base-2 system is a place value system, but not the Gumulgal’s.

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:
Do it yourself.

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:
Suppose if we had only 8-fingers then we might have adopted a base-8 system.
We would have 8 symbols 0, 1, 2, 3, 4, 5, 6, 7 and the landmark numbers would be 8⁰, 8¹, 8², 8³,….
Also, if we were using base-5 system then we would have 5 symbols 0, 1, 2, 3,4 and the landmark numbers would be 5°, 5¹, 5², 5³, 5⁴.
Now, we write the Hindu numeral 25 as base-8 system as follows.
25 = 3 × 8¹ + 1 × 8°
So, 25 can be written as 318 in base- 8 number system. Also, we write the Hindu numeral 25 as base-5 system as follows.
25 = 1 × 5² + 0 × 5¹ + 0 × 5°
So, 25 can be written as 100 5 in base-5 number system. Next, in base- 2 number system,
25 = 1 × 2⁴ + 1 × 2³ + 0 × 2² + 0 × 2¹ + 1 × 2°
So, 25 = 110012