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Mathematics
A Story of Numbers
NCERT Solutions
NCERT Solutions
A Story of Numbers
27 Solutions
Exercise:
All Exercises
Figure it Out (Advantages of a Base-n System)
Figure it Out (End of Chapter)
Figure it Out (End of Section 3.2)
Figure it Out (End of Section 3.3)
Figure it Out (The Egyptian Number System)
Figure it Out (The Mayan Number System)
Figure it Out (The Mechanism of Counting)
Figure it Out (The Mesopotamian Number System)
Figure it Out (The Roman Numerals)
Figure it Out (Variations on the Egyptian System)
Q1
Figure it Out (Advantages of a Base-n System)
Add the following Egyptian numerals:
(i)
(2 coiled ropes, 3 hobbles, 4 sticks) + (3 coiled ropes, 8 hobbles, 7 sticks)
(ii)
(2 fingers, 3 coiled ropes, 5 hobbles, 6 sticks) + (4 fingers, 8 coiled ropes, 7 hobbles, 9 sticks)
Q2
Figure it Out (Advantages of a Base-n System)
Add the following numerals that are in the base-5 system that we created: ◯◯□△△ + ◯◯◯□□□△△
Q3
Figure it Out (Advantages of a Base-n System)
Now find the following products- (i) (999 99 ∩∩ ||) × ∩ (ii) (P 𓆼 ∩) × 9
Q1
Figure it Out (End of Chapter)
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?
Q2
Figure it Out (End of Chapter)
Form a base-2 place value system using 'ukasar' and 'urapon' as the digits. Compare this system with that of the Gumulgal's.
Q3
Figure it Out (End of Chapter)
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?
Q4
Figure it Out (End of Chapter)
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?
Q1
Figure it Out (End of Section 3.2)
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?
Q2
Figure it Out (End of Section 3.2)
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-ukasarukasar)
(iii)
(ukasar-ukasar-ukasar-ukasar-urapon) × (ukasar-ukasar)
(iv)
(ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar) ÷ (ukasar-ukasar)
Q3
Figure it Out (End of Section 3.2)
Identify the features of the Hindu number system that make it efficient when compared to the Roman number system.
Q4
Figure it Out (End of Section 3.2)
Using the ideas discussed in this section, try refining the number system you might have made earlier.
Q1
Figure it Out (End of Section 3.3)
Can there be a number whose representation in Egyptian numerals has one of the symbols occurring 10 or more times? Why not?
Q2
Figure it Out (End of Section 3.3)
Create your own number system of base 4, and represent numbers from 1 to 16.
Q3
Figure it Out (End of Section 3.3)
Give a simple rule to multiply a given number by 5 in the base-5 system that we created.
Q1
Figure it Out (The Egyptian Number System)
Represent the following numbers in the Egyptian system: 10458, 1023, 2660, 784, 1111, 70707.
Q2
Figure it Out (The Egyptian Number System)
What numbers do these numerals stand for? (a) one tadpole, two fingers, three coiled ropes, four hobbles, five sticks. (b) one man, one tadpole, one finger, one coiled rope, one hobble, one stick.
Q1
Figure it Out (The Mayan Number System)
Represent the following numbers using the Mayan system:
(i)
77
(ii)
100
(iii)
361
(iv)
721
Q1
Figure it Out (The Mechanism of Counting)
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.
Q2
Figure it Out (The Mechanism of Counting)
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!
Q3
Figure it Out (The Mechanism of Counting)
Try making your own number system.
Q1
Figure it Out (The Mesopotamian Number System)
Represent the following numbers in the Mesopotamian system -
(i)
63
(ii)
132
(iii)
200
(iv)
60
(v)
3605
Q1
Figure it Out (The Roman Numerals)
Represent the following numbers in the Roman system.
(i)
1222
(ii)
2999
(iii)
302
(iv)
715
Q2
Figure it Out (The Roman Numerals)
Do it yourself now: (b) LXXXVII + LXXVIII
Q3
Figure it Out (The Roman Numerals)
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.
Q1
Figure it Out (Variations on the Egyptian System)
Write the following numbers in the above base-5 system using the symbols in Table 2: 15, 50, 137, 293, 651.
Q2
Figure it Out (Variations on the Egyptian System)
Is there a number that cannot be represented in our base-5 system above? Why or why not?
Q3
Figure it Out (Variations on the Egyptian System)
Compute the landmark numbers of a base-7 system. In general, what are the landmark numbers of a base-n system?
More from this chapter
Chapter overview
Important Points
Practice Questions
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