Chapter 3 A Story of Numbers (Class 8 - Latest Maths NCERT (Ganita Prakash I) Solutions)
Looking for accurate and insightful NCERT Solutions for Chapter 3: A Story of Numbers? You’ve come to the right place! This page provides comprehensive, step-by-step answers to the exercises in your Ganita Prakash I textbook. We help you navigate the fascinating historical journey of mathematics, from the primitive one-to-one mapping of the Stone Age to the sophisticated symbols we use today. Our solutions are designed to show you that math is a creative invention, making the evolution of counting relatable and easy to understand.
Our solutions offer detailed breakdowns of various ancient systems, including the Tally Marks of the Ishango bone and the "landmark number" logic of the Roman Number System. We provide clear guidance on solving problems related to Base-n Systems, such as the Mesopotamian base-60, Egyptian base-10, and Mayan base-20 systems. By following our step-by-step explanations, you will understand the historical "need for a placeholder" and how different civilizations tackled mathematical ambiguity.
A key focus of our resources is the Hindu Number System and the revolutionary concept of zero. We provide historical and mathematical context for the Bakhshali Manuscript and the contributions of legendary Indian mathematicians like Aryabhata and Brahmagupta. These resources, curated by learningspot.co, explain how the decimal place-value system transformed global commerce and science, ensuring you master both the history and the logic of the numbers we use every day.
Figure It Out (Page No. 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.
Answer:
Given: A collection of sticks where each stick ($|$) represents a unit. This is a basic unary system often used in primary education and tallying in India.
To Find: A purely visual and physical method for performing Addition, Subtraction, Multiplication, and Division using only sticks.
Solution:
1. Addition (Samyojan)
To add two collections of sticks, we simply bring both collections together into a single new pile. The resulting pile is the sum.
For example, if we have a pile of three sticks and a pile of two sticks:
$\begin{array}{cc} & | & | & | \\ + & & | & | \\ \hline & | & | & | & | & | \\ \hline \end{array}$$||| + || = |||||$
... (i)
2. Subtraction (Vyavkalan)
To subtract one collection from another, place the sticks of both collections in two parallel rows. For every stick in the smaller row, remove one corresponding stick from the larger row. The remaining sticks in the larger row form the result.
For example, taking away two sticks from five sticks:
$\begin{array}{cc} & | & | & | & | & | \\ - & & & & | & | \\ \hline & & & | & | & | \\ \hline \end{array}$$||||| - || = |||$
... (ii)
3. Multiplication (Gunan)
Multiplication can be treated as repeated addition. To multiply, lay down the sticks of the first collection. For each stick in that collection, place a group of sticks equal to the second collection.
For example, multiplying two sticks by three sticks:
$\begin{array}{cc}& & | & | & | \\ \times & & & & | & | \\ \hline && | & | & | \\ & & | & | & | \\ \hline & | & | & | & | & | & | \\ \hline \end{array}$The total sticks in all groups combined is the product.
4. Division (Bhajan)
Division is the process of repeated subtraction or grouping. To divide a large collection by a smaller one, keep taking away groups of sticks equal to the smaller collection from the large one until no more full groups can be taken.
For example, dividing six sticks by two sticks:
$\begin{array}{r} | & | & | \phantom{||||||)} \\ ||{\overline{\smash{\big)}\,||||||\phantom{)}}} \\ \underline{-~\phantom{()}(||)\phantom{)}} \\ ||||\phantom{)} \\ \underline{-~\phantom{()}(||)} \\ ||\phantom{)} \\ \underline{-~\phantom{()}(||)} \\ 0\phantom{)} \end{array}$The number of groups we successfully removed is the result (quotient). If any sticks are left over that cannot form a full group, they are the remainder.
Alternate Solution for Multiplication:
We can use the Intersection Method. Lay the first collection of sticks horizontally and the second collection of sticks vertically across them. The total number of points where the sticks cross (overlap) gives the result.
If we cross $||$ (two horizontal) with $|||$ (three vertical):
| Horizontal Sticks | Vertical Sticks | Intersections (Total Sticks) |
| $||$ | $|||$ | $||||||$ |
This provides a visual way to find the product without needing to count verbally.
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!
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Question 3. Try making your own number system.
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Figure It Out (Page No. 59)
Question. Represent the following numbers in the Roman system.
(i) $1222$
(ii) $2999$
(iii) $302$
(iv) $715$
Answer:
Figure It Out (Page No. 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?
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Question 2. Consider the extension of the Gumulgal number system beyond $6$ in the same way of counting by $2$s. Come up with ways of performing the different arithmetic operations ($+$, $-$, $\times$, $\div$) 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) $$ \times (ukasar-ukasar)$
(iv) $(ukasar-ukasar-ukasar-ukasar-ukasar-ukasar $$-ukasar-ukasar) \div (ukasar-ukasar)$
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Question 3. Identify the features of the Hindu number system that make it efficient when compared to the Roman number system.
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Question 4. Using the ideas discussed in this section, try refining the number system you might have made earlier.
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Figure It Out (Page No. 62)
Question 1. Represent the following numbers in the Egyptian system: $10458$, $1023$, $2660$, $784$, $1111$, $70707$.
Answer:
Question 2. What numbers do these numerals stand for?
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Figure It Out (Page No. 63)
Question 1. Write the following numbers in the above base-$5$ system using the symbols in Table $2$: $15$, $50$, $137$, $293$, $651$.
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Question 2. Is there a number that cannot be represented in our base-$5$ system above? Why or why not?
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Question 3. Compute the landmark numbers of a base-$7$ system. In general, what are the landmark numbers of a base-$n$ system?
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Figure It Out (Page No. 65)
Question 1. Add the following Egyptian numerals:
(i)
(ii)
Answer:
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!
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Figure It Out (Page No. 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?
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Question 2. Create your own number system of base $4$, and represent numbers from $1$ to $16$.
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Question 3. Give a simple rule to multiply a given number by $5$ in the base-$5$ system that we created.
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Figure It Out (Page No. 73)
Question. Represent the following numbers in the Mesopotamian system —
(i) $63$
(ii) $132$
(iii) $200$
(iv) $60$
(v) $3605$
Answer:
Figure It Out (Page No. 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?
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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.
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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?
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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-$2$?
The map shows the locations of the different civilisations. They existed in different time periods.
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