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Overview

Step-by-step NCERT solutions for A Story of Numbers (Chapter 3, NCERT Class 8 Mathematics) — the full working for every question, not just the final answer. You can also read the A Story of Numbers textbook chapter.

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What these solutions cover

All 21 questions in A Story of Numbers are solved in the PDF. Here's what's inside, exercise by exercise:

The Mechanism of Counting

  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.
  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!

The Roman Numerals

  1. Represent the following numbers in the Roman system.
    • (i) 1222
    • (ii) 2999
    • (iii) 302
    • (iv) 715

Some Early Number Systems

  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?
  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) –…
  3. Identify the features of the Hindu number system that make it efficient when compared to the Roman number system.

The Egyptian Number System

  1. Represent the following numbers in the Egyptian system: 10458, 1023, 2660, 784, 1111, 70707.
  2. What numbers do these numerals stand for?
    • (i) [2 coil-symbols, 7 arch-symbols, 6 stroke-symbols and a few more — see textbook image]
    • (ii) [4 lotus-symbols, 3 coil-symbols, 2 arch-symbols, 2 stroke-symbols — see textbook image]

Variations on the Egyptian System and the Notion of Base

  1. Write the following numbers in the above base-5 system using the symbols in Table 2: 15, 50, 137, 293, 651.
  2. Is there a number that cannot be represented in our base-5 system above? Why or why not?
  3. Compute the landmark numbers of a base-7 system. In general, what are the landmark numbers of a base-n system?

Advantages of a Base-n System

  1. Add the following Egyptian numerals:
    • (i) [two Egyptian numerals shown in the textbook, page 65]
    • (ii) [two more Egyptian numerals shown in the textbook, page 65]
  2. Add the following numerals that are in the base-5 system that we created: [circle + hexagon + pentagon + square + triangle + triangle] + [circle + circle + hexagon + square + square + triangle + triangle] (Remember that in this system, 5 times a landmark number gives the next one!)

Shortcomings of the Egyptian System

  1. Can there be a number whose representation in Egyptian numerals has one of the symbols occurring 10 or more times? Why not?
  2. Create your own number system of base 4, and represent numbers from 1 to 16.
  3. Give a simple rule to multiply a given number by 5 in the base-5 system that we created.

The Mesopotamian Number System

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

The Hindu Number System

  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?
  2. Form a base-2 place value system using 'ukasar' and 'urapon' as the digits. Compare this system with that of the Gumulgal's.
  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?
  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…
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More solutions in Ganita Prakash

01A Square and a Cube02Power Play04Quadrilaterals05Number Play06We Distribute, Yet Things Multiply07Proportional Reasoning-108Fractions in Disguise09The Baudhayana-Pythagoras Theorem10Proportional Reasoning-211Exploring Some Geometric Themes12Tales by Dots and Lines13Algebra Play14Area
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More NCERT Solutions for Class 8

Read the A Story of Numbers textbook chapter / PDF, or browse all NCERT Class 8 Mathematics solutions.

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