CLASS 8 NCERT WORKSHEET  CH-3 A STORY OF NUMBERS WITH ANSWER KEY

A Journey Through Numbers: Exploring Early Number Systems

Instructions: Answer the following questions based on the provided text, "A Story of Numbers." Ensure your answers are directly supported by the source material.

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Section 1: Foundations of Counting & Early Methods

1. When did humans first feel the need to count, and for what purposes?

   • Humans had the need to count as early as the Stone Age.
   • They counted to determine the quantity of food, the number of animals in their livestock, details regarding trades of goods, and the number of offerings given in rituals. They also wanted to keep track of passing days to predict important events like the new moon, full moon, or onset of a season.

2. What is a "one-to-one mapping" in the context of early counting? Provide an example from the text.

   • A one-to-one mapping is a way of associating each object to be counted with a distinct item, such as a stick or a sound, such that no two objects are associated or mapped to the same stick.
   • For example, early humans could keep one stick for every cow in a herd; the final collection of sticks would then represent the number of cows, which could be used to check if any cows were missing.

3. Name two of the oldest known mathematical artifacts featuring tally marks and state their approximate age and discovery location.

   • The Ishango bone, discovered in the Democratic Republic of Congo, dates back 20,000 to 35,000 years.
   • The Lebombo bone, discovered in South Africa, is an even older tally stick with 29 notches, estimated to be around 44,000 years old.

4. Describe the counting method of the Gumulgal people. What was a significant observation about their system compared to other cultures?

   • The Gumulgal, an indigenous group in Australia, formed their number names by counting in 2s. For instance, 3 was "ukasar-urapon" (2+1), and 4 was "ukasar-ukasar" (2+2). Any number greater than 6 was called "ras".
   • A significant and puzzling observation is that despite being geographically far apart and having no trace of contact, the Gumulgal, Bakairi (South America), and Bushmen (South Africa) developed equivalent number systems.

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Section 2: The Roman Numerals

1. List the "landmark numbers" and their associated symbols in the Roman numeral system up to 1,000.

   • I = 1
   • V = 5
   • X = 10
   • L = 50
   • C = 100
   • D = 500
   • M = 1,000

2. Represent the number 2367 in Roman numerals.

   • 2367 = 1000 + 1000 + 100 + 100 + 100 + 50 + 10 + 5 + 1 + 1
   • In Roman numerals, this is MMCCCLXII.

3. What was a significant drawback of the Roman numeral system, especially for arithmetic operations?

   • Despite its relative efficiency, the Roman system does not lend itself to an easy performance of arithmetic operations, particularly multiplication and division. People using this system often relied on a calculating tool called the abacus for computations, which only specially trained people used.

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Section 3: The Idea of a Base & Egyptian System

1. Define a "base-n number system." What are its landmark numbers?

   • A base-n number system is a system where the first landmark number is 1 (n⁰), and every subsequent landmark number is obtained by multiplying the current landmark number by a fixed number 'n'.
   • Its landmark numbers are therefore the powers of n: n⁰, n¹, n², n³, and so on.

2. What is the base of the Egyptian number system? What are its landmark numbers?

   • The Egyptian number system is a base-10 system, also known as a decimal system.
   • Its landmark numbers are powers of 10: 1, 10, 100, 1,000, 10,000, 100,000, 1,000,000 (10⁰, 10¹, 10², etc.).

3. Represent the number 324 in the Egyptian system using its principles (you can describe the symbols if you cannot draw them).

   • 324 = 100 + 100 + 100 + 10 + 10 + 1 + 1 + 1 + 1.
   • In Egyptian numerals, this would be represented by three symbols for 100, two symbols for 10, and four symbols for 1. (The source image shows coiled ropes for 100, heel bones for 10, and single strokes for 1).

4. What is a significant advantage of a base-n system, particularly for arithmetic operations, compared to systems like Roman numerals?

   • The primary advantage is that it simplifies arithmetic operations.
   • Because the landmark numbers in a base-n system are powers of a single number (n), the product of any two landmark numbers is always another landmark number. This property significantly streamlines multiplication, unlike the Roman system where multiplication is difficult due to inconsistent grouping sizes for landmark numbers.

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Section 4: Place Value Representation & The Hindu Number System

1. Define a "positional number system" or "place value system."

   • A positional number system, also known as a place value system, is a number system with a base that uses the position of each symbol to determine the landmark number that it is associated with. This innovative idea allows for representing an unending sequence of numbers using only a finite number of different symbols.

2. Which ancient civilizations are mentioned as having used place value representations?

   • Mesopotamian (Babylonian)
   • Mayan
   • Chinese
   • Indian (Hindu)

3. What was the base of the Mesopotamian (Babylonian) number system? What was a key ambiguity in their early system, and how did later Mesopotamians attempt to address it?

   • The Mesopotamian system was a base-60 system, also called the sexagesimal system.
   • A key ambiguity was that a blank space was used to indicate the absence of a power of 60 in a position. This made it difficult to maintain consistent spacing and led to confusion, as the same numeral could be read in different ways.
   • To address this, later Mesopotamians used a "placeholder" symbol to denote a blank space, similar to our modern zero (0).

4. Describe the key features of the modern Hindu number system that make it highly efficient and widely used today.

   • It is a place value system.
   • It uses ten symbols (digits 0 through 9).
   • Crucially, it incorporates the digit 0 (zero), which functions both as a placeholder and as a number in its own right.
   • The use of 0 as a digit and the use of a single digit in each position ensures that this system does not lead to any ambiguity when reading or writing numerals.
   • This system allows for unambiguous representation of all numbers using a finite set of symbols and facilitates efficient computation.
   • It is considered one of the greatest, most creative, and most influential inventions of all time, forming the basis of much of modern science, technology, and mathematics.

5. Who were some key figures involved in the development and transmission of the Hindu number system, as mentioned in the text?

   • The Bakhshali manuscript (c. 3rd century CE) contains the first known instance of numbers written using ten digits, including 0.
   • Aryabhata (c. 499 CE) was the first mathematician to fully explain and perform elaborate scientific computations with the Indian system.
   • Al-Khwārizmī (c. 825 CE), a Persian mathematician, popularized the Hindu numerals in the Arab world through his book On the Calculation with Hindu Numerals.
   • Al-Kindi (c. 830 CE), a noted philosopher, also contributed to popularizing these numerals in the Arab world.
   • The Italian mathematician Fibonacci (c. 1200 CE) significantly advocated for Europe to adopt the Indian numerals.
   • Brahmagupta (628 CE) codified the use of 0 as a number like any other, on which basic arithmetic operations could be performed, laying foundations for modern mathematics.

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Section 5: Reflection & Comparison

1. Explain why the term "Hindu numerals" or "Indian numerals" is considered more accurate than "Arabic numerals" for the system we use today.

   • The modern number system originated and was developed in India around 2000 years ago.
   • While European scholars learned these numerals from the Arab world and called them "Arabic numerals," Arab scholars themselves, such as Al-Khwārizmī and Al-Kindi, referred to them as "Hindu numerals".
   • The term "Hindu" in this context refers to the geography and people from whom these numbers came, not a religion. The correction of this terminology is now occurring in many textbooks globally.

2. Based on the summary of ideas in number representation:

   • List the five key ideas in the evolution of number representation.

     1. Counting in groups of a single number (e.g., Gumulgal's system).
     2. Grouping using landmark numbers (e.g., Roman numerals).
     3. The idea of a base, by choosing powers of a number as landmark numbers (e.g., Egyptian system).
     4. The idea of a place value system, using positions to denote landmark numbers (e.g., Mesopotamian, Chinese).
     5. The idea of 0 as a positional digit and as a number (e.g., Hindu system).

   • Which of these ideas represents the "highest point" in the history of the evolution of number systems, and why?

     • The idea of place value marks the highest point in the history of the evolution of number systems.

When combined with the idea of 0 as a positional digit and as a number, it offers a remarkably elegant solution to the problem of representing the unending sequence of numbers using only a finite number of distinct symbols. Furthermore, it vastly simplifies computations and laid the foundations for modern mathematics and science.

A Journey Through Numbers: Exploring Early Number Systems - ANSWER KEY

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Section 1: Foundations of Counting & Early Methods

When did humans first feel the need to count, and for what purposes?

Humans had the need to count as early as the Stone Age.

They counted to determine the quantity of food, the number of animals in their livestock, details regarding trades of goods, and the number of offerings given in rituals. They also wanted to keep track of passing days to predict important events like the new moon, full moon, or onset of a season.

What is a "one-to-one mapping" in the context of early counting? Provide an example from the text.

A one-to-one mapping is a way of associating each object to be counted with a distinct item, such as a stick or a sound, such that no two objects are associated or mapped to the same stick.

For example, early humans could keep one stick for every cow in a herd; the final collection of sticks would then represent the number of cows, which could be used to check if any cows were missing.

Name two of the oldest known mathematical artifacts featuring tally marks and state their approximate age and discovery location.

The Ishango bone, discovered in the Democratic Republic of Congo, dates back 20,000 to 35,000 years.

The Lebombo bone, discovered in South Africa, is an even older tally stick with 29 notches, estimated to be around 44,000 years old.

Describe the counting method of the Gumulgal people. What was a significant observation about their system compared to other cultures?

The Gumulgal, an indigenous group in Australia, formed their number names by counting in 2s. For instance, 3 was "ukasar-urapon" (2+1), and 4 was "ukasar-ukasar" (2+2). Any number greater than 6 was called "ras".

A significant and puzzling observation is that despite being geographically far apart and having no trace of contact, the Gumulgal, Bakairi (South America), and Bushmen (South Africa) developed equivalent number systems.

---

• Section 2: The Roman Numerals

List the "landmark numbers" and their associated symbols in the Roman numeral system up to 1,000.

I = 1
V = 5
X = 10
L = 50
C = 100
D = 500
M = 1,000

Represent the number 2367 in Roman numerals.

2367 = 1000 + 1000 + 100 + 100 + 100 + 50 + 10 + 5 + 1 + 1
In Roman numerals, this is MMCCCLXII.

What was a significant drawback of the Roman numeral system, especially for arithmetic operations?

Despite its relative efficiency, the Roman system does not lend itself to an easy performance of arithmetic operations, particularly multiplication and division. People using this system often relied on a calculating tool called the abacus for computations, which only specially trained people used.

• ---

• Section 3: The Idea of a Base & Egyptian System

Define a "base-n number system." What are its landmark numbers?

A base-n number system is a system where the first landmark number is 1 (n⁰), and every subsequent landmark number is obtained by multiplying the current landmark number by a fixed number 'n'.
Its landmark numbers are therefore the powers of n: n⁰, n¹, n², n³, and so on.

What is the base of the Egyptian number system? What are its landmark numbers?

The Egyptian number system is a base-10 system, also known as a decimal system.
Its landmark numbers are powers of 10: 1, 10, 100, 1,000, 10,000, 100,000, 1,000,000 (10⁰, 10¹, 10², etc.).

Represent the number 324 in the Egyptian system using its principles (you can describe the symbols if you cannot draw them).

324 = 100 + 100 + 100 + 10 + 10 + 1 + 1 + 1 + 1.
In Egyptian numerals, this would be represented by three symbols for 100 (coiled ropes), two symbols for 10 (heel bones), and four symbols for 1 (single strokes).

What is a significant advantage of a base-n system, particularly for arithmetic operations, compared to systems like Roman numerals?

The primary advantage is that it simplifies arithmetic operations.
Because the landmark numbers in a base-n system are powers of a single number (n), the product of any two landmark numbers is always another landmark number. This property significantly streamlines multiplication, unlike the Roman system where multiplication is difficult due to inconsistent grouping sizes for landmark numbers.

• ---

• Section 4: Place Value Representation & The Hindu Number System

Define a "positional number system" or "place value system."

A positional number system, also known as a place value system, is a number system with a base that uses the position of each symbol to determine the landmark number that it is associated with. This innovative idea allows for representing an unending sequence of numbers using only a finite number of different symbols.

Which ancient civilizations are mentioned as having used place value representations?

Mesopotamian (Babylonian)
Mayan
Chinese
Indian (Hindu)

What was the base of the Mesopotamian (Babylonian) number system? What was a key ambiguity in their early system, and how did later Mesopotamians attempt to address it?

The Mesopotamian system was a base-60 system, also called the sexagesimal system.
A key ambiguity was that a blank space was used to indicate the absence of a power of 60 in a position. This made it difficult to maintain consistent spacing and led to confusion, as the same numeral could be read in different ways (e.g., 60 and 3600 had ambiguous representations).
To address this, later Mesopotamians used a "placeholder" symbol to denote a blank space, similar to our modern zero (0).

Describe the key features of the modern Hindu number system that make it highly efficient and widely used today.

It is a place value system.
It uses ten symbols (digits 0 through 9).
Crucially, it incorporates the digit 0 (zero), which functions both as a placeholder and as a number in its own right.
The use of 0 as a digit and the use of a single digit in each position ensures that this system does not lead to any ambiguity when reading or writing numerals.
This system allows for unambiguous representation of all numbers using a finite set of symbols and facilitates efficient computation.
It is considered one of the greatest, most creative, and most influential inventions of all time, forming the basis of much of modern science, technology, and mathematics.

Who were some key figures involved in the development and transmission of the Hindu number system, as mentioned in the text?

The Bakhshali manuscript (c. 3rd century CE) contains the first known instance of numbers written using ten digits, including 0.
Aryabhata (c. 499 CE) was the first mathematician to fully explain and perform elaborate scientific computations with the Indian system.
Brahmagupta (628 CE) codified the use of 0 as a number like any other, on which basic arithmetic operations could be performed, laying foundations for modern mathematics.
Al-Khwārizmī (c. 825 CE), a Persian mathematician, popularized the Hindu numerals in the Arab world through his book On the Calculation with Hindu Numerals.
Al-Kindi (c. 830 CE), a noted philosopher, also contributed to popularizing these numerals in the Arab world.
The Italian mathematician Fibonacci (c. 1200 CE) significantly advocated for Europe to adopt the Indian numerals.

• ---

• Section 5: Reflection & Comparison

Explain why the term "Hindu numerals" or "Indian numerals" is considered more accurate than "Arabic numerals" for the system we use today.

The modern number system originated and was developed in India around 2000 years ago.
While European scholars learned these numerals from the Arab world and called them "Arabic numerals," Arab scholars themselves, such as Al-Khwārizmī and Al-Kindi, referred to them as "Hindu numerals".
The term "Hindu" in this context refers to the geography and people from whom these numbers came, not a religion. The correction of this terminology is now occurring in many textbooks globally.

Based on the summary of ideas in number representation:

• List the five key ideas in the evolution of number representation.

  • Counting in groups of a single number.
  • Grouping using landmark numbers.
  • The idea of a base, by choosing powers of a number as landmark numbers.
  • The idea of a place value system, using positions to denote landmark numbers.
  • The idea of 0 as a positional digit and as a number.

• Which of these ideas represents the "highest point" in the history of the evolution of number systems, and why?

  • The idea of place value marks the highest point in the history of the evolution of number systems.
  • When combined with the idea of 0 as a positional digit and as a number, it offers a remarkably elegant solution to the problem of representing the unending sequence of numbers using only a finite number of distinct symbols. Furthermore, it vastly simplifies computations and laid the foundations for modern mathematics and science.