Question bank 1 A story of numbers 8 maths worksheet 2025-2026

 Question bank 1 A story of numbers 8 maths worksheet 2025-2026

Class 8 Maths (Ganita Prakash) - Chapter 3: A Story of Numbers

Study Material & Competency-Based Worksheet

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Multiple Choice Questions (1 Mark Each)

1. The Ishango bone, discovered in Congo, is one of the oldest known mathematical artifacts. What does it feature?
a) Roman numerals
b) Tally marks arranged in columns
c) Egyptian hieroglyphs
d) Mesopotamian cuneiform
Answer: b) Tally marks arranged in columns
(Competency: Historical Awareness)

2. Which civilization first developed the number system using ten symbols (0-9) with place value?
a) Mesopotamian
b) Roman
c) Indian
d) Egyptian
Answer: c) Indian
(Competency: Conceptual Understanding)

3. In the Roman numeral system, what does the symbol 'L' represent?
a) 50
b) 100
c) 500
d) 1000
Answer: a) 50
(Competency: Recall)

4. The Mayan number system was primarily a base-____ system with a modification for the third place value.
a) 10
b) 20
c) 5
d) 60
Answer: b) 20
(Competency: Conceptual Understanding)

5. What is the value of the Roman numeral MCMXCIX?
a) 1999
b) 1899
c) 1989
d) 1889
Answer: a) 1999
(Competency: Application)

6. Which number system used a seashell-like symbol as a placeholder for zero?
a) Egyptian
b) Roman
c) Mayan
d) Chinese
Answer: c) Mayan
(Competency: Historical Awareness)

7. The Chinese rod numeral system alternated between Zong and Heng orientations to:
a) Make it look artistic
b) Prevent ambiguity in place value
c) Follow ancient traditions
d) Represent odd and even numbers
Answer: b) Prevent ambiguity in place value
(Competency: Reasoning)

8. In a base-5 number system, what is the value of '43' (base-5) in base-10?
a) 43
b) 23
c) 19
d) 53
Answer: b) 23
(Competency: Application)

9. Which of these is NOT a feature of the Hindu-Arabic number system?
a) Place value
b) Use of zero as a placeholder and a number
c) Additive principle for all numerals
d) Base-10 system
Answer: c) Additive principle for all numerals
(Competency: Conceptual Understanding)

10. The Lebombo bone, estimated to be around 44,000 years old, was discovered in:
a) India
b) South Africa
c) Iraq
d) Egypt
Answer: b) South Africa
(Competency: Recall)

11. The Roman numeral for 2999 is:
a) MMCMXCIX
b) MMDDDXXXXVIIII
c) MMIM
d) MCMXCIX
Answer: a) MMCMXCIX
(Competency: Application)

12. The Egyptian symbol for 100 was:
a) ∩
b) n
c) |
d) 9
Answer: d) 9
(Competency: Recall)

13. In the Mesopotamian number system, the base used was:
a) 10
b) 20
c) 60
d) 5
Answer: c) 60
(Competency: Recall)

14. The Fibonacci sequence is named after an Italian mathematician who popularized:
a) Roman numerals
b) Egyptian fractions
c) Hindu-Arabic numerals in Europe
d) Mayan calendar system
Answer: c) Hindu-Arabic numerals in Europe
(Competency: Historical Awareness)

15. The number 1010₂ in binary is equivalent to ____ in decimal.
a) 10
b) 20
c) 12
d) 15
Answer: a) 10
(Competency: Application)

16. Which ancient text first mentioned the names of numbers based on powers of 10?
a) Aryabhatiya
b) Bakhshali Manuscript
c) Yajurveda Samhita
d) Brahmasphutasiddhanta
Answer: c) Yajurveda Samhita
(Competency: Historical Awareness)

17. The distributive property of multiplication over addition holds in:
a) Only the Hindu number system
b) Only the Egyptian number system
c) All number systems
d) Only positional number systems
Answer: c) All number systems
(Competency: Reasoning)

18. In the Gumulgal number system, 'ukasar-ukasar-urapon' represents the number:
a) 5
b) 6
c) 7
d) 8
Answer: a) 5
(Competency: Application)

19. The Chinese rod numeral for the number 5 in the units place (Zong orientation) is:
a) 𝍬
b) 𝍣
c) 𝍮
d) 𝍥
Answer: b) 𝍣
(Competency: Recall)

20. The value of 7 in the number 375 represents:
a) 7
b) 70
c) 700
d) 7000
Answer: b) 70
(Competency: Conceptual Understanding)

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Assertion and Reasoning Questions (1 Mark Each)

Directions: Choose the correct option:
a) Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
b) Both A and R are true, but R is NOT the correct explanation of A.
c) A is true, but R is false.
d) A is false, but R is true.

1. Assertion (A): The Roman number system is difficult to use for multiplication.
Reason (R): The product of two Roman landmark numbers is not always a Roman landmark number.
Answer: a) Both A and R are true, and R is the correct explanation of A.
(Competency: Reasoning)

2. Assertion (A): The Hindu-Arabic number system is a positional number system.
Reason (R): It uses only ten symbols to represent all numbers.
Answer: b) Both A and R are true, but R is NOT the correct explanation of A.
(Competency: Reasoning)

3. Assertion (A): The Mayan number system used a shell symbol for zero.
Reason (R): They needed a placeholder to avoid ambiguity in their place value system.
Answer: a) Both A and R are true, and R is the correct explanation of A.
(Competency: Reasoning)

4. Assertion (A): The Egyptian number system is a base-10 system.
Reason (R): Its landmark numbers are powers of 10.
Answer: a) Both A and R are true, and R is the correct explanation of A.
(Competency: Reasoning)

5. Assertion (A): The Mesopotamian number system is called sexagesimal.
Reason (R): It was used by a civilization called Sexagia.
Answer: c) A is true, but R is false.
(Competency: Recall)

6. Assertion (A): Zero is indispensable in a place value system.
Reason (R): It acts as a placeholder and represents nothingness.
Answer: a) Both A and R are true, and R is the correct explanation of A.
(Competency: Conceptual Understanding)

7. Assertion (A): The Chinese rod numeral system alternated between Zong and Heng orientations.
Reason (R): This made the numerals look more beautiful.
Answer: c) A is true, but R is false.
(Competency: Reasoning)

8. Assertion (A): The Gumulgal number system is additive.
Reason (R): It is built by adding the words for 2 and 1.
Answer: a) Both A and R are true, and R is the correct explanation of A.
(Competency: Conceptual Understanding)

9. Assertion (A): The Bakhshali manuscript contains the first known use of zero as a digit.
Reason (R): It was written by Aryabhata in 499 CE.
Answer: c) A is true, but R is false.
(Competency: Historical Awareness)

10. Assertion (A): The distributive property holds for Egyptian numerals.
Reason (R): Egyptian numerals are based on the base-10 system.
Answer: b) Both A and R are true, but R is NOT the correct explanation of A.
(Competency: Reasoning)

11. Assertion (A): It is impossible to represent fractions in the Gumulgal number system.
Reason (R): The system was designed only for counting whole objects.
Answer: a) Both A and R are true, and R is the correct explanation of A.
(Competency: Reasoning)

12. Assertion (A): The Roman numeral for 40 is always written as XL.
Reason (R): Romans never used XXXX for 40.
Answer: c) A is true, but R is false.
(Competency: Historical Awareness)

13. Assertion (A): The number 60 in the Mesopotamian system was represented by the same symbol as 1.
Reason (R): They did not have a symbol for zero initially.
Answer: a) Both A and R are true, and R is the correct explanation of A.
(Competency: Reasoning)

14. Assertion (A): Al-Khwarizmi popularized the Indian number system in the Arab world.
Reason (R): He was an Indian mathematician who moved to Arabia.
Answer: c) A is true, but R is false.
(Competency: Historical Awareness)

15. Assertion (A): The number 25 in base-5 is written as 100.
Reason (R): In base-5, the place values are powers of 5.
Answer: a) Both A and R are true, and R is the correct explanation of A.
(Competency: Application)

16. Assertion (A): The Lebombo bone is considered a lunar calendar.
Reason (R): It has 29 notches, close to the lunar month cycle.
Answer: a) Both A and R are true, and R is the correct explanation of A.
(Competency: Reasoning)

17. Assertion (A): The Egyptian number system required an infinite number of symbols.
Reason (R): They had a unique symbol for every power of 10.
Answer: a) Both A and R are true, and R is the correct explanation of A.
(Competency: Conceptual Understanding)

18. Assertion (A): The Hindu-Arabic number system is used worldwide today.
Reason (R): It allows for efficient and unambiguous representation of numbers.
Answer: a) Both A and R are true, and R is the correct explanation of A.
(Competency: Conceptual Understanding)

19. Assertion (A): The Yajurveda Samhita lists names for numbers up to 10^12.
Reason (R): Ancient Indians needed to count very large numbers for astronomical calculations.
Answer: a) Both A and R are true, and R is the correct explanation of A.
(Competency: Historical Awareness)

20. Assertion (A): Brahmagupta defined zero as a number with specific arithmetic properties.
Reason (R): He wrote the Brahmasphutasiddhanta.
Answer: a) Both A and R are true, and R is the correct explanation of A.
(Competency: Historical Awareness)

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True or False (1 Mark Each)

1. The Roman numeral system is a place value system.
Answer: False
(Competency: Conceptual Understanding)

2. The digit zero was first used in India.
Answer: True
(Competency: Recall)

3. The Mesopotamian number system is a base-60 system.
Answer: True
(Competency: Recall)

4. The Mayan number system used bars and dots for representation.
Answer: True
(Competency: Recall)

5. The Egyptian number system is a base-5 system.
Answer: False
(Competency: Conceptual Understanding)

6. The Chinese rod numeral system used a blank space for zero.
Answer: True
(Competency: Recall)

7. The Gumulgal number system could represent fractions easily.
Answer: False
(Competency: Conceptual Understanding)

8. The Fibonacci sequence is related to the Hindu-Arabic numeral system.
Answer: True
(Competency: Historical Awareness)

9. The distributive property does not hold in the Egyptian number system.
Answer: False
(Competency: Reasoning)

10. The Ishango bone was discovered in South Africa.
Answer: False
(Competency: Recall)

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Short Answer Type Questions-I (2 Marks Each)

1. Convert the Roman numeral MCMXCIV to its Hindu-Arabic form.
Answer: M = 1000, CM = 900, XC = 90, IV = 4. So, 1000 + 900 + 90 + 4 = 1994
(Competency: Application)

2. Why did the Chinese rod numeral system use alternating orientations for digits?
Answer: To prevent ambiguity in place value. The alternation clearly separated units, tens, hundreds, etc., especially when blank spaces were used for zero.
(Competency: Reasoning)

3. Represent the number 73 in the Mayan number system. (Describe the representation)
Answer: 73 = 3×20 + 13×1. So, it would have 3 dots in the 20s place and 13 in the 1s place (13 is represented as 2 bars [10] and 3 dots [3]).
(Competency: Application)

4. What is the primary reason for the difficulty in multiplying Roman numerals?
Answer: The product of two Roman landmark numbers is not always a Roman landmark number, and there is no efficient positional algorithm for multiplication.
(Competency: Reasoning)

5. Express the number 18 in base-2 (binary).
Answer: 18 = 16 + 2 = 1×2⁴ + 0×2³ + 0×2² + 1×2¹ + 0×2⁰ = 10010₂
(Competency: Application)

6. How did the Egyptians represent the number 3045?
Answer: 3045 = 3×1000 + 0×100 + 4×10 + 5×1. So, 3 ∩, 0 9, 4 n, 5 |
(Competency: Application)

7. What is the value of the Mayan numeral that has 4 dots in the 360s place and 2 dots in the 1s place?
Answer: (4×360) + (2×1) = 1440 + 2 = 1442
(Competency: Application)

8. Why is the Hindu-Arabic number system considered more efficient than the Roman system?
Answer: Because it is positional, uses only ten digits including zero, and allows for efficient algorithms for all arithmetic operations.
(Competency: Conceptual Understanding)

9. Convert 43₅ to base-10.
Answer: 43₅ = (4×5¹) + (3×5⁰) = 20 + 3 = 23
(Competency: Application)

10. What was the significance of the placeholder in the Mesopotamian number system?
Answer: It helped to avoid ambiguity by marking empty place values in their positional (place value) system.
(Competency: Historical Awareness)

11. Represent the number 7 using tally marks.
Answer: ~~////~~ //
(Competency: Application)

12. How would you represent the number 12 in the Gumulgal number system?
Answer: Since numbers greater than 6 were called 'ras', 12 would be represented by a word meaning 'many' or by repeated addition of 'ukasar'.
(Competency: Application)

13. What is the largest number that can be represented using only the English letters a-z in Method 2?
Answer: 26 (z)
(Competency: Application)

14. Why did the Mayans use 360 instead of 400 for their third place value?
Answer: It is believed to be related to their calendar system, which had 360 days in a year.
(Competency: Historical Awareness)

15. Write 100₁₀ in Roman numerals.
Answer: C
(Competency: Application)

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Short Answer Type Questions-II (3 Marks Each)

1. Describe the evolution of the idea of number representation from tally marks to the place value system.
Answer:

• Started with tally marks (one-to-one correspondence).

• Moved to grouping (e.g., Gumulgal by 2s, Romans by 5s and 10s).

• Introduction of landmark numbers (e.g., Roman I, V, X, L, C, D, M).

• Use of a base (e.g., Egyptian base-10, powers of 10 as landmarks).

• Finally, the place value system (Mesopotamian, Mayan, Indian) using position to denote value and a placeholder (zero) to avoid ambiguity.
  (Competency: Conceptual Understanding)

2. Multiply XXIII (23) by V (5) in Roman numerals by first converting to Hindu-Arabic, then convert the product back to Roman.
Answer:
XXIII = 23, V = 5.
23 × 5 = 115.
115 in Roman numerals: C = 100, XV = 15. So, CXV.
(Competency: Application)

3. Represent the number 357 in the Egyptian number system.
Answer:
357 = 3×100 + 5×10 + 7×1.
So, three 9 symbols (for 300), five n symbols (for 50), and seven | symbols (for 7).
(Competency: Application)

4. Explain with an example why zero is necessary in a place value system.
Answer:
Without zero, numbers like 205 and 25 become ambiguous. In 205, zero indicates no tens. In Mesopotamian system,  2 5 could mean 2×60 + 5×1 = 125, or 2×3600 + 5×1 = 7205, etc. Zero (or a placeholder) specifies the place clearly.
(Competency: Reasoning)

5. Convert the binary number 11011₂ to base-10.
Answer:
1×2⁴ + 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 16 + 8 + 0 + 2 + 1 = 27
(Competency: Application)

6. How did the abacus help in performing calculations with Roman numerals?
Answer:
The abacus had grooves for different values (units, tens, hundreds, etc.). Counters were placed in these grooves to represent numbers. Calculations were done by moving these counters according to set rules, which made addition and subtraction easier than with written numerals.
(Competency: Historical Awareness)

7. Describe the number system of the Gumulgal people.
Answer:
It was an additive system based on grouping by twos.
1 = urapon, 2 = ukasar, 3 = ukasar-urapon, 4 = ukasar-ukasar, 5 = ukasar-ukasar-urapon, 6 = ukasar-ukasar-ukasar. Numbers greater than 6 were called 'ras'.
(Competency: Conceptual Understanding)

8. What are the advantages of a base-n number system?
Answer:

• Efficient representation of numbers using a limited set of symbols.

• Simplifies arithmetic operations like addition and multiplication.

• Easy to extend to represent very large numbers.
  (Competency: Conceptual Understanding)

9. Write 89 in base-4.
Answer:
4³=64, 4²=16, 4¹=4, 4⁰=1.
89 ÷ 64 = 1 remainder 25.
25 ÷ 16 = 1 remainder 9.
9 ÷ 4 = 2 remainder 1.
So, 89 = 1×64 + 1×16 + 2×4 + 1×1 = 1121₄
(Competency: Application)

10. Why is the Hindu-Arabic system called a place value system?
Answer:
Because the value of a digit depends on its position in the number. For example, in 373, the first '3' represents 300 (hundreds), the '7' represents 70 (tens), and the last '3' represents 3 (units).
(Competency: Conceptual Understanding)

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Long Answer Type Questions (5 Marks Each)

1. Compare and contrast the Roman and Hindu-Arabic number systems. Highlight at least three points of difference with examples.
Answer:

• Representation: Roman is additive (e.g., XX=20), Hindu-Arabic is positional (e.g., 20 means 2 tens + 0 units).

• Symbols: Roman uses many symbols (I,V,X,L,C,D,M), Hindu-Arabic uses only ten digits (0-9).

• Zero: Roman has no zero, Hindu-Arabic has zero as a placeholder and a number.

• Calculations: Arithmetic is complex in Roman (e.g., XXIII × V requires conversion), but efficient in Hindu-Arabic due to algorithms.

• Large Numbers: Representing large numbers in Roman requires new symbols or many repetitions, while Hindu-Arabic can represent any number with the same ten digits.
  (Competency: Conceptual Understanding)

2. Explain the evolution of the concept of zero from a placeholder to a number. Mention the contributions of Indian mathematicians.
Answer:

• Zero first emerged as a placeholder in positional systems (e.g., Mesopotamian blank space, Mayan shell symbol).

• Indian mathematicians transformed it into a number with defined properties.

• The Bakhshali manuscript (c. 3rd century CE) used a dot for zero as a digit.

• Aryabhata (499 CE) used the place value system and performed operations involving zero.

• Brahmagupta (628 CE) in Brahmasphutasiddhanta defined zero as a number resulting from subtracting a number from itself. He gave rules: a + 0 = a, a - 0 = a, a × 0 = 0, and discussed division by zero.

• This treatment of zero as a number was a groundbreaking contribution to mathematics.
  (Competency: Historical Awareness & Conceptual Understanding)

3. Describe the Mesopotamian number system. How did they represent numbers? What was the major ambiguity in their system, and how was it later resolved?
Answer:

• It was a base-60 (sexagesimal) system.

• They used cuneiform symbols: wedge (∀) for 1, angled wedge (‹) for 10.

• Numbers were represented using place value: the rightmost position for 1s, next for 60s, next for 3600s, etc.

• Initially, they used blank spaces for empty places, which caused ambiguity. For example, 1 1 could mean 1×60 + 1×1 = 61 or 1×3600 + 1×1 = 3601.

• Later, they introduced a placeholder symbol (similar to zero) to mark empty positions and resolve this ambiguity.
  (Competency: Conceptual Understanding)

4. Convert the decimal number 253 to base-7. Show all steps.
Answer:
7³=343 > 253, so use 7²=49.
253 ÷ 49 = 5 (5×49=245), remainder 8.
8 ÷ 7 = 1 (1×7=7), remainder 1.
1 ÷ 1 = 1 (1×1=1), remainder 0.
So, 253 = 5×49 + 1×7 + 1×1 = 511₇
(Competency: Application)

5. How does the Chinese rod numeral system work? How is the number 2,634 represented in it?
Answer:

• It is a decimal place value system that uses alternating orientations for digits to avoid ambiguity.

• Units place: Zong (vertical) digits.

• Tens place: Heng (horizontal) digits.

• Hundreds place: Zong again.

• Thousands place: Heng again, and so on.

• For 2,634:

• Thousands (Heng): 2 (二)

• Hundreds (Zong): 6 (𝍭 )

• Tens (Heng): 3 (三)

• Units (Zong): 4 (𝍣 )

• So, the numeral would be written from top to bottom: 二 (Heng), 𝍭 (Zong), 三 (Heng), 𝍣 (Zong).
  (Competency: Application)

6. Explain why the product of two landmark numbers in a base-n system is always another landmark number. Give an example from the Egyptian system.
Answer:

• In a base-n system, landmark numbers are powers of n: n⁰, n¹, n², n³, ...

• Multiplying two powers of n gives: nᵃ × nᵇ = nᵃ⁺ᵇ, which is another power of n and hence a landmark number.

• Example in Egyptian (base-10):

• Landmark numbers: 1 (|), 10 (n), 100 (9), 1000 (∩).

• 10 (n) × 100 (9) = 1000 (∩), which is a landmark number.

• 100 (9) × 100 (9) = 10000 (8), which is the next landmark number.

• This property simplifies multiplication in base-n systems.
  (Competency: Reasoning)

7. Represent the number 1,652 in the Mayan number system. (Show the calculation and the representation in words/drawing description)
Answer:

• Mayan levels: bottom (1s), middle (20s), top (360s).

• 1652 ÷ 360 = 4 (4×360=1440), remainder 212.

• 212 ÷ 20 = 10 (10×20=200), remainder 12.

• 12 ÷ 1 = 12.

• So:

• Top (360s): 4 dots.

• Middle (20s): 10 is two bars (since one bar=5).

• Bottom (1s): 12 is two bars (10) and two dots (2).

• Representation: At top: .... ; at middle: == (two bars); at bottom: == .. (two bars and two dots).
  (Competency: Application)

8. What are tally marks? How are they used for counting and performing basic operations?
Answer:

• Tally marks are lines used for counting, usually grouped for ease (e.g., in 5s: ~~////~~ ).

• Counting: One mark per object.

• Addition: Combine tally groups.

• Subtraction: Remove tallies from a group.

• Multiplication: Create multiple identical groups and combine.

• Division: Share tallies equally into groups.

• Example: 7 + 5: ~~////~~ // + ~~////~~ = ~~////~~ ~~////~~ // = 12.
  (Competency: Application)

9. Illustrate the addition of 47 and 36 using the Egyptian number system.
Answer:

• 47 = 4 n (40) + 7 | (7)

• 36 = 3 n (30) + 6 | (6)

• Combine: 7 n (70) + 13 | (13)

• Since 10 | = 1 n, convert 10 | to 1 n. Now total: 8 n (80) + 3 | (3)

• So, 47 + 36 = 83, represented as 8 n and 3 |.
  (Competency: Application)

10. How did the spread of the Hindu-Arabic number system impact Europe? Discuss the role of Fibonacci.
Answer:

• European scholars learned the system from Arabs and called it "Arabic numerals".

• Roman numerals were entrenched in Europe, so adoption was slow.

• Fibonacci (c. 1200) wrote "Liber Abaci", explaining the system and its advantages for calculation, commerce, and accounting.

• He demonstrated efficient algorithms for arithmetic operations.

• Despite resistance, the system gradually spread due to its utility, especially during the Renaissance, and eventually became essential for scientific progress.
  (Competency: Historical Awareness)

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Case-Based Questions (4 MCQs Each)

Case 1: The Gumulgal Number System

The Gumulgal people of Australia used a number system based on counting in twos. Their number names are:

• 1: urapon

• 2: ukasar

• 3: ukasar-urapon

• 4: ukasar-ukasar

• 5: ukasar-ukasar-urapon

• 6: ukasar-ukasar-ukasar
  Numbers greater than 6 were called 'ras'.

1. How would the Gumulgal represent the number 4?
a) ukasar-urapon
b) ukasar-ukasar
c) urapon-urapon
d) ras
Answer: b) ukasar-ukasar
(Competency: Application)

2. What is the value of 'ukasar-ukasar-urapon'?
a) 3
b) 4
c) 5
d) 6
Answer: c) 5
(Competency: Application)

3. Why is this system considered additive?
a) Because it uses addition to form number names
b) Because it uses multiplication
c) Because it has a base
d) Because it uses place value
Answer: a) Because it uses addition to form number names
(Competency: Conceptual Understanding)

4. What is a limitation of this system?
a) It cannot represent odd numbers
b) It becomes long and cumbersome for large numbers
c) It is not accurate
d) It uses too many symbols
Answer: b) It becomes long and cumbersome for large numbers
(Competency: Reasoning)