Chapter 3 A Story of Numbers - worksheet
Chapter Overview: This chapter explores the historical evolution of number systems, from early tally marks to the modern Hindu-Arabic numeral system. It covers concepts like one-to-one mapping, landmark numbers, base systems, place value, and the crucial role of zero.
PART 1: MULTIPLE CHOICE QUESTIONS (1 Mark Each)
Q1. The oldest known mathematical artifact, the Lebombo bone, is an example of:
a) Roman numerals b) Tally marks c) Egyptian numerals d) Mesopotamian numerals
(Competency: Remembering & Recalling Facts)
Q2. The Roman numeral for 49 is correctly written as:
a) IL b) XXXXIX c) XLIX d) VLIV (Competency: Applying Rules & Procedures)
Q3. The idea that the value of a digit depends on its position is called:
a) Landmark value b) Face value c) Place value d) Group value
(Competency: Understanding Core Concepts)
Q4. The number 101₅ in base-5, when converted to base-10, is equal to:
a) 15 b) 26 c) 101 d) 5 (Competency: Applying Rules & Procedures)
Q5. Which ancient civilization used a base-60 (sexagesimal) number system?
a) Egyptian b) Roman c) Mesopotamian (Babylonian) d) Mayan
(Competency: Remembering & Recalling Facts)
Q6. The modern Hindu-Arabic numeral system is a:
a) Base-5 system b) Base-10 system c) Base-20 system d) Base-60 system
(Competency: Remembering & Recalling Facts)
Q7. The number 0 is essential in our number system because it:
a) Makes numbers look larger
b) Acts as a placeholder and represents 'nothing'
c) Is the smallest natural number
d) Was invented by the Romans (Competency: Understanding Core Concepts)
Q8. Which number system used different orientations (Zong and Heng) of the same symbols to denote different place values?
a) Egyptian b) Roman c) Chinese Rod Numerals d) Mayan
(Competency: Remembering & Recalling Facts)
Q9. The number 23 represented in the Gumulgal system (which counts in 2s) would be:
a) ukasar-ukasar-urapon b) ras c) ukasar repeated 11 times and urapon
d) ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-urapon
(Competency: Applying Rules & Procedures)
Q10. The product of two landmark numbers in a base-n system is:
a) Always an odd number b) Always another landmark number c) Always a prime number d) Always an even number (Competency: Reasoning & Analysing)
Q11. To multiply any number by 10 in the Hindu-Arabic system, we:
a) Subtract a zero b) Append a zero to the right c) Divide the number by 10
d) Change the sign of the number
Answer: b) Append a zero to the right (Competency: Applying Rules & Procedures)
Q12. The Roman numeral D stands for:
a) 100 b) 500 c) 1000 d) 50 (Competency: Remembering & Recalling Facts)
Q13. The first civilization to treat zero as a number and define arithmetic operations with it was:
a) Mesopotamian b) Egyptian c) Indian d) Chinese
(Competency: Remembering & Recalling Facts)
Q14. The Mayan number system was primarily a base-20 system, but its third landmark number was 360 instead of 400. This was likely due to their interest in:
a) Astronomy and calendars b) Military strategies
c) Building pyramids d) Aquatic trade
(Competency: Understanding Core Concepts)
Q15. Performing division in a tally mark system is equivalent to:
a) Repeated subtraction b) Equal sharing c) Finding a remainder d) Both a) and b)
(Competency: Applying Rules & Procedures)
Q16. The Egyptian symbol for 100 was:
a) | b) n c) 9 d) ∩
(Competency: Remembering & Recalling Facts)
Q17. The main advantage of a place value system is that it:
a) Uses beautiful symbols b) Allows representation of all numbers using only a few symbols
c) Was used by many ancient civilizations d) Is very easy to learn
(Competency: Reasoning & Analysing)
Q18. The number 73 in Mayan numerals would require symbols in how many different place value levels?
a) 1 b) 2 c) 3 d) 4 (Competency: Applying Rules & Procedures)
Q19. The commutative property of addition holds for:
a) Only Hindu-Arabic numerals b) Only Roman numerals c) The concept of numbers, regardless of the system used to represent them
d) Only base-10 systems (Competency: Reasoning & Analysing)
Q20. The Roman numeral system is difficult to use for complex calculations because:
a) It has no symbol for zero
b) It is not a place value system
c) The product of two landmark numbers is not always a landmark number
d) All of the above (Competency: Reasoning & Analysing)
PART 2: ASSERTION AND REASONING
Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(a) Both A and 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 numeral for 40 is XL.
Reason (R): In Roman numerals, a smaller numeral placed before a larger numeral indicates subtraction.
(Competency: Reasoning & Analysing)
2. Assertion (A): The number 202 cannot be represented in Roman numerals without using the symbol 'C'.
Reason (R): The Roman numeral for 100 is C.
(Competency: Reasoning & Analysing)
3. Assertion (A): The Mesopotamian number system had ambiguities before the introduction of a placeholder zero.
Reason (R): The same cuneiform writing for '1' could represent 1, 60, or 3600 depending on its position, and blank spaces were hard to interpret.
(Competency: Reasoning & Analysing)
4. Assertion (A): The Chinese rod numeral system alternated between vertical (Zong) and horizontal (Heng) orientations.
Reason (R): This alternation made the numerals more aesthetically pleasing.
(Competency: Reasoning & Analysing)
5. Assertion (A): Multiplying by the base (like 10) is very easy in the Hindu-Arabic system.
Reason (R): Multiplying by the base simply shifts all digits to the left and adds a zero in the units place.
6. Assertion (A): The Gumulgal number system is inefficient for representing large numbers.
Reason (R): It is an additive system where the length of the number name increases with the size of the number.
7. Assertion (A): The distributive property of multiplication over addition holds for the Egyptian number system.
Reason (R): The distributive property is a fundamental property of numbers and is independent of the symbols used to represent them.
8. Assertion (A): The Mayan number system is considered an advanced intellectual achievement.
Reason (R): It was a place value system that included a symbol for zero, developed independently of Asian systems.
9. Assertion (A): The concept of 'landmark numbers' is crucial for building an efficient number system.
Reason (R): Landmark numbers serve as reference points, making it easier to represent and work with larger quantities.
10. Assertion (A): The Hindu-Arabic numeral system is now used all over the world.
Reason (R): It is perfectly suited for performing complex arithmetic operations efficiently.
PART 3: TRUE OR FALSE (1 Mark Each)
1. The Egyptian number system was a base-10 system. (Competency: Remembering & Recalling Facts)
2. The symbol 'X' can never be repeated more than three times in a Roman numeral. (e.g., XXX = 30 is valid) (Competency: Applying Rules & Procedures)
3. The number zero (0) was first used by the ancient Romans. (Competency: Remembering & Recalling Facts)
4. In a base-5 system, the number following 44₅ is 100₅. (44₅ + 1 = 100₅) (Competency: Applying Rules & Procedures)
5. The Ishango bone, found in Africa, is believed to be over 20,000 years old and contains tally marks. (Competency: Remembering & Recalling Facts)
6. The Roman number system is a place value system. (Competency: Understanding Core Concepts)
7. The Mesopotamian system's use of base-60 is the reason we have 60 minutes in an hour (Competency: Remembering & Recalling Facts)
8. In the Chinese rod numeral system, the symbol for 5 is the same in the units and the tens place. (Competency: Understanding Core Concepts)
9. The number 101₈ (base-8) is equal to 65₁₀ (base-10). (1×64 + 0×8 + 1×1 = 65) (Competency: Applying Rules & Procedures)
10. One-to-one mapping is the fundamental idea behind counting the number of objects in a collection. (Competency: Understanding Core Concepts)
PART 4: SHORT ANSWER QUESTIONS-I (2 Marks Each)
1. Express the Hindu-Arabic numeral 17 in the Roman numeral system.
(Competency: Applying Rules & Procedures)
2. Identify the number system that used a seashell shape as a symbol for zero.
(Competency: Remembering & Recalling Facts)
3. Represent the number 42 using Egyptian numerals. (Use | for 1, n for 10)
(Competency: Applying Rules & Procedures)
4. Convert the base-5 number 43₅ into its base-10 equivalent.
(Competency: Applying Rules & Procedures)
5. State one major advantage of the Hindu-Arabic numeral system over the Roman numeral system.
(Competency: Reasoning & Analysing)
6. What is the purpose of a 'placeholder' in a number system?
(Competency: Understanding Core Concepts)
7. If you have a collection of 38 sticks, how would you represent this number using the simple tally method?
(Competency: Applying Rules & Procedures)
8. Define 'landmark numbers' in the context of the evolution of number systems.
(Competency: Understanding Core Concepts)
9. Express the number 8 using the Gumulgal number system's logic.
(Competency: Applying Rules & Procedures)
10. Why is the number 60 considered a 'composite' base?
(Competency: Reasoning & Analysing)
PART 5: SHORT ANSWER QUESTIONS-II (3 Marks Each)
1. Compare the Roman and Egyptian number systems based on:
a) Base used b) Use of place value c) Ease of performing multiplication
(Competency: Analysing & Evaluating)
2. Represent the number 125 in the following systems:
a) Roman numerals
b) Egyptian numerals (Use ∩ for 1000, 9 for 100, n for 10, | for 1)
c) Base-5 system (using the concept of powers of 5)
(Competency: Applying Rules & Procedures)
3. Explain with an example why the Mesopotamian number system was ambiguous before the use of a placeholder zero.
(Competency: Reasoning & Analysing)
4. Describe the one-to-one mapping method of counting using a real-life example.
(Competency: Understanding Core Concepts)
5. Convert the following numbers as instructed:
a) 43₁₀ to base-2 (binary) b) 101101₂ to base-10 c) 77₁₀ to base-5
(Competency: Applying Rules & Procedures)
PART 6: LONG ANSWER QUESTIONS (5 Marks Each)
1. Explain the evolution of the idea of number representation from simple tally marks to the Hindu-Arabic numeral system. Discuss at least four key stages in this evolution.
(Competency: Analysing & Evaluating)
2. Describe the Mayan number system with the help of an example. What was its most significant feature and one puzzling aspect?
(Competency: Understanding Core Concepts, Remembering Facts)
3. Perform the addition of 47 and 65 using the method of grouping by 10s (like in the Egyptian system). Show each step clearly.
(Competency: Applying Rules & Procedures)
PART 7: CASE-BASED QUESTIONS (4 MCQ each)
CASE STUDY 1: The Roman Legacy
The Roman numeral system was used throughout Europe for centuries. It uses letters from the Latin alphabet as symbols to represent values.
I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1000.
Numbers are formed by combining these symbols additively. A smaller numeral placed before a larger numeral indicates subtraction of its value (e.g., IV = 4, IX = 9).
1. What is the value of the Roman numeral CDLXIX?
a) 466 b) 469 c) 571 d) 669
(Competency: Applying Rules & Procedures)
2. A clock face shows IIII for 4 instead of IV. This is an example of:
a) A common modern simplification
b) An error in the clock's design
c) An archaic form that is still sometimes used
d) Using addition instead of subtraction
(Competency: Understanding Core Concepts)
3. Why would calculating the product of LXX and X (70 × 10) be simpler than calculating the product of XLIX and IX (49 × 9) in this system?
a) Because LXX and X are both landmark numbers
b) Because 70 × 10 is a round number
c) Because the product of two landmark numbers (X and LXX) is another landmark number (DCC = 700)
d) All of the above
(Competency: Reasoning & Analysing)
4. The main reason the Roman numeral system was eventually replaced was that it:
a) Was too old
b) Was not a place value system, making complex calculations difficult
c) Used too many symbols
d) Could not represent fractions
(Competency: Reasoning & Analysing)
CASE STUDY 2: The Ingenious Hindu-Arabic System
The numeral system we use today has ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. It is a place-value, base-10 (decimal) system. This means the value of a digit depends on its place in the number. For example, in the number 555, the first 5 represents 500, the second 50, and the third 5. This system originated in India around 1500 years ago and was later transmitted to Europe via the Arab world.
1. The digit in the hundreds place in the number 8,675,439 is:
a) 8 b) 6 c) 5 d) 4
(Competency: Applying Rules & Procedures)
2. The historical name for this system, 'Hindu-Arabic numerals', reflects:
a) That it was invented by Arab mathematicians
b) The European perspective of learning it from the Arabs, though it originated in India
c) That it was a joint invention of Indian and Arab scholars
d) That the Arabs invented the digit 0
(Competency: Remembering & Recalling Facts)
3. Multiplying 234 by 10 gives 2340. This demonstrates the rule of:
a) Addition
b) Place value shift
c) Subtraction
d) Commutativity
(Competency: Applying Rules & Procedures)
4. The most revolutionary aspect of this system was the introduction of 0 as:
a) Just a placeholder
b) A placeholder and a number with defined properties
c) A symbol for nothing
d) A way to make numbers look larger
(Competency: Understanding Core Concepts)
CASE STUDY 3: The Egyptian Scribes
The ancient Egyptian number system, developed around 3000 BCE, was a decimal (base-10) system. It used specific hieroglyphic symbols for powers of ten and employed a simple additive principle.
| = 1
n = 10
9 = 100
∩ = 1,000
8 = 10,000
? = 100,000
? = 1,000,000
To represent a number, they would write the required multiple of each power of ten. For example, the number 3,422 was written as ∩∩∩ 9999 nn || (3*1000 + 4*100 + 2*10 + 2*1).
1. How would an Egyptian scribe represent the number 304?
a) 999 nnnn |||| b) 999 |||| c) 999 nnnn d) 999 ∩∩∩∩
(Competency: Applying Rules & Procedures)
2. What was a significant limitation of the Egyptian system for representing very large numbers?
a) It could not represent numbers beyond 1,000,000.
b) It required new, unique symbols to be invented for every higher power of 10.
c) It was not a base-10 system.
d) It did not have a symbol for 10.
(Competency: Reasoning & Analysing)
3. Adding 123 and 78 in the Egyptian system primarily involves:
a) Using an abacus b) Combining symbols and then grouping them into higher denominations
c) Multiplying the landmark numbers d) Using place value subtraction
(Competency: Applying Rules & Procedures)
4. The Egyptian system is best described as:
a) A place-value system b) An additive system using landmark numbers
c) A subtractive system like the Roman numerals d) A base-60 system
(Competency: Understanding Core Concepts)
CASE STUDY 4: The Mesopotamian Clay Tablets
The Mesopotamian (or Babylonian) civilization developed a sophisticated base-60 (sexagesimal) number system around 2000 BCE. They wrote on clay tablets using a stylus to create wedge-shaped marks (cuneiform). They used a place-value system where a vertical wedge ∀ meant '1' and a sideways wedge ‹ meant '10'. The value of a group of these symbols depended on its position relative to others. However, initially, they left a blank space to represent "zero" in a particular place value, which often led to confusion.
1. The enduring legacy of the Mesopotamian base-60 system can be seen today in our measurement of:
a) Temperature b) Time (hours, minutes) and angles (degrees)
c) Weight and volume d) Distance
(Competency: Remembering & Recalling Facts)
2. Why was the initial lack of a zero placeholder a major problem in this system?
a) It made the symbols difficult to draw.
b) It was impossible to represent the number zero.
c) It created ambiguity because the same numeral could represent different numbers (e.g., 61 vs. 3601).
d) It prevented them from representing any number over 60.
(Competency: Reasoning & Analysing)
3. How would the number 62 be represented in its simplest form in this system?
a) ‹‹ ∀∀ b) ∀∀ c) ‹ ‹‹ d) ∀ ‹‹
(Competency: Applying Rules & Procedures)
4. What was the key mathematical insight the Mesopotamians developed that was more advanced than the Egyptian system?
a) Using a base-10 system. b) Using hieroglyphics for numbers.
c) Using the principle of place value. d) Using only two symbols.
(Competency: Understanding Core Concepts)
CASE STUDY 5: The Gumulgal counters
The Gumulgal people of Australia used a number system based heavily on counting in pairs. Their number names were built from words for "one" (urapon) and "two" (ukasar). For example:
3 = ukasar-urapon (two and one)
4 = ukasar-ukasar (two and two)
5 = ukasar-ukasar-urapon (two and two and one)
6 = ukasar-ukasar-ukasar (two and two and two)
Any number larger than 6 was simply called "ras" (meaning "many").
1. What is the main structural principle of the Gumulgal number system?
a) Place Value b) Additive Grouping by 2s
c) Subtractive Grouping d) Multiplicative Grouping by 5s
(Competency: Understanding Core Concepts)
2. What is the most significant limitation of this system?
a) It cannot represent odd numbers. b) It becomes very long and impractical for numbers larger than 6.
c) It does not have a word for 2. d) It was only used for counting animals.
(Competency: Reasoning & Analysing)
3. How would the concept of "12" be expressed in this system, if they extended their logic beyond "ras"?
a) ras-ras b) ukasar-ukasar-ukasar-ukasar-ukasar-ukasar
c) urapon-urapon d) It would be a new, unique word.
(Competency: Applying Rules & Procedures)
4. The Gumulgal system is an example of how number systems are often shaped by:
a) Universal mathematical laws b) The specific needs and environment of a culture
c) Contact with more advanced civilizations d) The need to perform calculus
(Competency: Reasoning & Analysing)
QUESTIONS FROM TEXTBOOK
1. Represent the following numbers in the Roman system.
(i) 1222 (ii) 2999 (iii) 302 (iv) 715 (v) 27
2. Writing 2367 as a sum of landmark numbers starting from 1000 such that we take as many 1000s as possible, 500s as possible, and so on
3. Example: Try adding the following numbers without converting them to Hindu numerals: (a) CCXXXII + CCCCXIII
4. find the product of the following pairs of landmark numbers: V × L, L × D, V × D, VII × IX.
5. 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-ukasar ukasar) (iii) (ukasar-ukasar-ukasar-ukasar-urapon) × (ukasar-ukasar) (iv) (ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar) ÷ (ukasar-ukasar)
6. Represent the following numbers in the Egyptian system: 10458, 1023, 2660, 784, 1111, 70707, 324
7.What numbers do these numerals stand for?
8. Express the number 143 in this Egyptian new system
9. Write the following numbers in the above base-5 system using the symbols in Table 2: 15, 50, 137, 293, 651.
10. Add the following Egyptian numerals
11. 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
12. What is any landmark number multiplied by following products —Each landmark number is a power of 10 and so multiplying it with 10 increases the power by 1, which is the next landmark number.
13. Represent the number 640 & 7530 in this The Mesopotamian Number System
14. Represent the following numbers in the Mesopotamian system — (i) 63 (ii) 132 (iii) 200 (iv) 60 (v) 3605
15. What will be the representation for 3,600?
16. Represent the following numbers using the Mayan system: (i) 77 (ii) 100 (iii) 361 (iv) 721
17. 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?
18. 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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