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
Answer: b) Tally marks (Competency: Remembering & Recalling Facts)
Q2. The Roman numeral for 49 is correctly written as:
a) IL b) XXXXIX c) XLIX d) VLIV
Answer: c) XLIX (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
Answer: c) Place 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
Answer: b) 26 (1×25 + 0×5 + 1×1 = 26) (Competency: Applying Rules & Procedures)
Q5. Which ancient civilization used a base-60 (sexagesimal) number system?
a) Egyptian b) Roman c) Mesopotamian (Babylonian) d) Mayan
Answer: c) Mesopotamian (Babylonian) (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
Answer: b) Base-10 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
Answer: b) Acts as a placeholder and represents 'nothing' (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
Answer: c) Chinese Rod Numerals (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
Answer: d) ukasar repeated 11 times and 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
Answer: b) Always another landmark 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
Answer: b) 500 (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
Answer: c) Indian (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
Answer: a) Astronomy and calendars (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)
Answer: d) Both a) and b) (Competency: Applying Rules & Procedures)
Q16. The Egyptian symbol for 100 was:
a) | b) n c) 9 d) ∩
Answer: c) 9 (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
Answer: b) Allows representation of all numbers using only a few symbols (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
Answer: b) 2 (73 = 3×20 + 13×1) (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
Answer: c) The concept of numbers, regardless of the system used to represent them (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
Answer: 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.
Answer: (a) Both A and R are true and R is the correct explanation of A. (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.
Answer: (a) Both A and R are true and R is the correct explanation of A. (202 is CCII) (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.
Answer: (a) Both A and R are true and R is the correct explanation of A. (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.
Answer: (c) A is true but R is false. The alternation was to prevent ambiguity in reading place values, not for aesthetics. (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.
Answer: (a) Both A and R are true and R is the correct explanation of A. (Competency: Reasoning & Analysing)
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.
Answer: (a) Both A and R are true and R is the correct explanation of A. (Competency: Reasoning & Analysing)
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.
Answer: (a) Both A and R are true and R is the correct explanation of A. (Competency: Reasoning & Analysing)
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.
Answer: (a) Both A and R are true and R is the correct explanation of A. (Competency: Reasoning & Analysing)
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.
Answer: (a) Both A and R are true and R is the correct explanation of A. (Competency: Reasoning & Analysing)
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.
Answer: (a) Both A and R are true and R is the correct explanation of A. (Competency: Reasoning & Analysing)
PART 3: TRUE OR FALSE (1 Mark Each)
1. The Egyptian number system was a base-10 system. (True) (Competency: Remembering & Recalling Facts)
2. The symbol 'X' can never be repeated more than three times in a Roman numeral. (False) (e.g., XXX = 30 is valid) (Competency: Applying Rules & Procedures)
3. The number zero (0) was first used by the ancient Romans. (False) (Competency: Remembering & Recalling Facts)
4. In a base-5 system, the number following 44₅ is 100₅. (True) (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. (True) (Competency: Remembering & Recalling Facts)
6. The Roman number system is a place value system. (False) (Competency: Understanding Core Concepts)
7. The Mesopotamian system's use of base-60 is the reason we have 60 minutes in an hour. (True) (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. (False - the orientation changes) (Competency: Understanding Core Concepts)
9. The number 101₈ (base-8) is equal to 65₁₀ (base-10). (True) (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. (True) (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.
Answer: 17 = 10 + 5 + 1 + 1 = X + V + I + I = XVII (Competency: Applying Rules & Procedures)
2. Identify the number system that used a seashell shape as a symbol for zero.
Answer: The Mayan number system (Competency: Remembering & Recalling Facts)
3. Represent the number 42 using Egyptian numerals. (Use | for 1, n for 10)
Answer: 42 = 40 + 2 = (4 × 10) + (2 × 1) = nnnn || (Competency: Applying Rules & Procedures)
4. Convert the base-5 number 43₅ into its base-10 equivalent.
Answer: 43₅ = (4 × 5¹) + (3 × 5⁰) = (4 × 5) + (3 × 1) = 20 + 3 = 23₁₀ (Competency: Applying Rules & Procedures)
5. State one major advantage of the Hindu-Arabic numeral system over the Roman numeral system.
Answer: It is a place value system, which makes arithmetic operations like multiplication and division much easier and efficient. (Competency: Reasoning & Analysing)
6. What is the purpose of a 'placeholder' in a number system?
Answer: A placeholder (like zero) is used to indicate the absence of a value in a specific place value position, ensuring the other digits are in their correct places and the number is read accurately. (Competency: Understanding Core Concepts)
7. If you have a collection of 38 sticks, how would you represent this number using the simple tally method?
Answer: I would draw 38 vertical marks: |||| |||| |||| |||| |||| |||| |||| || (Often grouped in fives for ease: ~~////~~ ~~////~~ ~~////~~ ~~////~~ ~~////~~ ~~////~~ ~~///) (Competency: Applying Rules & Procedures)
8. Define 'landmark numbers' in the context of the evolution of number systems.
Answer: Landmark numbers are special numbers that are easily recognizable and used as reference points or building blocks for representing other numbers in a system (e.g., 5, 10, 60, 100 in Roman numerals; powers of 10 in the Egyptian system). (Competency: Understanding Core Concepts)
9. Express the number 8 using the Gumulgal number system's logic.
Answer: The Gumulgal system counts in 2s. 8 = 2 + 2 + 2 + 2. Therefore, it would be represented as "ukasar-ukasar-ukasar-ukasar". (Competency: Applying Rules & Procedures)
10. Why is the number 60 considered a 'composite' base?
Answer: 60 is a composite base because it has many divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60). This made fractions easier to handle in the Mesopotamian system, which is why it was chosen. (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
Answer:
a) Base: The Roman system uses multiple bases (1, 5, 10, etc.) for its landmark numbers. The Egyptian system is a strict base-10 system.
b) Place Value: Neither system is a true place value system. Both are additive systems where the value of a symbol does not depend on its position.
c) Multiplication: Multiplication is difficult in both systems compared to the Hindu-Arabic system. However, it was relatively easier in the Egyptian system because the product of two landmark numbers (powers of 10) is another landmark number, which is not always the case in the Roman system (e.g., V × L = 250, which is not a standard Roman landmark numeral). (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)
Answer:
a) Roman: 125 = 100 + 10 + 10 + 5 = C + X + X + V = CXXV
b) Egyptian: 125 = 100 + 20 + 5 = (1 × 100) + (2 × 10) + (5 × 1) = 9 nn |||||
c) Base-5: 125₁₀ in base-5: 125 ÷ 25 = 5, remainder 0. 5 ÷ 5 = 1, remainder 0. 1 ÷ 1 = 1, remainder 0. ∴ 125₁₀ = 1000₅ (1×125 + 0×25 + 0×5 + 0×1) (Competency: Applying Rules & Procedures)
3. Explain with an example why the Mesopotamian number system was ambiguous before the use of a placeholder zero.
Answer: The Mesopotamian system was a place value system with base 60, but it initially used blank spaces to represent zero in a place. This led to ambiguity because it was difficult to distinguish how many empty places there were between symbols. For example, the symbols for '1' and '2' could represent:
1 × 60 + 2 × 1 = 62 (if they were in consecutive places)
1 × 3600 + 0 × 60 + 2 × 1 = 3602 (if there was an empty 60s place between them)
Without a clear placeholder, the same written numeral could be interpreted in multiple ways. (Competency: Reasoning & Analysing)
4. Describe the one-to-one mapping method of counting using a real-life example.
Answer: One-to-one mapping is the process of pairing each object in a set to be counted with exactly one object in a standard reference set.
Example: A teacher wants to count if all students are present in class. She has a attendance register with a list of names. As each student says "present", she puts a checkmark (✓) next to their name. Here, each student is mapped to one checkmark. At the end, if there are names without checkmarks, those students are absent. The number of checkmarks gives the number of present students. The sticks used in the Stone Age are another example of a standard reference set used for one-to-one mapping. (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
Answer:
a) 43 to Binary: 43 is odd -> 1 (LSB). 43/2=21. 21 is odd -> 1. 21/2=10. 10 is even -> 0. 10/2=5. 5 is odd -> 1. 5/2=2. 2 is even -> 0. 2/2=1 -> 1 (MSB). Read from bottom: 101011₂
b) 101101₂ to Decimal: 1×2⁵ + 0×2⁴ + 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 32 + 0 + 8 + 4 + 0 + 1 = 45₁₀
c) 77 to Base-5: 77 ÷ 25 = 3, remainder 2. 2 ÷ 5 = 0, remainder 2. 2 ÷ 1 = 2, remainder 0. ∴ 77₁₀ = 302₅ (3×25 + 0×5 + 2×1) (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.
Answer: The evolution of number systems was a long process with key breakthroughs:
Tally Marks (One-to-one mapping): The earliest method involved making a mark for each object counted (e.g., on bones, sticks). This was simple but inefficient for large numbers. (e.g., Ishango bone)
Grouping and Landmark Numbers: To improve efficiency, people started grouping counts (e.g., in 5s, 10s). This led to the creation of landmark numbers (like 5, 10, 100). Numbers were represented additively using these landmarks. (e.g., Roman numerals: V=5, X=10; Egyptian: n=10, 9=100)
The Idea of a Base: A major improvement was choosing a specific number (the base) and defining landmark numbers as its powers (e.g., base-10: 1, 10, 100, 1000...). This made arithmetic operations like multiplication simpler. (e.g., Egyptian system)
Place Value System: The most crucial step was using the position of a digit to indicate its value (units, tens, hundreds). This meant the same symbol could represent different values (e.g., the '2' in 23 and 32). (e.g., Mesopotamian, Mayan systems)
The Digit Zero: The final step was the invention of a symbol for zero as a placeholder. This resolved ambiguities in place value systems (e.g., distinguishing 25, 205, and 250) and allowed for the unambiguous representation of all numbers. The Hindu-Arabic system perfected this base-10, place-value system with ten digits (0-9), making it the powerful and efficient system we use today. (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?
Answer:
Description: The Mayan number system was a vigesimal (base-20) system with a modification. It used a place value system with levels stacked vertically. The bottom level represented units (20⁰), the level above represented 20s (20¹), the level above that represented 360s (20 × 18), and so on. They used a dot (•) for 1 and a horizontal bar (—) for 5. Numbers from 1 to 19 were formed using combinations of these dots and bars.
Example: Let's represent the number 87.
87 ÷ 20 = 4, remainder 7.
So, 87 = (4 × 20) + (7 × 1).
The Mayan numeral would have two levels:
Bottom level (1s): 7 = two bars (5+5=10 is too big), so we need 5+2. This is represented as one bar (—) and two dots (••).
Top level (20s): 4 is represented as four dots (••••).
Most Significant Feature: Its most significant feature was that it was a sophisticated place value system that included a symbol for zero (a shell-like shape), developed independently in the Americas.
Puzzling Aspect: A puzzling aspect is that the third place value was 360 (20 × 18), not 400 (20²). Historians believe this was likely due to the importance of their 360-day calendar (the Tun). (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.
Answer:
Step 1: Represent each number using groups of 10s and 1s.
47 = 4 tens + 7 ones -> nnnn |||||||
65 = 6 tens + 5 ones -> nnnnnn |||||
Step 2: Combine all symbols.
Tens: nnnn + nnnnnn = nnnnnnnnnn (10 tens)
Ones: ||||||| + ||||| = |||||| ||||| (12 ones)
Step 3: Group the symbols.
Group the tens: 10 tens (nnnnnnnnnn) can be grouped into 1 hundred (9). (Since 10 × 10 = 100).
Group the ones: 12 ones (||||||||||||) can be grouped into 1 ten (n) and 2 ones (||). (Since 10 ones = 1 ten).
Step 4: Combine the new groups.
We now have: 1 hundred (9), 1 ten from the original groups? Wait no, we grouped all 10 original tens into 1 hundred. We also got 1 new ten from grouping the ones.
So final tally: 1 hundred (9), 1 ten (n), and 2 ones (||).
Step 5: Final Answer.
1 hundred + 1 ten + 2 ones = 100 + 10 + 2 = 112.
Egyptian representation: 9 n || (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
Answer: b) 469 (CD=400, LX=60, IX=9) (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
Answer: c) An archaic form that is still sometimes used (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
Answer: 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
Answer: b) Was not a place value system, making complex calculations difficult (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
Answer: c) 5 (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
Answer: b) The European perspective of learning it from the Arabs, though it originated in India (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
Answer: b) Place value shift (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
Answer: b) A placeholder and a number with defined properties (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 ∩∩∩∩
Answer: b) 999 |||| (3×100 + 0×10 + 4×1) (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.
Answer: b) It required new, unique symbols to be invented for every higher power of 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
Answer: b) Combining symbols and then grouping them into higher denominations (e.g., 10 | symbols become 1 n) (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
Answer: b) An additive system using landmark numbers (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
Answer: b) Time (hours, minutes) and angles (degrees) (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.
Answer: c) It created ambiguity because the same numeral could represent different numbers. (Competency: Reasoning & Analysing)
3. How would the number 62 be represented in its simplest form in this system?
a) ‹‹ ∀∀ b) ∀∀ c) ‹ ‹‹ d) ∀ ‹‹
Answer: a) ‹‹ ∀∀ (1×60 + 2×1). The ‹‹ represents one 60 (a single digit made of two 10s) and ∀∀ represents two 1s. (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.
Answer: c) Using the principle of place value. (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
Answer: b) Additive Grouping by 2s (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.
Answer: b) It becomes very long and impractical for numbers larger than 6. (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.
Answer: b) ukasar-ukasar-ukasar-ukasar-ukasar-ukasar (six repetitions of "ukasar" for 2+2+2+2+2+2) (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
Answer: b) The specific needs and environment of a culture (Competency: Reasoning & Analysing)
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