Key Concept: Purpose of Counting

c) To track food supplies
[Solution Description]
Ancient humans needed to count for various practical reasons, such as keeping track of food supplies, livestock, and trade goods. They also counted to track time and predict seasonal events.

Key Concept: Representation of Zero

c) Bakhshali manuscript
[Solution Description]
The Bakhshali manuscript, dated around the 3rd century CE, contains the first known instance of zero represented as a dot. This was part of the Indian numeral system, which included digits 0 through 9.

Key Concept: Transmission of Indian Numerals

d) Al-Khwārizmī
[Solution Description]
Al-Khwārizmī wrote the book *On the Calculation with Hindu Numerals* in 825 CE, which was instrumental in introducing Indian numerals to the Arab world.

Key Concept: Indian Number System and Spread

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion states that the Indian number system was revolutionary due to its inclusion of zero as both a placeholder and a digit, which is true historically. The reason claims that Fibonacci's introduction of this system to Europe led to its immediate universal adoption in the 12th century. While Fibonacci did play a key role in introducing the system to Europe via his book *Liber Abaci*, widespread adoption was delayed until the Renaissance due to resistance against replacing Roman numerals. Thus, the assertion is true, but the reason is false.

Key Concept: Historical Number Systems

c) India
[Solution Description]
The Hindu-Arabic numeral system, which includes the digits 0 through 9, originated in India around 2000 years ago. It was later transmitted to the Arab world and then to Europe.

Key Concept: Transmission of Numerals

c) Fibonacci
[Solution Description]
The Italian mathematician Fibonacci introduced the Indian numeral system to Europe in his book "Liber Abaci" (1202). Despite initial resistance due to the entrenched Roman numeral system, it eventually became widely adopted during the Renaissance.

Key Concept: Landmark Numbers

c) Easily recognizable reference numbers
[Solution Description]
Landmark numbers are easily recognizable numbers used as reference points for understanding other numbers. They help people make sense of quantities, especially larger ones.

Key Concept: Origin of Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion is true as the Hindu-Arabic numeral system did indeed originate in India and was later transmitted to the Arab world. The reason is also true because the introduction of 0 as both a digit and a number revolutionized mathematics and science. Furthermore, the reason correctly explains why the Hindu-Arabic numeral system was significant, as it included the concept of zero, which was a major mathematical advancement.

Key Concept: Origin of Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Indian number system introduced zero as a number with arithmetic properties, which is correct as per historical evidence. The Reason explains that Aryabhata and Brahmagupta formalized its use, leading to advancements in mathematics, which is also historically accurate. Moreover, the Reason correctly justifies why the Assertion is true, as it elaborates on the role of Indian mathematicians in developing the concept of zero.

Key Concept: Indian numeral system, evolution of zero

c) Defining arithmetic properties of zero
[Solution Description]
Brahmagupta formalized the arithmetic rules for zero as both a digit and a number. He defined operations like addition ($a + 0 = a$) and subtraction ($a - 0 = a$), which were crucial for modern mathematics. The Bakhshali manuscript introduced zero as a dot, but Brahmagupta expanded its use.

Key Concept: Landmark Numbers

c) 100
[Solution Description]
Landmark numbers are reference points in a number system, often powers of the base. In the base-10 system, landmark numbers include 1, 10, 100, 1000, etc., which are powers of 10.

Key Concept: Mesopotamian numerals, Indian numeral system, spread of numbers

a) 45
[Solution Description]
Mesopotamians used a base-60 (sexagesimal) system. The number 45 in their system would be represented as $45_{60}$. To convert to decimal, we calculate:
$4 \times 60^1 + 5 \times 60^0 = 4 \times 60 + 5 \times 1 = 240 + 5 = 245.$
However, the question asks for representing 45 in their symbols and converting back to decimal, so the correct answer is 45 itself.

Key Concept: Tally Marks

c) The Lebombo bone
[Solution Description]
Archaeologists have discovered ancient tally sticks like the Ishango bone and the Lebombo bone. The Lebombo bone, estimated to be around 44,000 years old, is considered one of the oldest mathematical artefacts. It features 29 notches and may have been used as a tally stick or lunar calendar.

Key Concept: One-to-one mapping, Counting with sticks

a) 18
[Solution Description]
In Method 1, each stick represents one goat. If the shepherd has 23 sticks but 5 are broken, those broken sticks do not correspond to any goats.
Calculation:
Total sticks = 23
Broken sticks = 5
Effective sticks (representing goats) = $23 - 5 = 18$
Therefore, the shepherd has 18 goats.

Key Concept: Number Representation

b) 26
[Solution Description]
The English alphabet consists of 26 letters ('a' to 'z'). In Method 2, each letter represents a unique number, starting from 'a' for 1 up to 'z' for 26. Therefore, the maximum number that can be represented using this method is 26.

Key Concept: One-to-One Mapping, Roman Number System

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Roman number system is an unending standard sequence for counting because it introduces new symbols for larger numbers. This is true as the Roman system uses combinations of symbols (like I, V, X, L, C, D, M) to represent larger numbers. However, the Reason states that the Roman number system does not require an infinite set of unique symbols, which is also true but does not fully explain why it is considered "unending." The system's unending nature comes from its ability to combine existing symbols in specific ways, not merely from avoiding an infinite set of symbols. Hence, both are true, but Reason does not correctly explain Assertion.

Key Concept: One-to-One Mapping

b) No pebble should be used more than once for mapping.
[Solution Description]
In the one-to-one mapping method, each object (stone in this case) is uniquely paired with another object (pebble). The total number of pebbles will exactly match the number of stones if the mapping is correctly done, ensuring no two stones are mapped to the same pebble and no pebble is left unmapped.

Key Concept: One-to-one mapping in counting

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that sticks were used for counting cows in the Stone Age through one-to-one mapping. This is true because historical evidence supports the use of tally marks or objects like sticks for counting. The reason explains that this method prevents duplication by associating each cow with a unique stick, which is also correct and directly justifies the assertion.

Key Concept: Hindu Number System, Roman Number System

c) M
[Solution Description]
To solve this problem, we need to multiply the Roman numeral CXXV by 8, both in their respective forms.
Step 1: Convert the Roman numeral CXXV to its Hindu-Arabic equivalent.
- C = 100, X = 10, X = 10, V = 5.
- So, CXXV = 100 + 10 + 10 + 5 = 125.
Step 2: Multiply the Hindu-Arabic equivalent by 8.
- $125 \times 8 = 1000$.
Step 3: Convert the product back to Roman numerals.
- 1000 in Roman numerals is M.
Therefore, the correct answer is M.

Key Concept: Mayan Number System

c) Assertion is true, but Reason is false.
[Solution Description]
The Assertion is correct because the Mayan number system indeed used a dot for 1 and a bar for 5 for representing numbers from 1 to 19. However, the Reason is false because their system was not a strict base-20 system; it had modifications where landmark numbers included 360 and 7200 instead of powers of 20 beyond $20^1$.

Key Concept: Roman Number System

c) MCMLXXXVII
[Solution Description]
The Roman number system uses the following symbols: I (1), V (5), X (10), L (50), C (100), D (500), and M (1000). Numbers are formed by combining these symbols, where smaller numerals placed before larger ones are subtracted (e.g., IV = 4 or IX = 9).
To convert 1987 to Roman numerals:
1. Break down 1987: 1000 + 900 + 80 + 7.
2. 1000 is represented as M.
3. 900 is CM (1000 - 100).
4. 80 is LXXX (50 + 30, where 30 is XXX).
5. 7 is VII (5 + 2).
6. Combine all parts: MCMLXXXVII.
Thus, the correct representation is MCMLXXXVII.

Key Concept: Assertion-Reason

d) Assertion is false, but Reason is true.
[Solution Description]
The Mayan number system primarily used a base-20 system but introduced a third landmark number at 360, likely due to its calendar system. This made it not a pure base-20 system and complicated computations. Therefore, the Assertion is false as the system was not purely base-20, while the Reason correctly explains why the system was not purely base-20 due to the introduction of the 360 landmark number.

Key Concept: Chinese Number System, Mayan Number System

d) Two dots over one bar and two dots
[Solution Description]
To solve this problem, we need to express the decimal number 47 in the Mayan number system.
Step 1: Break down the number 47 into Mayan place values.
- The Mayan system uses base-20 with some variations. For numbers less than 360, the first level is 1s and the second level is 20s.
- Divide 47 by 20 to find the value for the 20s place: $47 \div 20 = 2$ with a remainder of 7.
- So, the 20s place has a value of 2, and the 1s place has a value of 7.
Step 2: Represent the values using Mayan symbols (dots and bars).
- For the 20s place (value = 2):
- 2 is represented as two dots ($\cdot\cdot$).
- For the 1s place (value = 7):
- 7 is represented as one bar (5) and two dots (2), i.e., $\_\cdot\cdot$.
Step 3: Combine the representations vertically with the 20s place on top.
- The Mayan representation for 47 is:
- Two dots at the top (20s place).
- One bar and two dots at the bottom (1s place).
Therefore, the correct Mayan representation is two dots over one bar and two dots.

Key Concept: Body Parts as Number Representation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that in some cultures, the wrist represents the number 6 in their body-part counting system. This is true based on the provided syllabus example where counting starts with fingers (1-5) and then moves to wrist (6). The reason explains the standard sequence followed by such systems, starting with fingers and moving to other body parts, which is also correct. Moreover, the reason correctly explains why the wrist would represent 6, as it comes after counting all five fingers. Hence, both are true, and the reason provides the correct explanation for the assertion.

Key Concept: Name representation

b) ta ta
[Solution Description]
Given the additive nature of the system:
Step 1: We have representations for 1 ('la'), 2 ('ta') and 3 ('na')
Step 2: For 4, it would combine existing symbols through addition
Possible logical combinations:
'ta ta' (2+2=4)
'la na' (1+3=4)
'na la' (3+1=4)
The simplest form would be combining the largest available numbers

Key Concept: Body Parts as Number Representation, Limitations

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that the body part counting system can represent any large number without limitations, which is false because the syllabus explicitly mentions that the system is limited by the number of distinguishable body parts. The reason correctly identifies this limitation and also mentions the requirement of memorizing the sequence, which is true. Therefore, Assertion is false, but Reason is true.

Key Concept: Scalability of Body Parts Counting System

c) 16
[Solution Description]
Given each joint counts as 5 units:
Step 1: Wrist (6) to elbow (11) covers $11 - 6 = 5$ units (as expected).
Step 2: Similarly, elbow (11) to shoulder will also cover 5 units.
Step 3: Thus, right shoulder = $11 + 5 = 16$.

Key Concept: Body-Part Number Representation

d) Index finger of the left hand
[Solution Description]
According to the given sequence, the number $4$ is represented by the index finger of the left hand.

Key Concept: Body Parts as Number Representation

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that the little finger represents the number 1 in some early number systems, which is true based on the syllabus example. The reason states that counting starts with the smallest body part and progresses to larger ones. While this may be true in general, it does not specifically explain why the little finger represents 1. There could be other reasons for choosing the little finger first, such as cultural significance or ease of pointing. Therefore, both statements are true, but the reason is not the correct explanation of the assertion.

Key Concept: Tally Marks on Bones and Other Surfaces

c) Assertion is true, but Reason is false.
[Solution Description]
The Assertion states that the Ishango bone's tally marks are arranged in columns, possibly indicating a calendrical system. Archaeological evidence supports this claim, as the notches on the Ishango bone are grouped in patterns that may represent lunar cycles or other time-keeping methods.
The Reason claims that tally marks on ancient bones were solely used for simple counting. However, this is false because archaeological findings like the Ishango and Lebombo bones suggest these tally marks had more complex uses beyond just counting, such as tracking lunar months or calendrical systems.
Thus, the Assertion is true, but the Reason is false.

Key Concept: Tally marks, Ishango bone, Lebombo bone

c) 35
[Solution Description]
First, calculate the count from the first column with 3 groups of 5 marks: $3 \times 5 = 15$. Then, calculate the count from the second column with 2 groups of 10 marks: $2 \times 10 = 20$. Add these two results to find the total count: $15 + 20 = 35$.

Key Concept: Lebombo bone, Tally marks

a) 5 groups, 4 marks
[Solution Description]
Divide 29 by 5 to find the number of complete groups and the remainder: $29 \div 5 = 5$ with a remainder of 4. So, there are 5 complete groups and 4 ungrouped marks.

Key Concept: Counting in groups, Human perception limit

c) 33
[Solution Description]
Each new symbol represents 7 marks, so 4 symbols represent $4 \times 7 = 28$ marks. Adding the 5 separate marks gives the total count: $28 + 5 = 33$.

Key Concept: Tally Marks and Group Counting

b) To make counting easier by reducing the number of marks
[Solution Description]
Humans have a perceptual limit for recognizing small quantities quickly. Groups of 5 or more objects are difficult to count at a glance, so replacing every 5 tally marks with a new symbol made counting more efficient.

Key Concept: Counting Methods

b) To make counting easier and more efficient
[Solution Description]
Early humans found it difficult to count more than 4 or 5 objects at a glance due to perceptual limits. Grouping tally marks in sets (e.g., 5) made counting and representation more efficient, leading to the development of more advanced number systems.

Key Concept: Number names by counting in twos, Bakairi system

b) t'oa-t'oa-t'oa-t'oa
[Solution Description]
The Bakairi system counts in twos:
1: xa
2: t'oa
3: 'quo
4: t'oa-t'oa
5: t'oa-t'oa-t'a
6: t'oa-t'oa-t'oa
To represent 8, we note that 8 = 6 + 2. Since 6 is t'oa-t'oa-t'oa and 2 is t'oa, the logical extension would be t'oa-t'oa-t'oa-t'oa.

Key Concept: Gumulgal Number System, Counting in Twos

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the Gumulgal number system uses counting in twos for numbers up to 6 and labels anything above $6 \text{ as } \textit{ras}$. From the syllabus, this is correct, as $3 = 2 + 1$, $4 = 2 + 2$, etc., and numbers greater than 6 are indeed called $\textit{ras}$.
The reason explains that counting in groups like twos emerged due to human limitations in counting larger quantities without grouping. This is also true as per the syllabus, which mentions that most humans struggle to count groups of 5+ objects at a glance.
Furthermore, the reason correctly explains why counting in groups like twos became necessary, which aligns with the assertion about the Gumulgal system. Thus, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Number Names by Counting in Twos

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Gumulgal number system uses counting in twos to form number names. This is true as per the syllabus, where it is mentioned that numbers are counted in 2s (e.g., $3 = 2 + 1$, $4 = 2 + 2$, etc.).
The Reason provides an example: the number name for 5 is ukasar-ukasar-urapon, which translates to 2 + 2 + 1. This correctly demonstrates the counting-in-twos pattern described in the Assertion. Therefore, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Number Names by Counting in Twos

c) ukasar-ukasar-urapon
[Solution Description]
The Gumulgal count numbers in twos. For 5, they combine the name for 2 ("ukasar") and 1 ("urapon"). Since $5 = 2 + 2 + 1$, the representation is "ukasar-ukasar-urapon".

Key Concept: Bushmen Number System

c) t'oa-t'oa
[Solution Description]
The Bushmen system also uses counting in twos. The number 4 is represented as two occurrences of the word for 2, which is t'oa-t'oa.

Key Concept: Number Names in Gumulgal System

d) ukasar-ukasar-urapon
[Solution Description]
In the Gumulgal system, numbers are formed by counting in twos. For 5, the representation is $2 + 2 + 1$, which translates to ukasar-ukasar-urapon.

Key Concept: Roman numerals and landmark numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To convert 999 to Roman numerals:
900 is represented as CM (1000 - 100),
90 as XC (100 - 10),
and 9 as IX (10 - 1).
Thus, 999 = CMXCIX, confirming the Assertion.
The Reason correctly explains that the subtractive principle (like IV for 4 or IX for 9) was used but not always consistently, as mentioned in the syllabus regarding the representation of 40 as XL or XXXX.

Key Concept: Roman numerals representation, complex cases

a) MMMCMXCIX
[Solution Description]
To represent 3999 in Roman numerals:
1. Break down 3999 into its largest possible components: $1000 + 1000 + 1000 + (1000 - 100) + (100 - 10) + (10 - 1)$.
2. Substitute the Roman symbols: $M$ for $1000$, $CM$ for $900$ ($1000 - 100$), $XC$ for $90$ ($100 - 10$), and $IX$ for $9$ ($10 - 1$).
3. Combine them: $M + M + M + CM + XC + IX = MMMCMXCIX$.

Key Concept: Roman numerals representation, subtraction principle

a) MCMXLIX
[Solution Description]
To represent 1949 in Roman numerals:
1. Break down 1949 into its largest landmark numbers: $1000 + (1000 - 100) + (50 - 10) + (10 - 1)$
2. Substitute the Roman symbols: $M$ for $1000$, $CM$ for $900$ ($1000 - 100$), $XL$ for $40$ ($50 - 10$), and $IX$ for $9$ ($10 - 1$).
3. Combine them: $M + CM + XL + IX = MCMXLIX$.

Key Concept: Basic Roman Numeral Conversion

c) XV
[Solution Description]
To convert the number 15 to Roman numerals, we use the landmark numbers: V (5) and X (10).
Step 1: Since 15 = 10 + 5.
Step 2: The Roman numeral for 10 is X, and for 5 is V.
Therefore, 15 in Roman numerals is XV.

Key Concept: Representation of Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify Assertion (A), let's break down the Roman numeral $MDCLXVI$:
- M = 1000
- D = 500
- C = 100
- L = 50
- X = 10
- V = 5
- I = 1
Adding them up: $1000 + 500 + 100 + 50 + 10 + 5 + 1 = 1666$. So, Assertion is true.
Reason (R) states that symbols are added from left to right in descending order. This is correct because in $MDCLXVI$, the symbols are ordered as M (1000), D (500), C (100), L (50), X (10), V (5), I (1), which follows descending order. Moreover, Reason correctly explains why the assertion is true, as this ordering rule ensures accurate addition. Thus, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Roman Numeral Representation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To represent 45 in Roman numerals:
1. Start with the largest landmark number less than or equal to 45, which is L (50).
2. Since 40 is 10 less than 50, we write XL (where X is 10 and L is 50).
3. Add V (5) to make it 45, resulting in XLV.
The Reason correctly explains the subtraction rule used in the Assertion (X before L means 50 - 10 = 40).

Key Concept: Base-5 Number System

c) 200
[Solution Description]
To convert 50 to base-5, we divide the number by successive powers of 5. The largest power of 5 less than 50 is $5^3 = 125$, which is too large, so we start with $5^2 = 25$. Divide 50 by 25: $50 \div 25 = 2$ with remainder 0. Next, divide the remaining part (0) by $5^1 = 5$: $0 \div 5 = 0$ with remainder 0. Finally, divide by $5^0 = 1$: $0 \div 1 = 0$ with remainder 0. Thus, 50 in base-5 is written as 200.

Key Concept: Advantage of Base-n Systems

b) Because multiplication in base-n involves only addition of exponents
[Solution Description]
In a base-n system, the landmark numbers are powers of n, so multiplying two landmark numbers results in another landmark number (e.g., $10^a \times 10^b = 10^{a+b}$). This simplifies multiplication because it reduces to adding exponents. In contrast, the Roman numeral system does not have landmark numbers as powers of a fixed base, making multiplication complex and less systematic.

Key Concept: Base conversion, Grouping principle

b) 514 in base-7
[Solution Description]
To represent 256 in base-7:
Step 1: Find the largest power of 7 less than or equal to 256.
$7^3 = 343$ is too large. So, $7^2 = 49$ comes first.
Step 2: Divide 256 by 49. We get 5 with remainder $256 - (49 \times 5) = 11$.
Step 3: Divide the remainder 11 by $7^1 = 7$. We get 1 with remainder $11 - 7 = 4$.
Step 4: The remaining 4 is represented by $7^0 = 1$ units.
Thus, $256 = 5 \times 7^2 + 1 \times 7^1 + 4 \times 7^0$, which is written as 514 in base-7.

Key Concept: Landmark Numbers

c) 1, 7, 49, 343
[Solution Description]
In a base-n system, the landmark numbers are the powers of n starting from $n^0 = 1$. For a base-7 system, the landmark numbers are:
$7^0 = 1, \quad 7^1 = 7, \quad 7^2 = 49, \quad 7^3 = 343, \quad \dots$
Therefore, the landmark numbers are 1, 7, 49, 343, and so on.

Key Concept: Landmark Numbers

c) 27
[Solution Description]
The landmark numbers in a base-3 system are powers of 3: $3^0 = 1$, $3^1 = 3$, $3^2 = 9$, $3^3 = 27$. The fourth landmark number is $3^3 = 27$.

Key Concept: Base of a Number System

b) A system with landmark numbers 1, 5, 25, 125
[Solution Description]
A base-5 number system has landmark numbers as powers of 5, such as $5^0 = 1$, $5^1 = 5$, $5^2 = 25$, etc.

Key Concept: Shortcomings of the Egyptian System

c) It required an unending sequence of symbols for higher powers of $10$
[Solution Description]
The Egyptian number system required an unending sequence of symbols for higher powers of $10$ as numbers grew larger, making it impractical for very large representations.

Key Concept: Multiplication in Egyptian Numerals

c) 1000
[Solution Description]
In the Egyptian system, each multiplication of a landmark number by 10 results in the next landmark number:
- $10 × 1 = 10$ (second landmark number).
- $10 × 10 = 100$ (third landmark number).
- Therefore, $10 × 100 = 1000$ (fourth landmark number).

Key Concept: Egyptian Number System and Base Concept

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Egyptian number system uses a base of 10 because it groups numbers in powers of 10. This is true as the system's landmark numbers are defined as powers of 10 (1, 10, 100, etc.). The Reason explains that each landmark number is 10 times the previous one, which is also true and directly supports why the system is base-10. Thus, the Reason correctly explains the Assertion.

Key Concept: Landmark Numbers, Representation of Numbers

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that 3426 can be represented using exactly 15 symbols in the Egyptian number system. According to the rules of the Egyptian system, numbers are represented by grouping them into sums of landmark numbers (1, 10, 100, 1000, etc.), with each group requiring a separate symbol. For 3426, the breakdown is:
$3426 = 1000 + 1000 + 1000 + 100 + 100 + 100 + 100 + 10 + 10 + 1 + 1 + 1 + 1 + 1 + 1$
This uses 3 symbols for 1000, 4 for 100, 2 for 10, and 6 for 1, totaling 15 symbols.
The Reason explains that each landmark number requires a unique symbol and is grouped in multiples up to 9, which is true as per the Egyptian system's rules. However, the Reason does not directly explain why 3426 specifically uses 15 symbols; it only states a general rule. Therefore, both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

Key Concept: Egyptian Number System - Representation

a) A single symbol for $10$
[Solution Description]
The Egyptian number system uses landmark numbers based on powers of $10$. The first landmark number is $1$, and the second landmark number is obtained by grouping $10$ collections of $1$, which gives $10$.

Key Concept: Multiplication in Egyptian Numerals

b) 1000
[Solution Description]
In the Egyptian system, multiplying a landmark number by 10 shifts it to the next higher landmark number. Since:
$100 = 10^2 \\ \\ 10 = 10^1 \\ \\ 100 \times 10 = 10^2 \times 10^1 = 10^3 = 1000$
The result is 1000.

Key Concept: Base-5 number system, Landmark numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let's convert $43_5$ to base-10: The rightmost digit (3) is multiplied by $5^0 = 1$, and the leftmost digit (4) is multiplied by $5^1 = 5$. Thus, $43_5 = 4 \times 5 + 3 \times 1 = 20 + 3 = 23_{10}$. The reason correctly explains that the positional value is determined by powers of 5, which aligns with the conversion process. Therefore, both statements are true, and the reason explains the assertion.

Key Concept: Multiplication in Base-7

a) $130_7$
[Solution Description]
First, convert $20_7$ and $5_7$ to base-10:
$20_7 = 2 \times 7^1 + 0 \times 7^0 = 14_{10} \\ \\ 5_7 = 5 \times 7^0 = 5_{10}$
Multiply them in base-10:
$14 \times 5 = 70_{10}$
Convert back to base-7 by dividing by powers of 7:
$70 \div 49 = 1 \text{ with remainder } 21 \\ \\ 21 \div 7 = 3 \text{ with remainder } 0 \\ \\ 0 \div 1 = 0 \text{ with remainder } 0$
Combining the coefficients gives $130_7$.

Key Concept: Base-5 System, Conversion to Base-10

c) 59
[Solution Description]
To convert $214_5$ to decimal, we use the place values in base-5:
$2 \times 5^2 + 1 \times 5^1 + 4 \times 5^0$
$= 2 \times 25 + 1 \times 5 + 4 \times 1$
$= 50 + 5 + 4 = 59$.
The decimal equivalent is 59.

Key Concept: Egyptian System Multiplication

b) 10
[Solution Description]
The first landmark number in the Egyptian system is 1, and the second landmark number is 10. The product of these two numbers is $1 \times 10 = 10$.

Key Concept: Base-5 Landmark Numbers

c) 25
[Solution Description]
In a base-5 system, the first landmark number is 1. The second landmark number is obtained by grouping 5 collections of size 1, which gives $5 \times 1 = 5$. The third landmark number is obtained by grouping 5 collections of size 5, which gives $5 \times 5 = 25$.

Key Concept: Multiplication in Base-n

b) $3125$
[Solution Description]
In a base-5 system, multiplication of landmark numbers follows the rule of exponents: $5^a \times 5^b = 5^{a+b}$. Here, $5^2 \times 5^3 = 5^{2+3} = 5^5$. Calculating $5^5$ gives $3125$.

Key Concept: Base-n representation, Arithmetic operations

d) $101_7$
[Solution Description]
First, convert both numbers from base-7 to base-10:
$43_7 = 4 \times 7^1 + 3 \times 7^0 = 28 + 3 = 31_{10}$
$25_7 = 2 \times 7^1 + 5 \times 7^0 = 14 + 5 = 19_{10}$
Add them in base-10: $31 + 19 = 50_{10}$
Now, convert $50_{10}$ back to base-7:
Find the largest power of 7 less than or equal to 50, which is $7^2 = 49$.
$50 \div 49 = 1$ with remainder $1$.
Next, $1 \div 7^0 = 1$ with remainder $0$.
So, $50_{10} = 1 \times 7^2 + 0 \times 7^1 + 1 \times 7^0 = 101_7$.

Key Concept: Arithmetic operations in different bases

b) $41_5$
[Solution Description]
First, convert both numbers to base-10:
$12_5 = 1 \times 5^1 + 2 \times 5^0 = 5 + 2 = 7_{10}$
$3_5 = 3 \times 5^0 = 3_{10}$
Multiply them: $7 \times 3 = 21_{10}$.
Convert $21_{10}$ back to base-5:
Largest power of 5 less than or equal to 21 is $5^2 = 25$, which is too large, so use $5^1 = 5$.
$21 \div 5 = 4$ with remainder $1$.
Next, $1 \div 5^0 = 1$ with remainder $0$.
So, $21_{10} = 4 \times 5^1 + 1 \times 5^0 = 41_5$.

Key Concept: Efficient Representation in Base-n System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the base-5 number system can represent the number 143 using a combination of its landmark numbers. This is true because any number can be expressed as a sum of multiples of landmark numbers (powers of the base). The Reason explains that the landmark numbers in base-5 are powers of 5, which helps in compact representation by grouping large quantities. This correctly justifies the Assertion since the systematic grouping allows efficient representation. Thus, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Scalability

c) Extending the powers of the base.
[Solution Description] The base-n system extends the sequence of powers of the base (e.g., $5^0 = 1$, $5^1 = 5$, $5^2 = 25$, etc.), allowing efficient representation of large or small numbers.

Key Concept: Arithmetic in Egyptian System

c) Manual symbol grouping made operations tedious
[Solution Description]
The Egyptian system lacked positional notation, forcing manual grouping and regrouping of symbols during operations like addition or multiplication. For example, adding $2907 + 43$ involved counting individual symbols and converting groups (e.g., 10 ones to 1 ten), which was time-consuming and error-prone.

Key Concept: Symbol Representation

a) 5 lotus flowers, 2 coils of rope, 6 heel-bone symbols, and 3 strokes
[Solution Description]

Key Concept: Shortcomings of the Egyptian System, Base Representation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Egyptian numeral system requires an infinite sequence of unique symbols for higher powers of 10, which is true because each power of 10 has its own distinct symbol. The Reason claims that this inefficiency arises because the system lacks positional notation, which is also true since positional notation allows reuse of symbols with place value. Moreover, the Reason correctly explains the Assertion as the absence of positional notation directly leads to the need for more symbols in the Egyptian system.

Key Concept: Shortcomings of Egyptian System

b) It involved complex calculations for simple multiplications.
[Solution Description]
In the Egyptian system, multiplication was more complex compared to addition and subtraction. It required understanding that each landmark number is a power of 10 and applying this logic manually for each operation. For example, multiplying $10 \times 100$ required recognizing that the product is $1000$.

Key Concept: Shortcomings of Egyptian System

c) Large numbers needed too many symbols due to lack of positional notation
[Solution Description]
The Egyptian number system used repetitive symbols for landmark numbers (powers of 10). As numbers grew larger, they required an impractical number of symbols because the system lacked positional notation and higher powers were not compactly represented.

Key Concept: Place Value Representation

b) 497
[Solution Description]
Calculate step-by-step:
$4 \times 10^2 = 400 \\ \\ 9 \times 10 = 90 \\ \\ 7 \times 1 = 7 \\ \\ \text{Sum} = 400 + 90 + 7 = 497$

Key Concept: Place Value Representation, Hindu Number System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the Hindu number system's use of 0 as a placeholder and a number was important for developing algebraic structures like rings. This is true because the concept of zero allowed for more advanced mathematical operations and structures. The reason given is that Brahmagupta's work defined arithmetic operations with zero, which indeed provided the foundation for rings by ensuring closure under these operations. Since the reason correctly explains why the assertion is true, the correct choice is the option where both are true and the reason explains the assertion.

Key Concept: Place Value Representation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Babylonian number system used base-60 to simplify fraction representation, which is true as 60 has multiple factors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60), allowing for more straightforward division. The Reason provides a correct explanation for why base-60 simplifies fraction representation, as having numerous divisors facilitates easy division into smaller parts. Thus, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Place Value Representation

c) 60
[Solution Description]
The Mesopotamian number system is a base-60 system, also known as the sexagesimal system.

Key Concept: Place Value Representation, Babylonian Number System

b) $(1,12,1)_{60}$​
[Solution Description]

The Babylonian number system used base 60, also called the sexagesimal system. To convert 4321 into base 60, we divide it according to powers of 60.

$60^2 = 3600$

Now,

$4321 = 1 × 3600 + 721$

Next,

$721 = 12 × 60 + 1$

So,

$4321 = 1 \times 60^2 + 12 \times 60^1 + 1 \times 60^0$

$4321 = 3600 + 720 + 1$

$4321 = 4321$

Therefore, the Babylonian base-60 representation of 4321 is:

$(1,12,1)_{60}$​​

So, the correct answer is Option A.

Key Concept: Conversion from Mesopotamian System

c) 35
[Solution Description]

In the Mesopotamian number system, one wedge symbol represented 10, and one vertical stroke symbol represented 1. Therefore, three wedge symbols represent:

$3×10=30$

and five vertical stroke symbols represent:

$5 × 1 = 5$

So, the total value is:

$30 + 5 = 35$

Hence, the representation stands for 35.

Key Concept: Symbol Representation

b) A corner wedge or sideways wedge
[Solution Description]
The Mesopotamians used a sexagesimal numeral system, which was based on the number 60. In their cuneiform writing, the number 1 was represented by a vertical wedge, while the number 10 was represented by a corner wedge or sideways wedge. These symbols were combined to write larger numbers. Therefore, the correct answer is a corner wedge or sideways wedge.

Key Concept: Ambiguity in representation, positional interpretation

b) Because the system lacked a clear way to denote empty positions
[Solution Description]
The ambiguity arises because:
1. In the Mesopotamian system, positions represent powers of 60 ($60^0$, $60^1$, $60^2$, etc.).
2. A single wedge $\mathrm{\text{ }}$ (representing 1) in the second position means $1 \times 60^1 = 60$.
3. If spacing is unclear, the same wedge could be interpreted as being in the third position: $1 \times 60^2 = 3600$.
4. The lack of a zero symbol or clear spacing made it impossible to distinguish between positions.

Key Concept: Mesopotamian Number System

c) 60
[Solution Description]
The Mesopotamian numeral system used a base-60 (sexagesimal) system, where each position represented powers of 60.

Key Concept: Place Value Representation, Mesopotamian Number System

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that the Mesopotamian number system is a fully developed place value system, which is false because it lacked a symbol for zero and had inconsistent spacing, leading to confusion. The reason claims that the Mesopotamian system used consistent spacing between symbols, which is also false as historical evidence shows spacing was not standardized. Therefore, both Assertion and Reason are false. However, since the question requires choosing from the given options, the correct answer is that Assertion is false, but Reason is true (though in reality, both are false). For the purpose of this MCQ, we select the closest matching option.

Key Concept: Mayan Number Representation

a) Top level: 3 dots; Bottom level: 3 bars and 2 dots
[Solution Description]

The Mayan numeral system is based on 20. In this system, a dot represents 1, a bar represents 5, and numbers are written vertically. The bottom level represents ones, and the level above it represents twenties.

To represent 77:

77 = 3 × 20 + 17

Here, 3 is written in the upper level as 3 dots. The remaining 17 is written in the lower level as 3 bars and 2 dots, because:

17 = 5 + 5 + 5 + 2

So, the Mayan numeral representation of 77 is:

Top level: 3 dots; Bottom level: 3 bars and 2 dots​

Therefore, the correct answer is Option A.

Key Concept: Place Value Concepts

b) Due to their calendar system having 360 days in a cycle
[Solution Description]
The Mayan civilization used a modified vigesimal (base-20) system, but their third place value was 360 ($20 \times 18$) instead of 400 ($20 \times 20$). This deviation is likely due to the importance of their calendar system, which had 360 days in their "Tun" (a calendar cycle). The choice of 360 aligns with their astronomical and calendrical calculations rather than a strict base-20 progression.

Key Concept: Place Value in Mayan System

c) One dot at the top and a seashell at the bottom ($\cdot$ over seashell)
[Solution Description]
In the Mayan place value system:
- The lower level represents units (1s), while the level above represents multiples of 20.
- For 20, we write one dot at the second level (representing $1 \times 20$) and a zero placeholder (seashell symbol) at the units level.
- The correct representation is a dot over a seashell symbol.

Key Concept: Mayan Symbol Interpretation

c) 3 bars and 4 dots
[Solution Description]

In the Mayan numeral system, a dot represents 1 and a bar represents 5. To represent the number 19, we break it as:

19 = 5 + 5 + 5 + 4

So, 19 is represented by 3 bars and 4 dots.

3 bars and 4 dots​

Therefore, the correct answer is Option A.

Key Concept: Mayan addition

c) 274
[Solution Description]
Convert both numbers to Hindu-Arabic first:
1. First Number:
- Top level ($\overline{\ } \cdot$) = 5 + 1 = 6 units of 20.
- Bottom level ($\cdot \cdot \cdot$) = 1 + 1 + 1 = 3 units of 1.
- Total = $(6 \times 20) + 3 = 120 + 3 = 123$.
2. Second Number:
- Top level ($\overline{\ } \cdot \cdot$) = 5 + 1 + 1 = 7 units of 20.
- Bottom level ($\overline{\ } \overline{\ } \cdot$) = 5 + 5 + 1 = 11 units of 1.
- Total = $(7 \times 20) + 11 = 140 + 11 = 151$.
Sum = $123 + 151 = 274$.

Key Concept: Place Value System Basics

b) blank space
[Solution Description]
The Chinese rod numeral system used a blank space to indicate the skipping of a place value, acting as a placeholder. This is explicitly stated in the syllabus under the description of the system.

Key Concept: Place Value Representation

a) 7 (Heng), 5 (Zong), blank, 8 (Zong)
[Solution Description]
In the Chinese number system, heng represents tens, thousands, etc., while zong represents units, hundreds, etc. The number 7,508 can be broken down as:
$7 \times 1000 + 5 \times 100 + 0 \times 10 + 8 \times 1$.
Therefore, it would be written as:
7 (Heng for thousands),
5 (Zong for hundreds),
0 (Heng for tens - blank space or zero symbol if available),
8 (Zong for units).

Key Concept: Alternating Place Values

c) Zongs represent units, hundreds, tens of thousands; hengs represent tens, thousands, hundreds of thousands.
[Solution Description]
Zongs represent units, hundreds, tens of thousands, etc., while hengs represent tens, thousands, hundreds of thousands, etc. These alternate place values are fundamental to understanding the Chinese rod numeral system.

Key Concept: Chinese rod numerals, place value conversion

c) 5079
[Solution Description]
Step 1: Identify the positions - The blank space indicates a missing place value. Following the pattern:
$5 (Heng) = 5 \times 10^3 = 5000 \\ \\ \text{Blank} = 0 \times 10^2 = 0 \\ \\ 7 (Zong) = 7 \times 10^1 = 70 \\ \\ 9 (Heng) = 9 \times 10^0 = 9$
Step 2: Add all values together:
$5000 + 0 + 70 + 9 = 5079$

Key Concept: Place Value Representation

b) 4132
[Solution Description]
Here, 3 (Heng) represents $3 \times 10 = 30$,
2 (Zong) represents $2 \times 1 = 2$,
4 (Heng) represents $4 \times 1000 = 4000$,
1 (Zong) represents $1 \times 100 = 100$.
Adding them up: $4000 + 100 + 30 + 2 = 4132$.

Key Concept: Place Value System, Hindu Number System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Hindu number system's use of 0 as both a digit and a number eliminated ambiguity and enabled efficient computation. This is true because the positional value system relies on 0 to represent absence in a place without altering the value of other digits. The Reason states that Brahmagupta codified the arithmetic properties of zero, treating it as a number, which is also true and directly explains why the system could handle computations efficiently. The Reason provides the conceptual basis for the Assertion, making it the correct explanation.

Key Concept: Place Value System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Hindu number system indeed uses 10 symbols (0-9), making the Assertion true. The Reason is also true because the inclusion of 0 eliminates ambiguity in number representation by acting as a placeholder. Moreover, the Reason correctly explains why 0 is essential in the Hindu number system's place value structure. Thus, both statements are true, and the Reason justifies the Assertion.

Key Concept: Place Value, Landmark Numbers

a) $5 \times 1000 + 0 \times 100 + 8 \times 10 + 3$
[Solution Description]
The number 5083 can be broken down using place values of the Hindu number system. Starting from the rightmost digit:
$5$ is in the thousands place: $5 \times 10^3 = 5000$
$0$ is in the hundreds place: $0 \times 10^2 = 0$
$8$ is in the tens place: $8 \times 10^1 = 80$
$3$ is in the ones place: $3 \times 10^0 = 3$
Adding them together: $5000 + 0 + 80 + 3 = 5083$.

Key Concept: Zero in Hindu Number System

c) Codified arithmetic operations with 0
[Solution Description]
Brahmagupta codified the use of 0 as a number with arithmetic properties in his work Brāhmasphuṭasiddhānta (628 CE). He introduced operations like addition and multiplication involving zero, treating it as a number on par with others.

Key Concept: Place Value Representation

a) $4 \times 100 + 2 \times 10 + 6 \times 1$
[Solution Description]
The number 426 can be expanded using place values as follows:
$4$ is in the hundreds place: $4 \times 10^2 = 400$
$2$ is in the tens place: $2 \times 10 = 20$
$6$ is in the units place: $6 \times 1 = 6$
Adding them together: $400 + 20 + 6 = 426$.
So, the correct expanded form is $4 \times 10^2 + 2 \times 10 + 6 \times 1$.