Chapter 3 A Story of Numbers (Class 8 - Latest Maths NCERT (Ganita Prakash I) Solutions)

Given:

A number system where each number is represented by a collection of vertical sticks ($|$), such that one stick represents one unit.

To Find:

Methods for arithmetic operations (Addition, Subtraction, Multiplication, and Division) using only sticks.

Solution:

1. Addition:

To add two collections, simply place all the sticks from the first collection and the second collection together into a single pile. The total number of sticks in the combined pile represents the sum.

2. Subtraction:

To subtract a smaller collection from a larger one, line up the sticks from both collections side-by-side. For every stick in the smaller collection, remove or cross out one corresponding stick from the larger collection. The remaining sticks in the larger collection represent the difference.

3. Multiplication:

To multiply two collections, look at the number of sticks in the second collection. Place the entire first collection of sticks on the ground that many times. For example, if multiplying three sticks by two sticks, place a group of three sticks down twice. The total resulting sticks represent the product.

4. Division:

To divide a large collection (dividend) by a smaller collection (divisor), keep taking away a group of sticks equal to the divisor from the dividend until the remaining sticks are fewer than the divisor. The number of times you successfully took away a group is the quotient, and the leftover sticks are the remainder.

The total count of groups formed represents the result of the division.

Solution:

In this system, we assign numerical values to the English alphabets starting from $a$ up to $z$.

$a, b, c, \dots, z$ are 26 numbers representing values from $1$ to $26$.

$aa, bb, \dots, zz$ are 26 more numbers that extend the sequence.

Detailed Explanation:

To represent all numbers systematically, we can use a positional notation system similar to the base-10 (decimal) system we use in India, but here the base is $26$.

1. Single-letter strings: $a=1, b=2, c=3, \dots, z=26$.

2. Two-letter strings: After $z$ ($26$), we start with $aa$ ($27$), $ab$ ($28$), $ac$ ($29$), and so on, until $az$ ($52$). Then we continue with $ba, bb, \dots, bz$, and finally reaching $zz$.

The total number of two-letter combinations possible is $26 \times 26 = 676$.

3. Three-letter strings: After $zz$ is exhausted, we move to $aaa$, $aab$, and so on. This allows us to represent infinitely many numbers by simply increasing the length of the string.

Given:

The requirement is to design an original number system with unique symbols and rules for representation.

Construction (The Symbolic System):

In this new system, we will use symbols inspired by Indian nature to represent different values. This is an Additive Number System where the total value is found by adding the values of the individual symbols together.

The basic symbols are defined in the table below:

Solution:

Rule 1: Symbols are written from largest value to smallest value.

Rule 2: To find the total value, simply add the values of all symbols shown.

Rule 3: A symbol can be repeated as many times as needed to reach the desired amount.

For example, if we want to represent the price of a book which is $\textsf{₹} 123$, we would use:

1. One Lotus ($\clubsuit$) for $100$.

2. Two Lamps ($\triangle \triangle$) for $10 + 10 = 20$.

3. Three Petals ($\mid \mid \mid$) for $1 + 1 + 1 = 3$.

To Find:

The Roman numeral representation for the decimal numbers: (i) $1222$, (ii) $2999$, (iii) $302$, and (iv) $715$.

Solution:

In the Indian school curriculum, Roman numerals are taught as an ancient system of notation. The basic symbols used in this system are:

(i) For the number $1222$:

We expand the number according to its place values:

Thus, $1222$ in Roman system is MCCXXII.

(ii) For the number $2999$:

We expand the number as:

Converting each part to Roman numerals using the subtractive principle:

$2000 = \text{MM}$

$900 = \text{CM}$ (which is $1000 - 100$)

$90 = \text{XC}$ (which is $100 - 10$)

$9 = \text{IX}$ (which is $10 - 1$)

Thus, $2999$ in Roman system is MMCMXCIX.

(iii) For the number $302$:

Expanding the number:

Thus, $302$ in Roman system is CCCII.

(iv) For the number $715$:

Expanding the number:

Thus, $715$ in Roman system is DCCXV.

Solution:

The use of different number sequences for different objects by indigenous people is a fascinating cultural and linguistic phenomenon. There are several reasons why this might occur:

1. Contextual Significance: In many indigenous cultures, the nature of the object being counted is as important as the quantity. For example, counting sacred objects might require a formal or "higher" set of number names compared to counting everyday items like fish or coconuts.

2. Physical Counting Methods: Different objects may be counted using different parts of the body or different physical movements. If one set of objects is counted using fingers and another using joints, distinct names might evolve to distinguish these methods.

3. Grouping and Units: Different objects are often naturally grouped in different sizes. In an Indian perspective, we see this in traditional markets where items like bananas are counted in dozens (dastak), while grains might be measured by weight or volume units like paua or ser. Similarly, these islanders might have evolved names that inherently include the "unit" of the object.

4. Linguistic Evolution: Language often develops to be as descriptive as possible. Having specific number names for specific categories (like animate vs. inanimate objects) helps prevent confusion in communication within the community.

This practice reflects a holistic worldview where mathematics is not isolated from the environment but is deeply integrated into the daily life and spiritual beliefs of the people.

Given:

In the Gumulgal system:

$urapon = 1$

$ukasar = 2$

Rules for Arithmetic:

1. Addition: Combine all the terms from both numbers. If you have two $urapon$, replace them with one $ukasar$.

2. Subtraction: Remove the terms of the second number from the first number. If you need to subtract an $urapon$ but only have $ukasar$, decompose one $ukasar$ into two $urapon$.

3. Multiplication: Repeat the sequence of the first number as many times as there are units in the second number.

4. Division: See how many times the sequence of the divisor fits into the sequence of the dividend.

Solution (i): Addition

$(ukasar-ukasar-ukasar-ukasar-urapon) $$ + (ukasar-ukasar-ukasar-urapon)$

Combining them: $ukasar-ukasar-ukasar-ukasar $$-ukasar-ukasar-ukasar-urapon-urapon$

Result: $ukasar-ukasar-ukasar-ukasar-ukasar $$-ukasar $$-ukasar-ukasar$ (Total eight $ukasar$s)

Solution (ii): Subtraction

$(ukasar-ukasar-ukasar-ukasar-urapon) $$ - (ukasar-ukasar-ukasar)$

We remove three $ukasar$s from the first group.

Remaining: $ukasar-urapon$

Result: $ukasar-urapon$

Solution (iii): Multiplication

$(ukasar-ukasar-ukasar-ukasar-urapon) $$ \times (ukasar-ukasar)$

The second number is $ukasar-ukasar$, which is $4$ in decimal. We repeat the first sequence four times.

Total count of $ukasar$s: $4 \times 4 = 16$.

Total count of $urapon$s: $1 \times 4 = 4$ (which becomes $2$ $ukasar$s).

Result: $ukasar-ukasar-ukasar-ukasar-ukasar-ukasar $$-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar $$-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar$ (Total eighteen $ukasar$s)

Solution (iv): Division

$(ukasar-ukasar-ukasar-ukasar-ukasar-ukasar $$-ukasar-ukasar) \div (ukasar-ukasar)$

We group the eight $ukasar$s into sets of two $ukasar$s.

There are $4$ such sets. In this system, $4$ is $ukasar-ukasar$.

Result: $ukasar-ukasar$

Solution:

The Hindu-Arabic number system (which originated in India) is considered the most efficient system in the world due to the following features:

1. The Concept of Zero (Shunya): The inclusion of '0' as a placeholder and a value is the greatest contribution of Indian mathematicians like Aryabhata and Brahmagupta. It allows us to distinguish between numbers like $1, 10,$ and $100$ easily, whereas the Roman system has no symbol for zero.

2. Place Value System: In the Hindu system, the position of a digit determines its value (Units, Tens, Hundreds, etc.). In the Roman system, the value is mostly additive (e.g., $\text{III} = 1+1+1$) or subtractive ($\text{IV} = 5-1$), which makes representing large numbers extremely cumbersome.

3. Limited Symbols: We only need ten symbols ($0, 1, 2, 3, 4, 5, 6, 7, 8, 9$) to represent any number, no matter how large. The Roman system requires new symbols for larger values ($\text{I, V, X, L, C, D, M}$) and eventually runs out of standard symbols.

4. Ease of Calculation: Performing arithmetic operations like multiplication and long division is straightforward in a positional system. In the Roman system, even a simple multiplication like $\text{XVIII} \times \text{XXIV}$ is nearly impossible to do without converting it to another system first.

Comparison Table:

Earlier System (Simple Additive):

In the previous version, we used symbols: Lotus ($\clubsuit = 100$), Lamp ($\triangle = 10$), and Petal ($\mid = 1$). To write $111$, we had to write $\clubsuit \triangle \mid$.

Refined System (Positional with Zero):

Taking inspiration from the Hindu-Arabic system, I will refine the "Nature System" into a Positional Base-10 system. Instead of the symbols having fixed values, their value will now depend on their position.

1. New Symbols: We use 10 nature-based digits: $\text{Seed } (0), \text{Leaf } (1), \text{Bud } (2), \text{Flower } (3), \dots$ and so on.

2. Introduction of Shunya (Zero): The Seed ($\cdot$) will represent Zero. This is the most critical refinement.

3. Place Value Columns:

Example of Refinement:

To represent the number $102$ (e.g., $\textsf{₹} 102$ in an Indian shop):

Old System: $\clubsuit \mid \mid$ (Lotus + Petal + Petal)

Refined System: $\text{Leaf} \text{ Seed} \text{ Bud}$

Here, the Leaf is in the hundreds place, the Seed acts as a placeholder for zero in the tens place, and the Bud is in the units place.

Conclusion:

This refinement makes the system much more efficient. We no longer need to invent new symbols like "Sun" for 1000 or "Mountain" for 10,000. By using just 10 symbols and the concept of Shunya, we can represent infinite quantities, making it as powerful as the modern system used across India today.

Given:

The numbers to be converted into the Egyptian hieroglyphic system are $10458, 1023, 2660, 784, 1111,$ and $70707$.

To Find:

The Egyptian numeral representation for each of the given decimal numbers.

Solution:

The ancient Egyptian number system was an additive system with specific symbols for powers of $10$. In the Indian context, we can compare this to how we use distinct place values, although the Egyptians did not have a "zero" and simply repeated symbols.

(i) $10458$

Expansion: $10000 + 400 + 50 + 8$

Representation: 1 pointing finger, 4 coils, 5 heel bones, and 8 strokes.

(ii) $1023$

Expansion: $1000 + 20 + 3$

Representation: 1 lotus flower, 2 heel bones, and 3 strokes.

(iii) $2660$

Expansion: $2000 + 600 + 60$

Representation: 2 lotus flowers, 6 coils, and 6 heel bones.

(iv) $784$

Expansion: $700 + 80 + 4$

Representation: 7 coils, 8 heel bones, and 4 strokes.

(v) $1111$

Expansion: $1000 + 100 + 10 + 1$

Representation: 1 lotus flower, 1 coil, 1 heel bone, and 1 stroke.

(vi) $70707$

Expansion: $70000 + 700 + 7$

Representation: 7 pointing fingers, 7 coils, and 7 strokes.

To Find:

The decimal (Hindu-Arabic) equivalent of the given Egyptian numerals in images (i) and (ii).

Solution:

We use the Additive Principle where the total value is the sum of all individual symbols. This logic was also prevalent in ancient Indian trade before the decimal system was standardized.

Part (i):

By observing the symbols in image (i):

1. There are 2 spirals (coils of rope), each representing $100$. So, $2 \times 100 = 200$.

2. There are 7 heel bones, each representing $10$. So, $7 \times 10 = 70$.

3. There are 6 vertical strokes, each representing $1$. So, $6 \times 1 = 6$.

Total Value = $200 + 70 + 6 = 276$.

Therefore, the first numeral stands for $276$.

Part (ii):

By observing the symbols in image (ii):

1. There are 4 pointing fingers (hooks), each representing $10,000$. So, $4 \times 1,000 = 4,000$.

2. There are 3 spirals (coils of rope), each representing $100$. So, $3 \times 100 = 300$.

3. There are 2 heel bones, each representing $10$. So, $2 \times 10 = 20$.

4. There are 2 vertical strokes, each representing $1$. So, $2 \times 1 = 2$.

Total Value = $4,000 + 300 + 20 + 2 = 40,322$.

Therefore, the second numeral stands for $4,322$.

Given:

Numbers in the decimal system: $15, 50, 137, 293,$ and $651$.

System of conversion: Base-5 (Quinary System).

To Find:

The representation of these numbers in the base-$5$ system using the following symbolic mapping (as defined in our previous construction):

Solution:

To convert a decimal number to base-$5$, we divide the number by $5$ repeatedly and record the remainders. This is a method widely taught in Indian mathematics to understand the logic behind place values.

1. For $15$:

$15 = (3 \times 5) + 0$

Base-5 value: $(30)_5$. In symbols: $\triangle \circ$

2. For $50$:

$50 = (2 \times 5^2) + (0 \times 5^1) + (0 \times 5^0)$

Base-5 value: $(200)_5$. In symbols: $\wedge \circ \circ$

3. For $137$:

$137 = (1 \times 125) + (0 \times 25) + (2 \times 5) + 2$

Base-5 value: $(1022)_5$. In symbols: $\shortmid \circ \wedge \wedge$

4. For $293$:

$293 = (2 \times 125) + (1 \times 25) + (3 \times 5) + 3$

Base-5 value: $(2133)_5$. In symbols: $\wedge \shortmid \triangle \triangle$

5. For $651$:

$651 = (1 \times 625) + (0 \times 125) + (1 \times 25) + (0 \times 5) + 1$

Base-5 value: $(10101)_5$. In symbols: $\shortmid \circ \shortmid \circ \shortmid$

Solution:

No, there is no natural number that cannot be represented in the base-$5$ system.

Reasoning:

The base-$5$ system is a Positional Number System, much like the decimal system we use in India. Just as we can represent any number in base-$10$ using digits $0-9$, we can represent any number in base-$5$ using digits $0-4$.

1. Infinite Positions: We can always add a new place value to the left (powers of $5$: $5^0, 5^1, 5^2, \dots$) to represent larger and larger quantities.

2. Completeness: The digits $0, 1, 2, 3,$ and $4$ cover all possible remainders when any integer is divided by $5$. This ensures that every number has a unique and valid representation.

Indian Perspective:

Ancient Indian mathematicians like Pingala (who worked with binary/base-2) understood that any quantity could be expressed by varying the base and the position of symbols. This universality is what makes positional systems superior to additive ones like the Roman system.

To Find:

The landmark numbers for base-$7$ and the general formula for base-$n$.

Solution:

In any positional system, Landmark Numbers are those numbers that represent the start of a new place value (the powers of the base). In our common decimal system, the landmarks are $10, 100, 1000, \dots$.

1. Landmark Numbers for Base-$7$:

These are calculated by finding the powers of $7$:

$7^1 = 7$

$7^2 = 7 \times 7 = 49$

$7^3 = 49 \times 7 = 343$

$7^4 = 343 \times 7 = 2401$

The landmark numbers for base-$7$ are $7, 49, 343, 2401, \dots$

2. Landmark Numbers for Base-$n$:

In general, for any system with base $n$, the landmark numbers are the successive powers of that base:

Example Table:

Alternate Solution:

One can also think of landmark numbers as the values where the numeral representation "rolls over" to an extra digit. For example, in base-$7$, the number after $6$ is written as $10$, and the number after $66$ is $100$. These "round" numbers in their respective bases always correspond to the powers of $n$ in the decimal system.

Given:

Two sets of Egyptian numerals to be added in part (i) and part (ii).

To Find:

The sum of the Egyptian numerals and their equivalent decimal (Hindu-Arabic) values.

Solution (i):

First, we identify the values of the symbols provided in the first image:

1. Left Number: $1$ Pointing Finger ($10,000$) and $8$ Heel Bones ($8 \times 10 = 80$).

2. Right Number: $4$ Heel Bones ($4 \times 10 = 40$) and $6$ Strokes ($6 \times 1 = 6$).

Adding the decimal values:

$10,080 + 46 = 10,126$

In Egyptian numerals:

Combining the symbols, we get $1$ pointing finger, $12$ heel bones, and $6$ strokes. Since $10$ heel bones make $1$ coil ($\rho$), we carry over.

The final sum is: $1$ Pointing Finger, $1$ Coil, $2$ Heel Bones, and $6$ Strokes.

Solution (ii):

Identifying symbols from the second image:

1. Left Number: $9$ Pointing Fingers ($90,000$), $6$ Coils ($600$), and $8$ Strokes ($8$).

2. Right Number: $5$ Coils ($500$) and $7$ Strokes ($7$).

Adding the values:

$90,608 + 507 = 91,115$

In Egyptian numerals:

By combining, we have $9$ pointing fingers, $11$ coils, and $15$ strokes. Applying the carry-over rule ($10$ strokes = $1$ heel; $10$ coils = $1$ lotus flower):

The final sum is: $9$ Pointing Fingers, $1$ Lotus Flower, $1$ Coil, $1$ Heel Bone, and $5$ Strokes.

Indian Perspective:

In Indian arithmetic, this process is identical to the "carry-over" system used in addition. Just as we carry $1$ to the tens place when the sum of units exceeds $9$, Egyptians replaced $10$ identical symbols with $1$ symbol of the next higher landmark value.

Given:

Two base-$5$ numbers using symbols: Triangle ($\triangle = 1$), Square ($\square = 5$), Hexagon (Hex $= 25$), and Circle ($\bigcirc = 125$).

To Find:

The sum of the two symbolic numbers.

Solution:

We add the symbols from right to left (smallest to largest), similar to Indian place-value addition.

Step 1: Adding Triangles ($\triangle$)

Left side has $2$ $\triangle$, Right side has $2$ $\triangle$.

Total = $2 + 2 = 4$ $\triangle$. (No carry needed as $4 < 5$)

Step 2: Adding Squares ($\square$)

Left side has $1$ $\square$, Right side has $2$ $\square$.

Total = $1 + 2 = 3$ $\square$.

Step 3: Adding Hexagons (Hex)

Left side has $2$ Hex, Right side has $1$ Hex.

Total = $2 + 1 = 3$ Hexagons.

Step 4: Adding Circles ($\bigcirc$)

Left side has $1$ $\bigcirc$, Right side has $4$ $\bigcirc$.

Total = $1 + 4 = 5$ Circles.

Step 5: Applying the Base-$5$ rule

Since $5$ Circles = $1$ of the next landmark symbol (let us call it a "Star" or simply the next place value):

Final Result:

The sum is $1$ [Next Symbol], $0$ Circles, $3$ Hexagons, $3$ Squares, and $4$ Triangles.

In decimal values, this is:

$(1 \times 625) + (0 \times 125) + (3 \times 25) + (3 \times 5) + (4 \times 1) $$ = 625 + 0 + 75 + 15 + 4 = 719$.

Alternate Solution:

Convert both numbers to decimal first:

Number 1: $125 + 50 + 5 + 2 = 182$

Number 2: $500 + 25 + 10 + 2 = 537$

Sum = $182 + 537 = 719$.

Converting $719$ back to base-$5$ gives $10334_5$, which matches our symbolic result.

Given:

The structure of the ancient Egyptian numeral system, which uses symbols for powers of $10$.

Solution:

In the standard Egyptian numeral system, a symbol cannot occur $10$ or more times in a properly simplified number.

Reason:

The Egyptian system is based on the substitutional principle. Whenever a count of a particular symbol reaches $10$, those $10$ symbols are grouped together and replaced by a single symbol of the next higher power of $10$.

If we allowed a symbol to appear $10$ times, the number would not be in its simplest form. This is exactly like the Indian decimal system where we cannot have the digit '$10$' in a single place-value column; we must carry over '$1$' to the next column and leave a '$0$'.

Indian Perspective:

This is very similar to the concept of 'Haasil' (Carry) that Indian students learn in primary school arithmetic. Just as we carry over to the tens or hundreds place, the Egyptians carried over by changing the symbol entirely.

Construction:

We will create a Base-4 system using symbols inspired by the four directions (North, South, East, West), which are significant in Indian Vastu Shastra.

Solution:

In a base-$4$ system, the place values are $4^0=1, 4^1=4, 4^2=16,$ and so on. Below is the representation of numbers from $1$ to $16$:

Observation:

Notice that for the number $16$, we have reached the second landmark ($4^2$), which requires a three-digit string in our base-4 system.

To Find:

A simple rule for multiplication by the base value ($5$) in a base-$5$ system.

Solution:

In any positional number system, multiplying a number by its Base is extremely simple. In our base-$5$ system, the rule is:

"Shift all the symbols one place to the left and add the symbol for Zero ($\circ$) at the units place."

Explanation:

Consider a number in the decimal system (base-$10$). To multiply $12$ by $10$, we simply write $120$. We shifted the digits and added a zero.

Similarly, in our base-$5$ system, if we have the number $3$ (represented as $\triangle$):

Decimal: $3 \times 5 = 15$.

In base-5, $15$ is represented as $30_5$, which is $\triangle \circ$.

The symbol $\triangle$ moved from the $5^0$ place to the $5^1$ place, and a $\circ$ was added to the $5^0$ place.

Indian Perspective:

This is a fundamental property of Sthana-Mana (Place Value) developed by Indian mathematicians. Because the value of a position increases by a factor of the base as we move left, adding a 'zero' at the end automatically scales the entire number by the base value.

Given:

The numbers to be converted into the Mesopotamian (Babylonian) system are $63, 132, 200, 60,$ and $3605$.

To Find:

The symbolic representation of these numbers using the base-$60$ (sexagesimal) positional system.

Solution:

The Mesopotamian system uses a Base-60 structure. In the Indian perspective, this is very similar to how we measure time (hours, minutes, seconds) and angles in geometry, where $60$ units make one larger unit.

The two basic symbols used are:

(i) Representation of $63$:

We divide the number by $60$ to find the place values.

$63 = (1 \times 60) + 3$

This consists of $1$ unit in the sixty’s place and $3$ units in the one’s place.

Result: $\text{Y} \quad \text{YYY}$

(ii) Representation of $132$:

$132 = (2 \times 60) + 12$

This consists of $2$ units in the sixty’s place and $12$ (one ten and two units) in the one’s place.

Result: $\text{YY} \quad \text{

(iii) Representation of $200$:

$200 = (3 \times 60) + 20$

This consists of $3$ units in the sixty’s place and $20$ (two tens) in the one’s place.

Result: $\text{YYY} \quad \text{<<}$

(iv) Representation of $60$:

$60 = (1 \times 60) + 0$

This consists of $1$ unit in the sixty’s place and nothing in the unit's place. The Mesopotamians often used a space or a special placeholder to indicate the empty place.

Result: $\text{Y}$ (followed by a gap)

(v) Representation of $3605$:

Since $60^2 = 3600$, we use the third place value.

$3605 = (1 \times 3600) + (0 \times 60) + 5$

This consists of $1$ unit in the $3600$’s place, a gap for the $60$’s place, and $5$ units in the one’s place.

Result: $\text{Y} \quad \quad \text{YYYYY}$

Indian Perspective:

Ancient Indian Astronomers also used a sexagesimal system for precise calculations in Siddhantic Astronomy. While the symbols were different, the logic of using base $60$ allowed for very accurate tracking of planetary movements and time, which is still reflected in our Panchangs today.

Solution:

The Chinese rod numeral system was a decimal positional system. To distinguish between adjacent place values (like units, tens, hundreds), they used two different sets of symbols: Zong (vertical) and Heng (horizontal).

1. Reason for Alternation:

The alternation served as a visual boundary between digits. Since the ancient Chinese did not initially have a symbol for Shunya (Zero), they left a blank space. Alternating between vertical and horizontal rods helped the reader identify where one digit ended and the next began, preventing two separate digits from being read as a single grouped number.

2. Representing $41$ with only Zong symbols:

If we used only Zong (vertical) symbols:

The digit $4$ would be represented as: $||||$

The digit $1$ would be represented as: $|$

Without a clear separator, the number $41$ would appear as: $|||||$

3. Misinterpretation:

Yes, this numeral could be easily misinterpreted. In the Zong system, five vertical strokes ($|||||$) represent the number $5$. Therefore, without the alternating Heng symbols or significant spacing, $41$ would be indistinguishable from $5$. This could lead to massive errors in Indian-style calculations or commercial trade where accuracy is vital.

Indian Perspective:

This problem highlights why the Indian invention of '0' as a symbol was so revolutionary. While the Chinese used rod orientations to manage empty places, the Indian system used a specific character ($\cdot$ or $0$), which removed all ambiguity and allowed for the efficient arithmetic we use today across India.

Construction:

In a Base-2 (Binary) Place Value System, we only need two digits. Let us assign them as follows:

Digit $1$ = $urapon$

Digit $0$ = (a placeholder, let's use a blank or 'zero')

Solution:

To represent numbers in this refined system, the position of 'urapon' will determine its value based on powers of $2$ ($2^0, 2^1, 2^2, 2^3, \dots$).

Comparison with Original Gumulgal System:

The original Gumulgal system was primarily additive, whereas the new system is positional.

Conclusion:

The refined system is much more efficient. In the original system, the name for a large number like $32$ would be extremely long (sixteen $ukasar$s), but in the positional system, it would be a simple string of $6$ digits. This reflects the same shift India made from early symbolic counting to the sophisticated Decimal Place Value system which simplified global mathematics.

Solution:

In our daily lives in India, Hindu-Arabic numerals and the digit $0$ (Shunya) are indispensable. We use them for:

1. Financial Transactions: Every time we pay for groceries or check our balance in $\textsf{₹}$, we are using the decimal system. From a small Kirana store to the Reserve Bank of India, every ledger is based on these numerals.

2. Communication: Our mobile numbers, pin codes (like 110001), and vehicle registration plates are all composed of these digits.

3. Time and Measurement: Checking the time on a digital clock, measuring our height in centimeters, or weighing vegetables in kilograms requires these numbers.

Professional Importance:

Certain professions would simply cease to function without the Hindu numeral system:

1. Software Engineering and IT: The entire digital world runs on Binary Code (0 and 1). Without the concept of $0$, modern computing and the internet would not exist.

2. Engineering and Science: Agencies like ISRO calculate complex orbital trajectories using decimal mathematics. Without a placeholder system, large-scale calculations would be impossible.

3. Medicine: Monitoring heart rates, blood pressure levels, and calculating medicine dosages depends on the precision of our number system.

Life Without $0$ and Hindu Numerals:

If the Indian mathematicians like Aryabhata or Brahmagupta hadn't conceived of $0$ and the place-value system, our lives would be vastly different:

1. Mathematical Stagnation: We might still be using cumbersome systems like Roman numerals, where writing a large number like one crore ($\textsf{₹} 1,00,00,000$) would take up several lines of text.

2. No Industrial Revolution: Most scientific discoveries depend on calculus and higher mathematics, which are only possible due to the efficiency of the decimal system.

3. Manual Trade: Global trade would be extremely slow, as calculating interest, taxes, and exchange rates would be an arduous manual task prone to many errors.

Given:

A decimal (Base-10) number: $25$.

To Find:

Representation of $25$ in Base-8, Base-5, and Base-2.

Solution:

If we had only $8$ fingers, we would likely have developed an Octal System (Base-8). Instead of our digits $0$ to $9$, we would only use $0, 1, 2, 3, 4, 5, 6,$ and $7$.

1. Converting $25$ to Base-8:

We divide $25$ by $8$ and check the remainder:

So, $25_{10} = 31_8$.

2. Converting $25$ to Base-5:

We divide $25$ by $5$ repeatedly:

$\begin{array}{c|cc} 5 & 25 \\ \hline 5 & 5 & \text{rem } 0 \\ \hline & 1 & \text{rem } 0 \end{array}$

Reading the quotients and remainders from bottom to top:

So, $25_{10} = 100_5$.

3. Converting $25$ to Base-2 (Binary):

We use the successive division by 2 method:

$\begin{array}{c|cc} 2 & 25 \\ \hline 2 & 12 & \text{rem } 1 \\ \hline 2 & 6 & \text{rem } 0 \\ \hline 2 & 3 & \text{rem } 0 \\ \hline 2 & 1 & \text{rem } 1 \\ \hline & 0 & \text{rem } 1 \end{array}$

So, $25_{10} = 11001_2$.

Summary Table:

Indian Perspective:

The beauty of the Hindu-Arabic system is its adaptability. Whether the base is $10$, $8$, or $2$, the logic of place value (Sthana-mana) remain the same. This flexibility is why Indian mathematical concepts eventually replaced the rigid additive systems used in the Egyptian and Roman civilizations shown on the map.