NCERT Solutions for Class 8 Maths Chapter 3 A Story of Numbers – Ganita Prakash Part 1

This worksheet provides complete and accurate NCERT Solutions for Class 8 Maths Chapter 3 A Story of Numbers from Ganita Prakash Part 1. This chapter takes students on a fascinating journey through the history and development of number systems across different civilisations. From tally marks and sticks to Roman numerals, Egyptian hieroglyphs, Mayan symbols, Mesopotamian place value, Chinese rod numerals, and finally the Hindu number system, students explore how humans invented ways to count and calculate. This chapter is important because it helps Class 8 students understand why our current number system is so powerful and efficient, building a strong conceptual foundation in mathematics.

Chapter summary: stories, poems & themes

Chapter 3 of Ganita Prakash Part 1 is an inquiry-based, activity-driven chapter. It does not contain stories or poems. Instead, it is built around the central idea of how different human civilisations developed number systems over thousands of years.

The chapter begins with the most basic act of counting — using sticks or tally marks — and guides students to think about how arithmetic operations like addition, subtraction, multiplication, and division can be done without numerals. It then explores early systems such as the Gumulgal system from Australia (using ukasar for 2 and urapon for 1), Roman numerals, the Egyptian hieroglyphic number system, the Mesopotamian base-60 place value system, the Mayan base-20 system, and the Chinese rod numeral system.

The main theme of the chapter is the evolution of number systems and the eventual emergence of the Hindu number system as the most efficient, flexible, and universally applicable system — thanks to its base-10 structure, place value, and the invention of zero.

What this NCERT chapter covers?

The learning focus of Class 8 Maths Chapter 3 A Story of Numbers includes the following areas:

• Understanding the basics of counting and how early humans tracked quantities without written numerals
• Performing arithmetic using physical methods such as sticks and tally marks
• Learning about early number systems — Gumulgal, Roman, Egyptian, Mesopotamian, Mayan, and Chinese
• Understanding the concept of a base in a number system and landmark numbers as powers of a base
• Exploring base-5, base-7, base-8, base-10, and base-2 number systems
• Understanding place value and how it makes large number representation efficient
• Recognising the role of zero as a placeholder in the Hindu number system
• Converting numbers between different number systems
• Comparing the strengths and limitations of various historical number systems
• Reflecting on why the Hindu number system became the global standard

How to use these NCERT solutions?

Students should first attempt every Math Talk, Figure it Out, Try This, and Inline question on their own before referring to the answers in this worksheet. Trying the problem independently builds reasoning skills and helps students understand their own thought process.

Parents and teachers can use this worksheet to verify answers, identify gaps in understanding, and guide students through the step-by-step working shown for each solution. Since all solutions follow the exact order of the NCERT Ganita Prakash Part 1 textbook, it is easy to match each answer to the corresponding page and question.

This worksheet is also very useful during revision, unit test preparation, and for clearing doubts about how different number systems work. The detailed explanations for activity-based questions help students understand not just the answer but the reasoning behind it.

Student tips & learning tricks

• For Roman numeral conversions, always break the number into its place value parts first (thousands, hundreds, tens, ones) before assigning Roman symbols. This avoids errors.
• Remember that in the Roman system, subtractive notation is used — IX means 9, XC means 90, CM means 900. Do not simply string symbols together without checking this rule.
• When working with the Egyptian system, remember that each symbol can appear at most 9 times. If a symbol appears 10 times, it must be replaced by the next higher landmark symbol.
• For the Gumulgal system, treat ukasar as 2 and urapon as 1, and build numbers additively by combining these words in sequence.
• When converting numbers to base-5 or base-7 systems, always identify the highest power of the base that fits into the number first, then work downwards.
• A very common mistake is forgetting the role of zero as a placeholder. In any place value system, zero is not just "nothing" — it holds a position and gives meaning to the digits around it.
• For Mayan numbers, remember that the system is base-20, dots represent ones (up to 4), bars represent 5, and the seashell symbol represents 0.
• In the Mesopotamian base-60 system, be careful about the ambiguity problem — without a placeholder, numbers like 60 and 3600 can look the same.

Why NCERT solutions are important?

NCERT-aligned solutions like these help Class 8 students build a solid conceptual understanding of mathematics rather than just memorising procedures. Chapter 3 is particularly important because it develops mathematical thinking — students learn to question, compare, and evaluate number systems rather than just compute answers.

These solutions are written strictly based on the NCERT textbook content and follow the same sequence and logic as the Ganita Prakash Part 1 chapter. This ensures that students are always learning within the NCERT framework, which is the standard for all school examinations.

By regularly practising with these solutions, students become more confident in handling unfamiliar types of questions, which is exactly the kind of thinking that Class 8 Maths and future competitive assessments demand. Strong foundational understanding of number systems also supports learning in computer science, where base-2 (binary) and other systems are widely used.

Complete answer key – NCERT solutions

Math Talk – Q1, Q2, Q3 (Page 51)

These are student-generated activity questions meant to be discussed in class. Students should think about how they would track a herd of cows, compare quantities, and find differences without using Hindu number names or written numerals. There are no fixed answers — any thoughtful, consistent method is acceptable.

Figure it Out (Page 54)

1. Addition using sticks: Place two collections of sticks side by side. The total number of sticks in the combined group gives the sum.
Subtraction using sticks: Remove as many sticks from one collection as there are in the second collection. The remaining sticks represent the difference.
Multiplication using sticks: Repeat one collection of sticks as many times as there are sticks in the second collection. The total sticks give the product.
Division using sticks: Repeatedly remove a group of sticks (equal to the divisor) from the dividend collection until no more complete groups can be removed. The number of times the group was removed is the quotient.

2. One way to extend Method 2 is to use single letters a–z for numbers 1–26. For 27 onwards, use two-letter combinations: aa = 27, ab = 28, ..., az = 52, ba = 53, and so on. For 703 onwards, use three-letter combinations: aaa = 703, etc. In general, using strings of length k with 26 letters each, we can represent 26^k numbers. Combining all lengths gives an unending system. (Many other valid systems are also acceptable.)

3. Student-generated activity. Students should try inventing their own symbols for numbers and define their own ordering. Any consistent system with a fixed sequence qualifies as a number system. For example, one could use shapes: ● = 1, ▲ = 2, ■ = 3, etc., and come up with a rule to represent larger numbers.

Some early number systems

Math Talk – "Can you see how their number names are formed?"
Already explained in the text: numbers in the Gumulgal system are formed by repeated use of ukasar (2) and urapon (1).

Math Talk – "Quickly count the number of objects in each of the following boxes"
Explanation: Students look at each box quickly and try to count without counting one by one. The expected observation is that most humans can instantly recognise groups of up to 4 objects, but find it difficult to count 5 or more at a single glance.

"What could be the difficulties with using a number system that counts only in groups of a single particular size?"
Explanation: If only groups of 5 are used, large numbers require writing very long sequences. For example, 1345 in a system counting only by 5s: 1345 = 269 groups of 5 = 53 groups of 25 = 10 groups of 125 + ... The representation becomes extremely long and cumbersome for large numbers.

Figure it Out – Roman Numerals (Page 54 onwards)

1. Convert to Roman numerals:
(i) 1222: 1222 = 1000 + 100 + 100 + 10 + 10 + 1 + 1. Answer: MCCXXII
(ii) 2999: 2999 = 1000 + 1000 + 900 + 90 + 9 = MM + CM + XC + IX. Answer: MMCMXCIX
(iii) 302: 302 = 300 + 2 = CCC + II. Answer: CCCII
(iv) 715: 715 = 500 + 100 + 100 + 10 + 5 = D + CC + X + V. Answer: DCCXV

Roman Numeral Exercises

Example (a): CCXXXII + CCCCXIII (Page 59). Answer: DCXLV

Do it yourself (b): LXXXVII + LXXVIII (Page 60)
LXXXVII = 50 + 10 + 10 + 10 + 5 + 1 + 1 = 87
LXXVIII = 50 + 10 + 10 + 5 + 1 + 1 + 1 = 78
Sum = 165. Grouping: 100 + 50 + 10 + 5. Answer: CLXV

Try This – Products of landmark numbers (Page 60):
V × L = 5 × 50 = 250. Answer: V × L = CCL
L × D = 50 × 500 = 25000. Answer: L × D = 25000 (The Roman system has difficulty expressing this beyond standard symbols.)
V × D = 5 × 500 = 2500. Answer: V × D = MMD
VII × IX = 7 × 9 = 63. Answer: VII × IX = LXIII

Daredevil Contest – Multiply CCXXXI and MDCCCLII (Page 60):
CCXXXI = 200 + 30 + 1 = 231
MDCCCLII = 1000 + 500 + 300 + 50 + 2 = 1852
231 × 1852 = 427,812. Answer (in Hindu numerals): 427812
(The Roman system makes multiplication extremely difficult without converting to Hindu numerals.)

Figure it Out (Pages 60–61)

1. A group of indigenous people in a Pacific island use different sequences of number names to count different objects because different objects may have different cultural or ritual significance. Separate counting sequences may have developed independently for different categories (e.g., people, animals, fish, ceremonial objects). Over time, these became fixed traditions within the community.

2. Extending the Gumulgal system beyond 6 by continuing the pattern of 2s:
7 = ukasar-ukasar-ukasar-urapon
8 = ukasar-ukasar-ukasar-ukasar
9 = ukasar-ukasar-ukasar-ukasar-urapon
10 = ukasar-ukasar-ukasar-ukasar-ukasar

Arithmetic in this system:
Addition: Combine both strings, then simplify (replace ukasar-urapon with the next ukasar step, following the 2+1 pattern).
Subtraction: Remove matching portions from the longer string.
Multiplication: Repeat one number's string as many times as the other number indicates.
Division: Repeatedly subtract the divisor string from the dividend string; count the number of times it fits.

(i) (ukasar-ukasar-ukasar-ukasar-urapon) + (ukasar-ukasar-ukasar-urapon) = 9 + 7 = 16 = ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar
(ii) (ukasar-ukasar-ukasar-ukasar-urapon) – (ukasar-ukasar-ukasar) = 9 – 6 = 3 = ukasar-urapon
(iii) (ukasar-ukasar-ukasar-ukasar-urapon) × (ukasar-ukasar) = 9 × 4 = 36 = (repeat ukasar 18 times, i.e., 18 pairs of ukasar)
(iv) (ukasar repeated 8 times) ÷ (ukasar-ukasar) = 16 ÷ 4 = 4 = ukasar-ukasar

3. Features of the Hindu number system that make it more efficient than Roman:
1. The Hindu system has a base (base-10), with landmark numbers as powers of 10 (1, 10, 100, 1000, ...). The Roman system uses irregular landmark numbers (I, V, X, L, C, D, M) that are not all powers of the same base.
2. The Hindu system is a place value system — the position of a digit determines its value. Roman numerals have no place value.
3. The Hindu system uses 0 as a digit, enabling unambiguous representation of any number with just 10 symbols. Roman numerals need new symbols for every new large number.
4. Arithmetic (especially multiplication and division) is straightforward in the Hindu system. It is extremely difficult in Roman numerals.
5. The Hindu system uses only 10 symbols to represent any number of any size. Roman numerals become impractical for very large numbers.

Figure it Out (Page 62) – Egyptian Number System

1. Writing numbers in the Egyptian system:
10458 = 1 × 10,000 + 4 × 100 + 5 × 10 + 8 × 1. Answer: [bent finger symbol] × 1, [rope coil] × 4, [arch] × 5, [stroke] × 8
1023 = 1 × 1000 + 2 × 10 + 3 × 1. Answer: [lotus] × 1, [arch] × 2, [stroke] × 3
2660 = 2 × 1000 + 6 × 100 + 6 × 10. Answer: [lotus] × 2, [rope coil] × 6, [arch] × 6
784 = 100 × 7 + 10 × 8 + 4. Answer: [rope coil] × 7, [arch] × 8, [stroke] × 4
1111 = 1000 + 100 + 10 + 1. Answer: [lotus] × 1, [rope coil] × 1, [arch] × 1, [stroke] × 1
70707 = 7 × 10,000 + 7 × 100 + 7 × 1. Answer: [bent finger] × 7, [rope coil] × 7, [stroke] × 7

2. Reading Egyptian numerals:
(i) 9 9 / ∩∩∩ / ∩∩∩ / ||| ||| ∩ = 2 × 100 + 3 × 10 + 3 × 10 + (6 × 1 + 10) = 200 + 30 + 30 + 6 + 10 = 276. Answer: 276
(ii) 4 × 10,000 + 3 × 1,000 + 3 × 100 + 1 × 10 + 2 × 1. Answer: 43,312

4. Student-generated activity. Using the ideas of base and landmark numbers from this section, students should revisit the number system they created in Q3 of the earlier Figure it Out (page 54) and try to improve it by: choosing a fixed base (e.g., base-5 or base-10), defining landmark numbers as powers of that base, introducing a place value or positional idea, and adding a placeholder symbol for zero.

Figure it Out (Page 63) – Base-5 System

1. Writing numbers in base-5 using symbols (△ = 1, □ = 5, ⬡ = 25, ○ = 125, ~ = 625):
15 = 3 × □ + 0 × △. Answer: □ □ □
50 = 2 × ⬡ + 0 × □ + 0 × △. Answer: ⬡ ⬡
137 = 1 × 125 + 0 × 25 + 2 × 5 + 2 × 1. Answer: ○ □ □ △ △
293 = 2 × 125 + 1 × 25 + 3 × 5 + 3 × 1. Answer: ○ ○ ⬡ □ □ □ △ △ △
651 = 1 × 625 + 0 × 125 + 1 × 25 + 0 × 5 + 1 × 1. Answer: ~ ⬡ △

2. No. Every positive integer can be represented in the base-5 system. This is because any number can be expressed as a sum of powers of 5 (with each power used at most 4 times before regrouping gives the next power). Since the landmark numbers (powers of 5) go on endlessly (5^0, 5^1, 5^2, 5^3, ...), every number, no matter how large, can always be represented by choosing enough higher powers.

3. Landmark numbers of a base-7 system:
7^0 = 1, 7^1 = 7, 7^2 = 49, 7^3 = 343, 7^4 = 2401, 7^5 = 16807 ... and so on.
In general, the landmark numbers of a base-n system are: n^0 = 1, n^1 = n, n^2, n^3, n^4, ... (all powers of n, starting from n^0 = 1).

Advantages of a base-n system

1. What is any landmark number multiplied by ∩ (that is, 10)?
(i) ∩ × ∩ = 10 × 10 = 100 = [rope coil symbol]
(ii) ? × ∩ = 100 × 10 = 1000 = [lotus]
(iii) [lotus with dot] × ∩ = 1000 × 10 = 10,000 = [bent finger]
(iv) [bent finger bent] × ∩ = 10,000 × 10 = 100,000 = [tadpole]
Answer: Multiplying any landmark number by 10 gives the next landmark number (the next power of 10).

2. What is any landmark number multiplied by ? (10^2 = 100)?
(i) ∩ × ? = 10 × 100 = 1000 = [lotus]
(ii) ? × ? = 100 × 100 = 10,000 = [bent finger]
(iii) [lotus with dot] × ? = 1000 × 100 = 100,000 = [tadpole]
(iv) [bent finger] × ? = 10,000 × 100 = 1,000,000 = [astonished man]
Answer: Multiplying any landmark number by 100 increases the power of 10 by 2, giving the landmark number two steps ahead.

Inline questions – Find the following products (Page 67):
(i) ∩ × [tadpole] = 10 × 100,000 = 1,000,000 = [astonished man]
(ii) ? × [lotus with dot] = 100 × 1,000 = 100,000 = [tadpole]
(iii) [lotus with dot] × [lotus with dot] = 1,000 × 1,000 = 1,000,000 = [astonished man]
(iv) [bent finger] × [sun] = 10,000 × 10,000,000 = 100,000,000 (beyond the seven standard Egyptian symbols — a new symbol would be needed)
Answer: The product of any two landmark numbers is another landmark number.

Math Talk – "Does this property hold true in the base-5 system?"
Yes. In any base-n system, all landmark numbers are powers of n. The product of two powers of n is also a power of n (since n^a × n^b = n^(a+b)). So the product of any two landmark numbers is always another landmark number. This holds for any number system with a base.

Inline – "What can we conclude about the product of a number and ∩ (10)?"
Answer: Multiplying any number by 10 (∩) in the Egyptian system shifts all its symbols to the next higher landmark number. That is, each landmark symbol in the number gets replaced by the next one — similar to appending a 0 at the end in the Hindu system.

Now find the following products (Page 68):
(i) Each symbol shifts one landmark up → answer is the original number × 10
(ii) [lotus] ∩ × ∩ = 1010 × 10 = 10100

Simple rule to multiply a number by ∩ (10) in the Egyptian system: Replace each symbol with the symbol for the next higher landmark number (i.e., shift each symbol one step up in the sequence). This is equivalent to appending a 0 in the Hindu number system.

Figure it Out (Page 65) – Adding Egyptian numerals

1. (i) Step 1: Count all strokes (|) from both numerals → total ones. Step 2: Count all arches (∩) → total tens. Step 3: Count all rope coils → total hundreds, etc. Step 4: Whenever a group of 10 same symbols appear, replace with 1 symbol of the next landmark. Step 5: Write the final simplified numeral.
Number A ≈ 43,623 and Number B ≈ 9,237 (verify from textbook image). (ii) Method is the same as above — count all symbols from both numerals, combine, regroup when 10 of any symbol accumulate.

2. Step 1: Count each symbol type:
△ (1s): 2 + 2 = 4 → write 4△
□ (5s): 1 + 2 = 3 → write 3□
⬡ (25s): 1 + 2 = 3 → write 3⬡
○ (125s): 1 + 2 = 3 → write 3○
~ (625s): 0 + 0 = 0
No regrouping needed (no symbol appears 5 times).
Answer: ○ ○ ○ ⬡ ⬡ ⬡ □ □ □ △ △ △ △
Verification: First number: 125 + 25 + 5 + 1 + 1 = 157. Second number: 125 + 125 + 25 + 25 + 5 + 5 + 1 + 1 = 312. Sum: 157 + 312 = 469. Answer in base-5: 3×125 + 3×25 + 3×5 + 4×1 = 375 + 75 + 15 + 4 = 469 ✓

Inline multiplication questions (Pages 66–68):
1. Products (Page 66):
(i) ∩ × ∩ = 10 × 10 = 100 (rope coil)
(ii) ? × ∩ = 100 × 10 = 1000 (lotus)
(iii) [lotus] × ∩ = 1000 × 10 = 10,000 (bent finger)
(iv) [bent finger] × ∩ = 10,000 × 10 = 100,000 (tadpole)

2. Products with 100 (Page 66):
(i) ∩ × ? = 10 × 100 = 1000 (lotus)
(ii) ? × ? = 100 × 100 = 10,000 (bent finger)
(iii) [lotus] × ? = 1,000 × 100 = 100,000 (tadpole)
(iv) [bent finger] × ? = 10,000 × 100 = 1,000,000 (astonished man)

Figure it Out (Page 69)

1. No. In the Egyptian system, a given number is represented by grouping into landmark numbers (powers of 10). If any symbol (say the arch ∩ = 10) appears 10 or more times, those 10 arches can be grouped together to form the next landmark number (one rope coil = 100). Similarly for all other symbols. So by definition of the representation method, each symbol can appear at most 9 times.

2. Student-generated activity. Using symbols: * = 1, # = 4, @ = 16. Landmark numbers: 4^0=1, 4^1=4, 4^2=16, 4^3=64 ...
1 = *, 2 = **, 3 = ***, 4 = #, 5 = # *, 6 = # **, 7 = # ***, 8 = # #, 9 = # # *, 10 = # # **, 11 = # # ***, 12 = # # #, 13 = # # # *, 14 = # # # **, 15 = # # # ***, 16 = @
(Students may use any symbols; the pattern above demonstrates the concept.)

3. In the base-5 system, multiplying any number by 5 shifts each symbol one step up to the next landmark number:
△ (1) × 5 = □ (5), □ (5) × 5 = ⬡ (25), ⬡ (25) × 5 = ○ (125), and so on.
Simple Rule: Replace each symbol in the number with the symbol for the next higher landmark number. This is equivalent to appending a '0' in the Hindu system, or multiplying by 10 in the Egyptian system.

Place value representation

"Can we represent this more compactly?"
Yes. Instead of writing the symbols for landmark numbers (60, 3600, etc.) explicitly, we can just write the count for each landmark in sequence from left to right. This is exactly the place value idea.
Example in textbook: 640 = 10 × 60 + 40 → written as [10] [40]
Example: 7530 = 2 × 3600 + 5 × 60 + 30 → written as [2] [5] [30]

Figure it Out (Page 73) – Mesopotamian base-60 system

1. Using the symbols: ᐯ = 1, ≺ = 10 (as used in the textbook). Landmark numbers: 1, 60, 3600 (60²), 216000 (60³) ...
(i) 63 = 1 × 60 + 3 × 1. Answer: [1] [3] — one group of 60s, three 1s = ᐯ | ᐯᐯᐯ
(ii) 132 = 2 × 60 + 12 × 1. Answer: [2] [12] — two 60s, twelve 1s = ᐯᐯ | ≺ᐯᐯ
(iii) 200 = 3 × 60 + 20. Answer: [3] [20] — three 60s, twenty 1s = ᐯᐯᐯ | ≺≺
(iv) 60 = 1 × 60 + 0 × 1. Answer: [1] [blank] — one 60, nothing in the 1s place = ᐯ (with a blank/space after it, or placeholder symbol)
(v) 3605 = 1 × 3600 + 0 × 60 + 5 × 1. Answer: [1] [blank] [5] — one 3600, skip 60s, five 1s = ᐯ [placeholder] ᐯᐯᐯᐯᐯ

"Look at the representation of 60. What will be the representation for 3600?"
60 is written as: ᐯ (one symbol in the 60s position, blank in the 1s position). 3600 = 1 × 60². It would be written as: ᐯ in the 3600s position, blank in the 60s position, blank in the 1s position. Without a placeholder symbol, 60 and 3600 look identical — both show just ᐯ. This is the ambiguity problem of the Mesopotamian system.

II. The Mayan number system

"Represent the following numbers using the Mayan system"
(i) 77 = 3 × 20 + 17. 17 = 3 × 5 + 2 = 3 bars + 2 dots
Answer (77): [20s place]: • • • (3 dots = 3 × 20 = 60). [1s place]: — — — • • (3 bars and 2 dots = 17)

(ii) 100 = 5 × 20 + 0. 5 at 20s position = 1 bar (5 × 20 = 100). 0 at 1s position = seashell symbol (0)
Answer (100): [20s place]: — (1 bar = 5 × 20 = 100). [1s place]: ⊘ (seashell = 0)

(iii) 361 = 1 × 360 + 0 × 20 + 1 × 1
Answer (361): [360s place]: • (1 dot = 1 × 360). [20s place]: ⊘ (seashell = 0). [1s place]: • (1 dot = 1)

(iv) 721 = 2 × 360 + 0 × 20 + 1 × 1 = 720 + 1
Answer (721): [360s place]: • • (2 dots = 2 × 360 = 720). [20s place]: ⊘ (seashell = 0). [1s place]: • (1 dot = 1)

III. The Chinese number system

"Where does the Hindu number system figure...?"
The Hindu number system is a base-10 (decimal) place value system. Its landmark numbers are: 1, 10, 10², 10³, 10⁴, ... (all powers of 10). Yes, it uses a place value system — each digit's position tells you which power of 10 it is associated with.

Figure it Out (Page 80) – Chinese rod numerals

1. The Chinese alternated between Zong (vertical rod) symbols and Heng (horizontal rod) symbols for alternate place values to avoid confusion between digits in adjacent positions. Without alternation, adjacent digits using the same type of symbols (e.g., two Zong symbols side by side) would be hard to distinguish as separate digits — especially if there was no clear boundary between them.
If only Zong symbols were used: 41 would be written as: 10s place: |||| (4 vertical rods), 1s place: | (1 vertical rod). This would look like ||||| (five rods in a row), which could be misread as 5. If there is no significant space, |||| | could be misread as 5 (a single group) instead of 41 (4 tens and 1 one). The alternating Zong/Heng system prevents this ambiguity.

2. In the Gumulgal system, ukasar = 2, urapon = 1. But those are number names, not digits. In a base-2 place value system: Let urapon = 0 and ukasar = 1 (as the two digits, representing 0 and 1). Landmark numbers (powers of 2): 1, 2, 4, 8, 16, 32, ...
1 = ukasar (binary: 1)
2 = ukasar urapon (binary: 10)
3 = ukasar ukasar (binary: 11)
4 = ukasar urapon urapon (binary: 100)
5 = ukasar urapon ukasar (binary: 101)
6 = ukasar ukasar urapon (binary: 110)
Comparison with Gumulgal's system: The Gumulgal system is NOT a place value system — it uses repeated words additively (ukasar-ukasar-urapon = 2+2+1 = 5). It also runs out at 6. The base-2 place value system can represent every number using only two symbols, and the position of each symbol gives it a different value. It is unending and far more efficient.

3. Hindu numerals and 0 are used in virtually every area of modern life: Education (all mathematics, science, and calculations), Commerce and banking (prices, account numbers, transactions, interest calculations), Science and technology (all measurements, formulas, data, and computations), Computing (computer science is built on binary/base-2, which itself is derived from the concept of place value invented in the Hindu system), Medicine (dosages, measurements, patient data), Engineering (blueprints, dimensions, calculations), Navigation and surveying (coordinates and distances), and Time (clocks, calendars, and schedules). Without 0 and the Hindu number system, representing large numbers would require countless new symbols, arithmetic would be very difficult, and modern science, technology, trade, and computing would be virtually impossible.

4. If humans had 8 fingers, we would likely use base-8 (octal). We would need 8 digit symbols: 0, 1, 2, 3, 4, 5, 6, 7. The landmark numbers would be powers of 8: 1, 8, 64, 512, ...
25 in base-8: 25 = 3 × 8 + 1 × 1 = 31 (base-8). Answer: 31₈
25 in base-5: 25 = 1 × 25 + 0 × 5 + 0 × 1 = 100 (base-5). Answer: 100₅
25 in base-2: 25 = 16 + 8 + 1 = 1 × 2⁴ + 1 × 2³ + 0 × 2² + 0 × 2¹ + 1 × 2⁰ = 11001 (base-2). Answer: 11001₂

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