Official NCERT 2026-27 Solutions
NCERT Solutions for Class 8 Maths Chapter 3: A Story of Numbers (Ganita Prakash)
Authentic handwritten solutions in Topper Notebook style. Covers every historical numeral system, base conversion, place-value exploration, and “Figure it Out” problem from Pages 69 to 80 of Ganita Prakash Part I.
Quick Chapter Navigation & Blueprint
- Part A: Ancient Landmark Systems (Pages 69–73) — Egyptian hieroglyphic rules, Base-4 positional design, and Mesopotamian sexagesimal (Base-60) representations.
- Part B: Mesoamerican & East Asian Systems (Pages 76–80) — Mayan vigesimal calendar tiers (Base 20/360) and Chinese Rod Numerals (Zong vs. Heng alternating patterns).
- Part C: Binary, Octal & Base Conversions (Page 80) — The Gumulgal counting system, Base-2 place-value tables, and converting 25 to Base-8, Base-5, and Base-2.
- Part D: Philosophical & Modern Impact of Zero — Role of Hindu-Arabic numerals in science, computing, trade, and daily life.
Official Textbook PDF Reference: Download NCERT Class 8 Maths Chapter 3 (hegp103.pdf)
Part A: Egyptian & Mesopotamian Systems (Pages 69–73)
Question 1 (Page 69–70):
Can there be a number whose representation in Egyptian numerals has one of the symbols occurring 10 or more times? Why not?
Can there be a number whose representation in Egyptian numerals has one of the symbols occurring 10 or more times? Why not?
Topper’s Historical & Mathematical Analysis:
- Structure of Egyptian Numerals: The Egyptian system was an additive decimal system with specific landmark symbols for powers of 10:
• Stroke (1) = 100
• Heel bone (10) = 101
• Coil of rope (100) = 102
• Lotus flower (1,000) = 103
• Pointing finger (10,000) = 104
• Tadpole/Frog (100,000) = 105
• Astonished man (1,000,000) = 106 - The Bundle Rule: Whenever 10 identical symbols accumulate, they are grouped (bundled) and replaced by 1 single symbol of the next landmark power of 10.
- For example, 10 strokes (||||||||||) are immediately replaced by 1 heel bone (∩). 10 heel bones are replaced by 1 coil of rope.
- Therefore, no single symbol ever needs to be written 10 or more times. The maximum count for any symbol in a properly written numeral is 9.
Final Answer: No. In the Egyptian numeral system, 10 copies of any landmark symbol are always bundled and replaced by a single symbol of the next higher landmark power of 10. Thus, any symbol occurs at most 9 times.
Question 2 (Page 70):
Create your own number system of base 4, and represent numbers from 1 to 16.
Create your own number system of base 4, and represent numbers from 1 to 16.
Designing a Base-4 Landmark System:
- In Base-4, the landmark powers are:
• 40 = 1 (Units): Let symbol be L (Corner stroke)
• 41 = 4 (Fours): Let symbol be Δ (Triangle)
• 42 = 16 (Sixteens): Let symbol be □ (Square) - Writing numbers from 1 to 16 by bundling at 4:
• 1 = L | 2 = LL | 3 = LLL
• 4 = Δ (bundle of four L’s becomes 1 Δ)
• 5 = Δ L | 6 = Δ LL | 7 = Δ LLL
• 8 = Δ Δ | 9 = Δ Δ L | 10 = Δ Δ LL | 11 = Δ Δ LLL
• 12 = Δ Δ Δ | 13 = Δ Δ Δ L | 14 = Δ Δ Δ LL | 15 = Δ Δ Δ LLL
• 16 = □ (bundle of four Δ’s becomes 1 □)
Base-4 Symbolic Scheme:
1 to 3: L, LL, LLL
4: Δ | 5 to 7: ΔL, ΔLL, ΔLLL
8 to 11: ΔΔ, ΔΔL, ΔΔLL, ΔΔLLL
12 to 15: ΔΔΔ, ΔΔΔL, ΔΔΔLL, ΔΔΔLLL
16: □
1 to 3: L, LL, LLL
4: Δ | 5 to 7: ΔL, ΔLL, ΔLLL
8 to 11: ΔΔ, ΔΔL, ΔΔLL, ΔΔLLL
12 to 15: ΔΔΔ, ΔΔΔL, ΔΔΔLL, ΔΔΔLLL
16: □
Question 3 (Page 73 – Mesopotamian Base-60 System):
Represent the following numbers in the Mesopotamian (Sexagesimal) system:
(i) 63 (ii) 132 (iii) 200 (iv) 60 (v) 3605
Represent the following numbers in the Mesopotamian (Sexagesimal) system:
(i) 63 (ii) 132 (iii) 200 (iv) 60 (v) 3605
Sexagesimal (Base-60) Place Value Decompositions:
- Mesopotamian numerals use base 60 with two primary wedge symbols:
• Vertical wedge (T) = 1
• Corner wedge (<) = 10
Positions from right to left correspond to: Units (600 = 1), Sixties (601 = 60), Thirty-Six Hundreds (602 = 3600). - (i) 63: 63 = 1 × 60 + 3 × 1 → Place 60: [1], Place 1: [3].
Symbols: [T] [TTT]. - (ii) 132: 132 = 2 × 60 + 12 × 1 → Place 60: [2], Place 1: [10 + 2].
Symbols: [TT] [< TT]. - (iii) 200: 200 = 3 × 60 + 20 × 1 → Place 60: [3], Place 1: [20].
Symbols: [TTT] [<<]. - (iv) 60: 60 = 1 × 60 + 0 × 1 → Place 60: [1], Place 1: [empty space / placeholder].
Symbols: [T] [space]. - (v) 3605: 3605 = 1 × 3600 + 0 × 60 + 5 × 1 → Place 3600: [1], Place 60: [0/space], Place 1: [5].
Symbols: [T] [space] [TTTTT].
Final Sexagesimal Expansions:
(i) 63 = (1 × 60) + 3
(ii) 132 = (2 × 60) + 12
(iii) 200 = (3 × 60) + 20
(iv) 60 = (1 × 60) + 0
(v) 3605 = (1 × 3600) + (0 × 60) + 5
(i) 63 = (1 × 60) + 3
(ii) 132 = (2 × 60) + 12
(iii) 200 = (3 × 60) + 20
(iv) 60 = (1 × 60) + 0
(v) 3605 = (1 × 3600) + (0 × 60) + 5
Part B: Mayan & Chinese Rod Numerals (Pages 76–80)
Question 4 (Page 76 – Mayan System):
Represent the following numbers using the Mayan vertical calendar system:
(i) 77 (ii) 100 (iii) 361 (iv) 721
Represent the following numbers using the Mayan vertical calendar system:
(i) 77 (ii) 100 (iii) 361 (iv) 721
Mayan Vertical Tier Decomposition:
- Mayan System Rules:
• Dot (•) = 1
• Horizontal Bar (—) = 5
• Shell symbol = 0
• Vertical positions: Bottom tier = 1s, Middle tier = 20s, Top tier = 360s (18 × 20, aligned with the solar calendar year). - (i) 77:
77 = (3 × 20) + (17 × 1)
• Upper tier (20s): 3 → three dots (•••)
• Lower tier (1s): 17 = 3 bars + 2 dots (•• over ———) - (ii) 100:
100 = (5 × 20) + (0 × 1)
• Upper tier (20s): 5 → one bar (—)
• Lower tier (1s): 0 → shell symbol - (iii) 361:
361 = (1 × 360) + (0 × 20) + (1 × 1)
• Top tier (360s): 1 → one dot (•)
• Middle tier (20s): 0 → shell symbol
• Bottom tier (1s): 1 → one dot (•) - (iv) 721:
721 = (2 × 360) + (0 × 20) + (1 × 1)
• Top tier (360s): 2 → two dots (••)
• Middle tier (20s): 0 → shell symbol
• Bottom tier (1s): 1 → one dot (•)
Mayan Tiers:
(i) 77 = (3) × 20 + (17) × 1
(ii) 100 = (5) × 20 + (0) × 1
(iii) 361 = (1) × 360 + (0) × 20 + (1) × 1
(iv) 721 = (2) × 360 + (0) × 20 + (1) × 1
(i) 77 = (3) × 20 + (17) × 1
(ii) 100 = (5) × 20 + (0) × 1
(iii) 361 = (1) × 360 + (0) × 20 + (1) × 1
(iv) 721 = (2) × 360 + (0) × 20 + (1) × 1
Question 5 (Page 80 – Chinese Rod Numerals):
Why do you think the Chinese alternated between the Zong and Heng symbols? If only the Zong symbols were to be used, how would 41 be represented? Could this numeral be interpreted in any other way if there is no significant space between two successive positions?
Why do you think the Chinese alternated between the Zong and Heng symbols? If only the Zong symbols were to be used, how would 41 be represented? Could this numeral be interpreted in any other way if there is no significant space between two successive positions?
Topper’s Analysis of Alternating Orientations:
- Zong (Vertical) vs Heng (Horizontal): In ancient China, counting rods were placed on boards. Zong rods were vertical, while Heng rods were horizontal.
- The Alternation Rule: Units, hundreds, and ten-thousands used Zong; tens and thousands used Heng.
- Preventing Confusion: If only Zong (vertical) rods were used:
• 4 is written as 4 vertical strokes: ||||
• 1 is written as 1 vertical stroke: |
Writing 41 would look like: |||| | (5 vertical strokes). - Without a clear blank space or zero symbol, 5 adjacent vertical strokes could easily be misread as:
• 5 (|||||)
• 23 (|| followed by |||)
• 32 (||| followed by ||)
• 14 (| followed by ||||)
• 122 (| followed by || followed by ||) - By strictly alternating vertical and horizontal rods, the reader instantly identifies where each decimal place starts and ends!
Final Answer: The Chinese alternated between Zong and Heng rods to clearly demarcate adjacent place values and prevent ambiguous readings. If only Zong were used, 41 (|||| |) could be confused with 5, 23, 32, 14, or 122.
Part C: Modern Base Conversions & Binary System (Page 80)
Question 6 (Page 80 – Base-2 System & Gumulgal Language):
Form a base-2 place value system using ‘ukasar’ (1) and ‘urapon’ (0) as digits. Compare this system with the indigenous Gumulgal counting system.
Form a base-2 place value system using ‘ukasar’ (1) and ‘urapon’ (0) as digits. Compare this system with the indigenous Gumulgal counting system.
Binary Place Value vs. Tally Addition:
- Let digit 0 = urapon (ur) and digit 1 = ukasar (uk).
- In a Base-2 Place Value System, digits multiply powers of 2 (20 = 1, 21 = 2, 22 = 4, 23 = 8):
• 1 = 1 × 20 = uk
• 2 = (1 × 21) + (0 × 20) = uk ur
• 3 = (1 × 21) + (1 × 20) = uk uk
• 4 = (1 × 22) + (0 × 21) + (0 × 20) = uk ur ur
• 5 = (1 × 22) + (0 × 21) + (1 × 20) = uk ur uk
• 6 = (1 × 22) + (1 × 21) + (0 × 20) = uk uk ur
• 7 = (1 × 22) + (1 × 21) + (1 × 20) = uk uk uk
• 8 = (1 × 23) + (0 × 22) + (0 × 21) + (0 × 20) = uk ur ur ur - Comparison with Gumulgal System:
The traditional Gumulgal system was additive (tally-like), simply repeating “ukasar-ukasar” for 4, and “ukasar-ukasar-urapon” for 5.
In contrast, the Base-2 place-value system uses positional exponentiation, allowing compact representation of arbitrarily large numbers!
Key Difference: The true Base-2 positional system uses place values (powers of 2) with a zero placeholder (urapon), growing logarithmically, whereas the Gumulgal tally system was purely additive and grew linearly.
Question 7 (Page 80 – Converting Number 25 across Bases):
Convert the Base-10 number 25 into:
(a) Base-8 (Octal)
(b) Base-5 (Quinary)
(c) Base-2 (Binary)
Convert the Base-10 number 25 into:
(a) Base-8 (Octal)
(b) Base-5 (Quinary)
(c) Base-2 (Binary)
Topper’s Step-by-Step Successive Division Method:
- (a) Base-8 Conversion:
25 ÷ 8 = 3, with remainder 1 (units place).
3 ÷ 8 = 0, with remainder 3 (eights place).
Verification: (3 × 81) + (1 × 80) = 24 + 1 = 25.
→ Base-8 representation: 318. - (b) Base-5 Conversion:
25 ÷ 5 = 5, remainder 0 (units place).
5 ÷ 5 = 1, remainder 0 (fives place).
1 ÷ 5 = 0, remainder 1 (twenty-fives place).
Verification: (1 × 52) + (0 × 51) + (0 × 50) = 25 + 0 + 0 = 25.
→ Base-5 representation: 1005. - (c) Base-2 Conversion:
25 ÷ 2 = 12, remainder 1
12 ÷ 2 = 6, remainder 0
6 ÷ 2 = 3, remainder 0
3 ÷ 2 = 1, remainder 1
1 ÷ 2 = 0, remainder 1
Reading remainders from bottom to top: 1, 1, 0, 0, 1.
Verification: 24 + 23 + 20 = 16 + 8 + 1 = 25.
→ Base-2 representation: 110012.
Final Representations of 25:
• In Base-8: 318
• In Base-5: 1005
• In Base-2: 110012
• In Base-8: 318
• In Base-5: 1005
• In Base-2: 110012
Question 8 (Page 80 – Role of Zero & Hindu Numerals):
Where in your daily lives, and in which professions, do the Hindu numerals and ‘0’ play an important role? How might our lives have been different if our number system and ‘0’ hadn’t been invented?
Where in your daily lives, and in which professions, do the Hindu numerals and ‘0’ play an important role? How might our lives have been different if our number system and ‘0’ hadn’t been invented?
Topper’s Real-World Reflection:
- Essential Everyday Roles:
• Digital Technology & Computing: All modern computers, smartphones, and the internet operate on binary (0 and 1). Without the concept of 0 as a placeholder and state, modern computing would not exist.
• Banking & Commerce: Account balances, transactions, credit cards, and digital currencies rely on decimal place value.
• Science & Engineering: Coordinates, temperatures (positive and negative scales passing through zero), calculus, and space navigation all demand zero and positional notation. - Life Without Zero:
Calculations would be excruciatingly slow, requiring physical abacuses or cumbersome Roman/Greek numeral additions (e.g., trying to multiply MDCCCLXXXVIII by CCXLIV). Scientific revolutions in astronomy, physics, and medicine would have been delayed by centuries.
Conclusion: The invention of the digit 0 and the decimal place-value system by ancient Indian mathematicians (Aryabhata, Brahmagupta, Bhaskara) is widely regarded as one of humanity’s greatest intellectual breakthroughs, forming the bedrock of modern civilization and computing.