NCERT Solutions for Class 8 Mathematics Ganita Prakash I Chapter 3
NCERT solutions for Class 8 Mathematics Chapter 3: A Story of Numbers – complete answers and explanations
Class 8 Mathematics Ganita Prakash Part 1, Chapter 3 – A Story of Numbers – is one of the most fascinating chapters in the entire Grade 8 curriculum. It takes students on a journey through thousands of years of human history, exploring how different civilisations across the world invented their own unique ways of counting and writing numbers. From the Gumulgal people of Australia counting in twos, to the powerful Roman numerals used in ancient Europe, to the Mayan, Egyptian, Mesopotamian, and Chinese number systems, and finally to the Hindu number system that the entire world now uses – this chapter covers it all. Understanding this chapter helps students appreciate why the Hindu number system, which originated in India, is so brilliantly efficient: it uses just ten symbols, a place value structure, and the revolutionary concept of zero to represent any number imaginable. Solving the questions in this chapter sharpens logical thinking, deepens number sense, and helps students see mathematics as a living, evolving human story. This blog provides clear and reliable NCERT solutions for every question and activity in Chapter 3, following the exact worksheet order, so students and parents can check answers with confidence. Download the worksheet and practice alongside these solutions for better clarity. Whether your child is stuck on Roman numeral conversions, base-n systems, or the Mesopotamian place value system, expert guidance is available – book a free trial now to get expert guidance.
What this NCERT chapter covers?
1. The chapter begins by exploring
the earliest human need for counting – tracking herds, food, trade, and the passage of time – and introduces three foundational methods of counting: using physical objects like sticks (one-to-one mapping), using a sequence of sounds or names, and using written symbols.
2. Students learn about the concept of a number system as a standard, ordered sequence of objects, names, or symbols, and why such a sequence needs to be unending to represent all numbers.
3. The chapter covers early number systems from world history: the Gumulgal (Australia), Bakairi (South America), and Bushmen (South Africa) – three geographically separate groups who independently developed nearly identical counting systems based on twos.
4. The Roman numeral system is studied in depth, including landmark numbers (I, V, X, L, C, D, M), conversion of numbers up to the thousands into Roman numerals, and how addition is performed in this system.
5. The Egyptian number system is introduced as a base-10 system with pictorial symbols for powers of 10 (stroke, arch, rope coil, lotus, bent finger, tadpole, astonished man), teaching students how to write and add numbers using this ancient system.
6. The concept of a base-n number system is formally defined: a system where the first landmark number is 1 and every next landmark number is a fixed multiple (n) of the previous one, making all landmark numbers powers of n.
7. A complete exploration of the base-5 system is provided, with students learning to represent numbers using five symbols (triangle, square, hexagon, circle, wave) and perform addition in this system.
8. The advantages of a base-n system over the Roman system are clearly established: multiplying landmark numbers becomes simple because the product of two powers of n is always another power of n.
9. The Mesopotamian (Babylonian) sexagesimal (base-60) number system is explored, including the idea of place value representation and the need for a placeholder symbol (like zero) to avoid ambiguity.
10. The Mayan number system (almost base-20), the Chinese rod numeral system (base-10), and finally the Hindu number system are studied, leading students to appreciate why the Hindu system – with its ten digits, place value, and the digit zero – is the most efficient system ever invented.
11. The chapter concludes with the five-step evolution of number representation: counting in groups, grouping by landmark numbers, choosing powers of a base, using positional/place value notation, and introducing 0 as both a positional digit and a number.
How to use these NCERT solutions?
1. Always attempt each question on
your own first, without looking at the answer. This chapter requires a lot of creative thinking, so try working through the logic yourself before checking.
2. Use these solutions to check your approach and answer after you have attempted each question. If your method is different but your answer matches, that is perfectly fine – this chapter often has multiple valid approaches.
3. All solutions in this blog follow the exact order of sections and questions as they appear in the worksheet – from Math Talk questions (Page 51) all the way to Figure it Out (Page 80). You will never have to hunt for an answer.
4. For questions marked as Math Talk or discussion activities, note that these are meant to be discussed in class. The explanations given here will help you understand the expected reasoning and form your own thoughts on the topic.
5. Parents can use this blog as a reference to help their child understand any concept they are stuck on. The solutions are written in simple, clear language that does not require any additional textbook knowledge to understand.
6. For open-ended activities like Try This or questions that ask students to create their own number system, students are encouraged to try their own version first and then refer to the example given in the solutions for guidance.
Important tips and tricks for students
1. When converting numbers to Roman numerals, always group from the largest landmark number downwards. Start with M (1000), then D (500), then C (100), then L (50), then X (10), then V (5), then I (1). Never skip the order.
2. A very common mistake in Roman numerals is forgetting to apply the subtraction rule: IX = 9, XL = 40, XC = 90, CM = 900. Always check if a subtraction form is needed after you have done the initial grouping.
3. In the Egyptian system, remember that each symbol can appear at most 9 times. If a symbol appears 10 or more times, group those 10 to form the next landmark symbol. This is the key rule for Egyptian addition.
4. In base-n systems, multiplying any number by the base shifts every symbol to the next higher landmark – exactly like adding a zero in the Hindu system. Use this rule whenever a multiplication-by-base question appears.
5. For the Gumulgal system, remember: ukasar = 2 and urapon = 1. When performing arithmetic in this system, convert to Hindu numerals first, do the calculation, and then convert back to the Gumulgal format.
6. For Figure it Out activities that say Try This or Math Talk, you are not expected to have one exact right answer. Show your reasoning clearly and make sure your system is consistent – any system with a fixed base and a clear pattern is acceptable.
7. The most important concept in this chapter for exams is the definition of a base-n number system and the place value system. Be able to explain: (a) what a landmark number is, (b) what makes a system base-n, and (c) why a place value system is more efficient than the Egyptian or Roman systems.
8. When comparing the Hindu and Roman systems, remember the five key advantages of the Hindu system: base-10 structure, place value, only 10 symbols needed, zero as a digit, and ease of arithmetic operations (especially multiplication and division).
NCERT solutions – complete answer key
THE MECHANISM OF COUNTING
Math Talk – Q1, Q2, Q3 (Page 51)
Explanation:
These questions are 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.
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:
- 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 ways are also valid.)
3. Explanation:
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: circle = 1, triangle = 2, square = 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 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 (Page 59) – Roman numerals
Represent the following numbers in the Roman system.
(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 = CCL
L × D = 25000 (Roman system has difficulty expressing this)
V × D = MMD
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
(Roman system makes multiplication extremely difficult without converting.)
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 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.
4. Explanation:
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
- Adding a placeholder symbol for zero
THE IDEA OF A BASE – Egyptian number system
Figure it Out (Page 62)
1. Represent the following numbers in the Egyptian system: 10458, 1023, 2660, 784, 1111, 70707.
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. What numbers do these numerals stand for?
(i) 99 / arch arch arch / arch arch arch / ||| ||| arch
= 2 × 100 + 3 × 10 + 3 × 10 + (6 × 1 + 10)
= 200 + 30 + 30 + 6 + 10
= 276
(ii) as per textbook visual:
4 × 10,000 + 3 × 1,000 + 3 × 100 + 1 × 10 + 2 × 1
Answer (ii): 43,312
Figure it Out (Page 63) – base-5 system
1. Write the following numbers in the base-5 system:
15:
15 = 5 × 3 = 3 × square + 0 × triangle
Answer: square square square
50:
50 = 25 × 2 = 2 × circle + 0 × square + 0 × triangle
Answer: circle circle
137:
137 = 125 + 5 × 2 + 1 × 2 = 1 × 125 + 0 × 25 + 2 × 5 + 2 × 1
Answer: bigcircle square square triangle triangle
293:
293 = 250 + 43 = 2 × 125 + 1 × 25 + 3 × 5 + 3 × 1
Answer: bigcircle bigcircle hexagon square square square triangle triangle triangle
651:
651 = 625 + 25 + 1 = 1 × 625 + 0 × 125 + 1 × 25 + 0 × 5 + 1 × 1
Answer: wave hexagon triangle
2. Is there a number that cannot be represented in our base-5 system above? Why or why not?
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. Compute the landmark numbers of a base-7 system. In general, what are the landmark numbers of a base-n 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 arch (that is, 10)?
(i) arch × arch = 10 × 10 = 100 (rope coil)
(ii) rope coil × arch = 100 × 10 = 1000 (lotus)
(iii) lotus × arch = 1000 × 10 = 10,000 (bent finger)
(iv) bent finger × arch = 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 rope coil (10^2 = 100)?
(i) arch × rope coil = 10 × 100 = 1000 (lotus)
(ii) rope coil × rope coil = 100 × 100 = 10,000 (bent finger)
(iii) lotus × rope coil = 1,000 × 100 = 100,000 (tadpole)
(iv) bent finger × rope coil = 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) arch × tadpole = 10 × 100,000 = 1,000,000 (astonished man)
(ii) rope coil × lotus = 100 × 1,000 = 100,000 (tadpole)
(iii) lotus × lotus = 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 arch (10)?"
Answer: Multiplying any number by 10 (arch) 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 arch × arch = 1010 × 10 = 10100
Simple rule to multiply a number by arch (10) in 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) – Egyptian addition
1. (i) Step 1: Count all strokes (|) from both numerals → total ones
Step 2: Count all arches (arch) → 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 approximately 43,623 and Number B approximately 9,237 (approximate, verify from image)
(ii) Method is the same as above — count all symbols from both numerals, combine, regroup when 10 of any symbol accumulate.
2. Add the following numerals in the base-5 system:
Step 1: Count each symbol type:
triangle (1s): 2 + 2 = 4 → write 4 triangles
square (5s): 1 + 2 = 3 → write 3 squares
hexagon (25s): 1 + 2 = 3 → write 3 hexagons
bigcircle (125s): 1 + 2 = 3 → write 3 bigcircles
wave (625s): 0 + 0 = 0
No regrouping needed (no symbol appears 5 times).
Answer: bigcircle bigcircle bigcircle hexagon hexagon hexagon square square square triangle triangle triangle triangle
Verification in Hindu numerals:
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 ✓
Figure it Out (Page 69)
1. Can there be a number whose representation in Egyptian numerals has one of the symbols occurring 10 or more times? Why not?
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. Create your own number system of base 4, and represent numbers from 1 to 16.
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. Give a simple rule to multiply a given number by 5 in the base-5 system that we created.
In the base-5 system, multiplying any number by 5 shifts each symbol one step up to the next landmark number:
triangle (1) × 5 = square (5)
square (5) × 5 = hexagon (25)
hexagon (25) × 5 = bigcircle (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 – Mesopotamian number system
"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]
(Each group of symbols represents the count for a successive power of 60.)
Example: 7530 = 2 × 3600 + 5 × 60 + 30 → written as [2] [5] [30]
Figure it Out (Page 73)
1. Represent the following numbers in the Mesopotamian system.
(i) 63
63 = 1 × 60 + 3 × 1
Answer: [1] [3] — one group of 60s, three 1s
(ii) 132
132 = 2 × 60 + 12 × 1
Answer: [2] [12] — two 60s, twelve 1s
(iii) 200
200 = 3 × 60 + 20
Answer: [3] [20] — three 60s, twenty 1s
(iv) 60
60 = 1 × 60 + 0 × 1
Answer: [1] [blank] — one 60, nothing in the 1s place
(v) 3605
3605 = 1 × 3600 + 0 × 60 + 5 × 1
Answer: [1] [blank] [5] — one 3600, skip 60s, five 1s
"Look at the representation of 60. What will be the representation for 3600?"
60 is written as: V (one symbol in the 60s position, blank in the 1s position)
3600 = 1 × 60²
It would be written as: V 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 V.
This is the ambiguity problem of the Mesopotamian system.
II. THE MAYAN NUMBER SYSTEM
"Represent the following numbers using the Mayan system"
(i) 77
77 = 3 × 20 + 17
17 = 3 bars + 2 dots (15 + 2)
3 at the 20s position = 3 dots
Mayan representation of 77:
Top (20s level): 3 dots (3 × 20 = 60)
Bottom (1s level): bar bar bar dot dot (3 bars and 2 dots = 17)
Answer (77):
[20s place]: 3 dots
[1s place]: 3 bars and 2 dots (= 17)
(ii) 100
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 symbol (= 0)
(iii) 361
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
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)
1.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?
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 (single group) instead of 41 (4 tens and 1 one). The alternating Zong/Heng system prevents this ambiguity.
2. Form a base-2 place value system using 'ukasar' and 'urapon' as the digits. Compare this system with that of the Gumulgal's.
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, ...
Some numbers written using positions (from right: 1s, 2s, 4s, 8s...):
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. 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 or conceived of?
Hindu numerals and 0 are used in virtually every area of modern life:
- Education: All mathematics, science, and calculations use Hindu numerals.
- 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. Also, all programming and software use Hindu numerals.
- Medicine: Dosages, measurements, patient data.
- Engineering: Blueprints, dimensions, calculations.
- Navigation and surveying: Coordinates and distances.
- 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. The ancient Indians likely used base 10 for the Hindu number system because humans have 10 fingers. But what if we had only 8 fingers? How would we be writing numbers then?
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
Why NCERT solutions help students?
Having access to accurate NCERT-aligned answers builds exam readiness and concept clarity from the ground up. Chapter 3 – A Story of Numbers is especially important because it asks students to think, explore, and reason rather than just memorise formulas. Students who understand the why behind each number system – not just the what – will find it much easier to tackle related questions in higher classes and competitive exams. Clear and reliable NCERT solutions allow students to self-assess honestly, catch mistakes early, and approach even the trickiest questions with confidence. For parents, these solutions provide peace of mind that their child is on the right track.
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