NCERT Solutions Class 8 Maths Chapter 03 A Story of Numbers

Page 54

Question 1. Suppose you are using the number system that uses sticks to represent numbers, as in Method 1. Without using either the number names or the numerals of the Hindu number system, give a method for adding, subtracting, multiplying and dividing two numbers or two collections of sticks.
Answer:
Method 1: Addition (Putting Together)
►Gather sticks that stand for the first amount.
►Gather another group of sticks that stand for the second amount.
►Put both groups into one larger group.
►The total sticks in the larger group show the sum.

Method 2: Subtraction (Taking Away)
►Begin with the group of sticks that stands for the bigger amount.
►Remove or take away the sticks that stand for the smaller amount.
►The sticks left over show the result of subtraction.

Method 3: Multiplication (Repeated Addition)
►Form several groups of sticks, where each group has the same number.
►Add all the sticks from every group together.
►The total sticks show the product.

Method 4: Division (Equal Sharing or Grouping)
►Take the total sticks.
►Divide them into equal groups.
Either:
• Find how many sticks fit in each group (equal sharing), or
• Find how many such groups you can form (repeated subtraction).
In simple words: You can add by putting groups together, subtract by taking some away, multiply by making repeated groups, and divide by splitting into equal parts.

Exam Tip: When explaining stick-based methods, always show clear steps and give a concrete example with actual stick counts to demonstrate your understanding.

Question 2. One way of extending the number system in Method 2 is by using strings with more than one letter - for example, we could use 'aa' for 27. How can you extend this system to represent all the numbers? There are many ways of doing it!
Answer: Think of it as a base-26 system using letters. Each letter works like a digit and we treat letter sequences as base-26 numbers, where: 'a' = 1, 'b' = 2, …, 'z' = 26. After 'z', we move to: 'aa' = 27, 'ab' = 28 …, 'az' = 52, 'ba' = 53 …, 'zz' = 26×26 = 676, 'aaa' = 677, and continuing in this pattern.

Alternate methods:
►Repeating letters: a = 1, b = 2, …, z = 26; aa = 27, bb = 28, … but this method gets used up fast and isn't reliable.
►Insert separators: After 26, start using combinations like 'a-a', 'a-b', 'a-c', etc. But this is less effective and harder to grow.
►Numeric suffix: a1 = 1, a2 = 2, …, z1 = 26, a1a = 27, etc. More complex but able to be changed.
In simple words: Treat letter patterns like numbers in base-26. Single letters stand for 1-26, then double letters stand for 27 and higher.

Exam Tip: The base-26 method is most efficient because it scales to any size number and stays consistent, unlike the repeating letter method which runs out quickly.

Question 3. Try making your own number system.
Answer: I created a system called the "ABC Number System". In this system, I use the letters A, B, C, D and E to stand for normal digits. Each letter means a number: A = 0, B = 1, C = 2, D = 3 and E = 4. This lets me count using only these five letters, just like we normally count with digits 0 to 9.

I also follow place value - the rightmost letter has worth 1s, then 5s, then 25s, and so on (because this is a base-5 system). For example, the code BD means B = 1 (in the 5s place) and D = 3 (in the 1s place). So BD = (1 × 5) + 3 = 8. This system is fun and feels like a secret code! I can count and do Maths using only letters, which helps me see how numbers can be shown in many different ways.
In simple words: Use five symbols (A-E for 0-4) to count in base-5, where each spot means powers of 5, just like our base-10 system means powers of 10.

Exam Tip: A good custom number system must include place value and a clear set of symbols for digits 0 to (base-1), otherwise it cannot represent large numbers well.

Page 59

Question 1. Represent the following numbers in the Roman system. (i) 1222
Answer: Split into parts: 1000 + 200 + 20 + 2 = M + CC + XX + II = MCCXXII
In simple words: Break the number into parts, turn each part into Roman symbols, then put them together.

Exam Tip: Always break large numbers into place values first (thousands, hundreds, tens, ones) and convert each part separately.

Question 1. (ii) 2999
Answer: Split into parts: 1000 + 1000 + 900 + 90 + 9 = M + M + CM + XC + IX = MMCMXCIX
In simple words: When a smaller symbol goes before a larger one, subtract it. So CM means 1000 - 100 = 900, and IX means 10 - 1 = 9.

Exam Tip: Watch for subtractive notation - whenever a smaller Roman numeral appears before a larger one, you must subtract, not add.

Question 1. (iii) 302
Answer: Split into parts: 300 + 2 = CCC + II = CCCII
In simple words: Write C three times for 300, then II for 2.

Exam Tip: Remember that a symbol can be repeated at most three times in a row - never write four identical symbols together.

Question 1. (iv) 715
Answer: Split into parts: 700 + 10 + 5 = DCC + X + V = DCCXV
In simple words: D (500) plus CC (200) gives 700, then X is 10, and V is 5.

Exam Tip: For hundreds, remember: C = 100, CC = 200, CCC = 300, CD = 400, D = 500, DC = 600, DCC = 700, DCCC = 800, CM = 900.

Page 60

Question 1. A group of indigenous people in a Pacific island use different sequences of number names to count different objects. Why do you think they do this?
Answer: Indigenous people on a Pacific island might use different sequences of number names for different objects because their language and culture connect deeply to daily life and nature. They may count coconuts, fish, people or days differently because each thing matters in a different way and may follow different customs.

For example, they might use one type of number word for things that are alive and another for things that are not alive, or they may count pairs of items (like eyes or shoes) instead of single things. Using different number systems helps them learn, sort and recall things more easily in their own way.
In simple words: Different objects matter in different ways in their life, so they created different counting words to match each type of thing.

Exam Tip: When asked why a culture uses a specific counting method, always link the answer to their daily activities, environment, and cultural values rather than describing only the method itself.

Question 2. Consider the extension of the Gumulgal number system beyond 6 in the same way of counting by 2s. Come up with ways of performing the different arithmetic operations (+, -, ×, ÷) for numbers occurring in this system, without using Hindu numerals. Use this to evaluate the following: (i) (ukasar-ukasar-ukasar-ukasar-urapon) + (ukasar-ukasarukasar-urapon) (ii) (ukasar-ukasar-ukasar-ukasar-urapon) - (ukasar-ukasarukasar) (iii) (ukasarukasar-ukasar-ukasar-urapon) × (ukasar-ukasar) (iv) (ukasar-ukasar-ukasar-ukasarukasar-ukasar-ukasar-ukasar) ÷ (ukasar-ukasar)
Answer:
Understanding the Gumulgal number system, which counts in groups of 2 using the following number names:
urapon = 1
ukasar = 2
ukasar-urapon = 3 (2 + 1)
ukasar-ukasar = 4 (2 + 2)
ukasar-ukasar-urapon = 5 (2 + 2 + 1)
ukasar-ukasar-ukasar = 6 (2 + 2 + 2) and so on

►Converting Gumulgal terms to Hindu numerals:
(ukasar-ukasar-ukasar-ukasar-urapon) = 2 + 2 + 2 + 2 + 1 = 9
(ukasar-ukasar-ukasar-urapon) = 2 + 2 + 2 + 1 = 7
(ukasar-ukasar) = 2 + 2 = 4
(ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar) = 8 ukasar = 8 × 2 = 16

►Performing Arithmetic Operations:
(i) Addition: 9 + 7 = 16 → Convert 16 back into Gumulgal style: 8 ukasar = ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar

(ii) Subtraction: 9 - 6 = 3 → 2 + 1 = ukasar-urapon

(iii) Multiplication: 9 × 4 = 36 → Break 36 as 2 + 2 + 2 + ... 18 times = 18 ukasar → ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar

(iv) Division: 16 ÷ 4 = 4 → 2 + 2 = ukasar-ukasar
In simple words: Convert the Gumulgal words to regular numbers, do the maths, then change the answer back into Gumulgal words by building it from 2s and 1s.

Exam Tip: When working with non-standard number systems, always convert to a familiar system first (like base-10), perform the operation, then convert back - this reduces errors.

Question 3. Identify the features of the Hindu number system that make it efficient when compared to the Roman number system.
Answer: The Hindu number system has several features that make it work better than the Roman system. It uses a place value system where the spot of a digit tells you its worth. It includes zero (0) as both a digit and a placeholder. It needs only 10 symbols (0-9) to write any number. It allows for quick and simple arithmetic work (adding, subtracting, etc.). It gives an unambiguous and short way to show even very big numbers. It forms the ground for modern maths and science. It is recognised and used around the world in all fields today.
In simple words: The Hindu system uses place value and the digit zero, which makes large numbers easy to write and maths easy to do.

Exam Tip: Always mention "place value" and "zero" as the two most critical features that distinguish the Hindu system from older additive systems like Egyptian and Roman numerals.

Question 4. Using the ideas discussed in this section, try refining the number system you might have made earlier.
Answer: After learning from this chapter, I refined my number system in the following ways:
►I turned my number system into a base-5 system.
►It uses five symbols: A, B, C, D, E (where A = 0, B = 1, …, E = 4).
►Each spot from right to left shows powers of 5 (1, 5, 25, 125…).
►I added place value, so the same symbol has different worth based on where it sits.
►Including A as zero stops confusion and lets me write big numbers with ease.
►This system is now more compact, clear and good for doing maths, just like the Hindu number system.
In simple words: I added place value and zero to my system, so now it works like the Hindu system but in base-5 instead of base-10.

Exam Tip: A complete number system must have: (1) a set of symbols, (2) place value, (3) a zero symbol, and (4) a clear base - all of these together make a system efficient.

Page 62

Question 1. Represent the following numbers in the Egyptian system: 10458, 1023, 2660, 784, 1111, 70707.
Answer: In the Egyptian system, each symbol stands for a fixed value and symbols are repeated to build up the total. The key symbols are: 𓏺 (1), ∩ (10), 𓍢 (100), 𓎆 (1000), and higher powers as needed. Each number is shown by adding up the counts of each symbol type. For example, 10458 is shown as nine lotus flowers (9000), four coiled ropes (400), five arches (50), and eight single marks (8), all combined together in Egyptian notation.
In simple words: Write each place value as a group of Egyptian symbols - ones as marks, tens as arches, hundreds as coiled ropes, thousands as lotus flowers.

Exam Tip: In Egyptian notation, order doesn't matter - you can arrange symbols in any way. Always count how many of each symbol type you need before writing them down.

Question 2. What numbers do these numerals stand for? (i) Two spirals, one arch symbol, six ones arranged as shown
Answer: The numerals stand for: 100 + 100 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 1 + 1 + 1 + 1 + 1 + 1 = 200 + 70 + 6 = 276
In simple words: Add all the symbols: two hundreds, seven tens, and six ones give you 276.

Exam Tip: Simply add up the value of each symbol, counting how many times each one appears in the numeral.

Question 2. (ii) Four lotus flowers, three spirals, one arch, two ones arranged as shown
Answer: The numerals stand for: 1000 + 1000 + 1000 + 1000 + 100 + 100 + 100 + 10 + 10 + 1 + 1 = 4000 + 300 + 20 + 2 = 4322
In simple words: Add up the totals: four thousands, three hundreds, two tens, and two ones give 4322.

Exam Tip: Count the quantity of each symbol type separately, then add all the subtotals together to find the final number.

Page 63

Question 1. Write the following numbers in the above base-5 system using the symbols in Table 2: 15, 50, 137, 293, 651.
Answer: In base-5 using the symbols (circles, hexagons, squares, triangles):

15 in base-5: Break 15 = 1×(5²) + 3×(5¹) + 0×(5⁰) - but let me recalculate: 15 ÷ 5 = 3 remainder 0, and 3 ÷ 5 = 0 remainder 3. So 15 = 30 in base-5 (represented as 3 hexagons and 1 circle).

50 in base-5: 50 ÷ 5 = 10 remainder 0, 10 ÷ 5 = 2 remainder 0, 2 ÷ 5 = 0 remainder 2. So 50 = 200 in base-5 (represented as 2 circles, with 0 hexagons, with 0 single circles).

137 in base-5: 137 ÷ 5 = 27 remainder 2, 27 ÷ 5 = 5 remainder 2, 5 ÷ 5 = 1 remainder 0, 1 ÷ 5 = 0 remainder 1. So 137 = 1022 in base-5 (represented as 1 large circle, 0 hexagons, 2 squares, 2 triangles).

293 in base-5: 293 ÷ 5 = 58 remainder 3, 58 ÷ 5 = 11 remainder 3, 11 ÷ 5 = 2 remainder 1, 2 ÷ 5 = 0 remainder 2. So 293 = 2133 in base-5 (represented as 2 large circles, 1 hexagon, 3 squares, 3 triangles).

651 in base-5: 651 ÷ 5 = 130 remainder 1, 130 ÷ 5 = 26 remainder 0, 26 ÷ 5 = 5 remainder 1, 5 ÷ 5 = 1 remainder 0, 1 ÷ 5 = 0 remainder 1. So 651 = 10101 in base-5 (represented as 1 very large circle, 0 large circles, 1 hexagon, 0 squares, 1 triangle).
In simple words: Divide the number by 5 over and over, keep the remainders, then read them backwards to get the base-5 form.

Exam Tip: Always use the division-by-base method to convert from base-10 to any other base - it's the most reliable approach.

Question 2. Is there a number that cannot be represented in our base-5 system above? Why or why not?
Answer: No, there is no number that cannot be shown in our base-5 system. Here's why: A base-5 system uses digits A, B, C, D, E (which stand for 0 to 4). Any number, no matter how big, can be written using mixes of these symbols and place values based on powers of 5 (1, 5, 25, 125…). Since there is no upper limit on how many spots we can use, we can show every whole number.
In simple words: Base-5 can represent any whole number because we can always add more digit spots if we need to write something bigger.

Exam Tip: Any complete place value system with a base n (where n ≥ 2) can represent all non-negative whole numbers - there is no theoretical upper limit.

Question 3. Compute the landmark numbers of a base-7 system. In general, what are the landmark numbers of a base-n system?
Answer: In a base-7 system, the landmark numbers are powers of 7:
7⁰ = 1
7¹ = 7
7² = 49
7³ = 343
7⁴ = 2401
7⁵ = 16807
…and so on.

In general, the landmark numbers of a base-n number system are the powers of n starting from n⁰ = 1, n¹, n², n³,…
In simple words: Landmark numbers are the place values in a number system - in base-10 they are 1, 10, 100, 1000; in base-7 they are 1, 7, 49, 343, and so on.

Exam Tip: The landmark numbers tell you the worth of each digit position - the rightmost position is always n⁰ = 1, and each position to the left multiplies by the base.

Page 65

Question 1. Add the following Egyptian numerals: (i) Three lotus flowers, three spirals, three arches (first number) plus two spirals, four arches, three ones (second number)
Answer: First number = 3000 + 300 + 30 = 3330
Second number = 200 + 40 + 3 = 243
Sum = 3330 + 243 = 3573

In Egyptian notation, the sum is shown as: three lotus flowers, five spirals, seven arches, and three ones.
In simple words: Add up the values of each group, then write the total using Egyptian symbols in the usual order.

Exam Tip: When adding Egyptian numerals, combine like symbols - pool all the ones, all the tens, all the hundreds, etc., then group any excess tens into higher symbols if needed.

Question 1. (ii) One lotus flower, three arches, two ones (first number) plus three arches, four ones (second number)
Answer: First number = 1000 + 30 + 2 = 1032
Second number = 30 + 4 = 34
Sum = 1032 + 34 = 1066

In Egyptian notation, the sum is shown as: one lotus flower, six arches, and six ones.
In simple words: Group all the same symbols together and count them - one thousand, six tens, and six ones.

Exam Tip: The Egyptian system makes addition simple - just collect all symbols of the same type and regroup if you have 10 or more of one kind.

Question 2. Add the following numerals that are in the base-5 system that we created: Circle, hexagon, hexagon, square, triangle, triangle (first number) plus circle, circle, circle, hexagon, square, square, triangle, triangle (second number)
Answer: First number in base-5 = 12322 (1 circle, 2 hexagons, 3 squares, 2 triangles)
Second number in base-5 = 31322 (3 circles, 1 hexagon, 3 squares, 2 triangles)

Converting to base-10 for addition:
First = 1×(5⁴) + 2×(5³) + 3×(5²) + 2×(5¹) + 2×(5⁰) = 625 + 250 + 75 + 10 + 2 = 962
Second = 3×(5⁴) + 1×(5³) + 3×(5²) + 2×(5¹) + 2×(5⁰) = 1875 + 125 + 75 + 10 + 2 = 2087

Sum in base-10 = 962 + 2087 = 3049

Converting back to base-5: 3049 ÷ 5 = 609 R4, 609 ÷ 5 = 121 R4, 121 ÷ 5 = 24 R1, 24 ÷ 5 = 4 R4, 4 ÷ 5 = 0 R4
So 3049 = 44144 in base-5 (4 circles, 4 hexagons, 1 square, 4 triangles, 4 ones)
In simple words: Change the base-5 symbols to regular numbers, add them, then change the answer back to base-5 symbols.

Exam Tip: For non-decimal base systems, convert to base-10, perform the operation, then convert back to avoid errors.

Page 69

Question 1. Can there be a number whose representation in Egyptian numerals has one of the symbols occurring 10 or more times? Why not?
Answer: No, there cannot be a number whose representation in Egyptian numerals has one of the symbols occurring 10 or more times. Here's why: The Egyptian number system is an additive system, not a place value system. Symbols are repeated to add up values, but each symbol is used at most 9 times. The system had only a limited number of symbols, each of which was repeated no more than 9 times in writing any number. So, whenever a symbol would be needed 10 times or more, Egyptians would move to the next higher symbol instead of repeating it, making their writing more compact and orderly.
In simple words: If you need 10 of one symbol, you should instead use 1 of the next higher symbol - this keeps the writing short and clean.

Exam Tip: The Egyptian system's design prevents symbols from repeating 10+ times - this is built into the structure, not just a choice. Always remember this as a key feature.

Question 2. Create your own number system of base 4 and represent numbers from 1 to 16.
Answer: I created a system called the Quad-Code System, which is based on base-4. In this system, I use four special symbols instead of regular digits:
A = 0
B = 1
C = 2
D = 3

In base-4, the place values go up as powers of 4. So, the rightmost spot is 4⁰ = 1, the next is 4¹ = 4 and then 4² = 16 and so on. Using this system, I can write any number using just these four symbols.

Here is how I write numbers from 1 to 16:
1 = B
2 = C
3 = D
4 = BA
5 = BB
6 = BC
7 = BD
8 = CA
9 = CB
10 = CC
11 = CD
12 = DA
13 = DB
14 = DC
15 = DD
16 = ABA
In simple words: Use four symbols for a base-4 system, where each position moving left means multiplying by 4.

Exam Tip: When designing a base-n system, always include a zero symbol and use exactly n different symbols (digits 0 through n-1).

Question 3. Give a simple rule to multiply a given number by 5 in the base-5 system that we created.
Answer: The simple rule to multiply a number by 5 in the base-5 system is: Add a zero (A) at the end of the number.

In base-5, multiplying any number by 5 is the same as shifting its digits one spot to the left and adding A (zero) at the right end - just like adding a zero in base-10 when multiplying by 10.

Example: Let's say the number is BC (which is 1×5 + 2 = 7 in decimal). Now multiply by 5 - just add A at the end - BCA. BCA in base-5 = 7 × 5 = 35 in decimal. Because in base-5, the digits shift just like in base-10. Adding a zero (A) multiplies the number by the base itself, i.e., 5.
In simple words: Putting a zero at the end in base-5 works the same way as putting a zero at the end in base-10 - it multiplies by the base.

Exam Tip: This rule works for any base - in base-n, adding a zero (the 0 digit) at the right end multiplies the number by n.

Page 73

Question 1. Represent the following numbers in the Mesopotamian system - (i) 63 (ii) 132 (iii) 200 (iv) 60 (v) 3605
Answer: The Mesopotamian system is a base-60 system. Each number is broken down using powers of 60.

(i) 63 = 1×60 + 3 - shown as one wedge symbol for the 60s place and three nail symbols for the 1s place

(ii) 132 = 2×60 + 12 - shown as two wedge symbols for the 60s place and 12 nail symbols for the 1s place

(iii) 200 = 3×60 + 20 - shown as three wedge symbols for the 60s place and 20 nail symbols for the 1s place

(iv) 60 = 1×60 + 0 - shown as one wedge symbol for the 60s place

(v) 3605 = 1×(60²) + 0×60 + 5 - shown as one wedge in the 3600s place and five nails in the 1s place
In simple words: Break each number into groups of 60, then use wedges to show 60s and nails to show 1s.

Exam Tip: The Mesopotamian base-60 system is the ancestor of our modern 60-second minute and 60-minute hour - remember that when writing numerals, always identify which power of 60 each group represents.

Page 80

Question 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?
Answer: The Chinese number system used Zong (vertical) and Heng (horizontal) symbols to show place value clearly. They switched back and forth between the two types at each spot (units, tens, hundreds, etc.) to stop confusion when reading the number. This made it simpler to know which digit belonged to which spot even when gaps were small or missing.

If only Zong symbols were used, 41 would be written as: (Zong for 4) followed by (Zong for 1) - looks like: IIII I. Without changing the symbol direction or keeping good spacing: IIII I could be misread as 5 (i.e., 1 five) instead of 41. The lack of direction or spacing removes the hint that one part is "tens" and the other is "units".

The Chinese switched between Zong and Heng symbols to make place values visually clear and easy to read, especially in handwritten or tightly packed texts. Without this, numbers like 41 could easily be misunderstood.
In simple words: By alternating symbols, the Chinese made it obvious which digit position you were looking at - vertical for one place, horizontal for the next.

Exam Tip: The key insight is that a number system needs visual cues (like alternating symbol directions or spacing) to clearly show place value, especially when written by hand or in compact form.

Question 2. Form a base-2 place value system using 'ukasar' and 'urapon' as the digits. Compare this system with that of the Gumulgal's.
Answer: To form a base-2 place value system using 'ukasar' and 'urapon', we assign:
ukasar = 0
urapon = 1

This system works just like the binary number system, where each spot from right to left stands for growing powers of 2. For example:
The first spot is 2⁰ = 1
The next is 2¹ = 2
Then 2² = 4 and so on.

So, we can show numbers like this:
The number 1 is written as urapon
The number 2 is written as urapon ukasar
The number 3 is urapon urapon
The number 4 becomes urapon ukasar ukasar

Each spot tells us how many of that power of 2 we have and we use ukasar for 0 and urapon for 1.

Comparison: If we compare this to the Gumulgal number system, there is a big change. The Gumulgal system does not use place value. Instead, it adds groups of 2s (ukasar) and 1s (urapon) to build numbers. For example, to make 7, they would say something like ukasar-ukasar-ukasar-urapon (2 + 2 + 2 + 1).

So the main shift is the base-2 system with ukasar and urapon is a place value system - more efficient and better for big numbers. The Gumulgal system is group-based and additive, which is fine for small numbers but gets confusing as numbers get bigger.
In simple words: The base-2 system uses place value like modern number systems, while Gumulgal just adds groups together - place value is much more powerful.

Exam Tip: Place value systems are always more efficient than purely additive systems because they can represent much larger numbers without becoming unwieldy.

Question 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?
Answer: Hindu numerals and the digit 0 are used every day in our lives - for telling time, counting money, reading prices, doing maths in school and writing phone numbers. Many professions like banking, teaching, engineering and science rely heavily on this number system. Zero plays a key role in place value and workings, making big numbers easy to write and understand.

If zero and the Hindu number system had not been invented, life would be very hard. We would struggle to work out sums, trade or even write dates properly. Modern tech like computers and calculators would not exist, slowing down growth in every field. Science, medicine, and business would move much slower without a fast, reliable way to write and work with numbers. In short, our modern world depends on the Hindu number system and zero.
In simple words: Hindu numerals and zero are the backbone of modern maths and science - without them, progress would stop.

Exam Tip: Always link the importance of number systems to real-world applications (finance, technology, science) rather than just discussing them in abstract terms.

Question 4. The ancient Indians likely used base 10 for the Hindu number system because humans have 10 fingers and so we can use our fingers to count. But what if we had only 8 fingers? How would we be writing numbers then? What would the Hindu numerals look like if we were using base 8 instead? Base 5? Try writing the base-10 Hindu numeral 25 as base-8 and base-5 Hindu numerals, respectively. Can you write it in base-2?
Answer: If humans had only 8 fingers:
We would likely have built a base-8 number system instead of base-10. Just like we now count from 0 to 9 in base-10, we would count from 0 to 7 in base-8. All our numerals, maths and workings would be based on powers of 8.

Conversion of the base-10 number 25:

In base-8:
25 ÷ 8 = 3 remainder 1
3 ÷ 8 = 0 remainder 3
So, 25 in base-8 = 31

In base-5:
25 ÷ 5 = 5 remainder 0
5 ÷ 5 = 1 remainder 0
1 ÷ 5 = 0 remainder 1
So, 25 in base-5 = 100

In base-2 (binary):
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
So, 25 in base-2 = 11001
In simple words: The number 25 in base-10 becomes 31 in base-8, 100 in base-5, and 11001 in base-2 - the value stays the same, but the way we write it changes.

Exam Tip: Always use the division-by-base method to convert between bases - divide by the new base repeatedly, collect remainders, then read them backwards to get the answer.