AP 8th Maths Textbook Solutions – 3. A STORY OF NUMBERS (2026-27)

3 A STORY OF NUMBERS

3.1 Hasantika's Curiosity (Math Talk)

• 1. How do we ensure that all cows have returned safely after grazing?
  We can use a one-to-one mapping method. For every cow that goes out to graze, we can keep one pebble or stick. When they return, we match each cow to a stick. If all sticks are matched with a cow, they have all returned safely.
• 2. Do we have fewer cows than our neighbour?
  We can compare our collection of sticks (representing our cows) with our neighbour's collection of sticks. We pair one stick from our pile with one stick from their pile. If our pile runs out first, we have fewer cows.
• 3. If there are fewer, how many more cows would we need so that we have the same number of cows as our neighbour?
  After pairing our sticks with the neighbour's sticks one by one, the number of unmatched sticks left in the neighbour's pile represents the exact number of additional cows we need.

• 1. How will you use such sticks to answer the other two questions (Q2 and Q3)?
  As mentioned above, we use the method of one-to-one correspondence. Pairing the sticks from two different herds allows us to see which pile is smaller (answering Q2) and counting the remaining unmatched sticks in the larger pile gives the exact difference (answering Q3).

• 1. How many numbers can you represent in this way using the sounds of the letters of your language?
  In the English alphabet, we can represent exactly 26 numbers because there are 26 letters (a to z). (In Telugu, we could represent 56 numbers using the alphabets).
• 2. Do you see a way of extending this method to represent bigger numbers as well? How?
  Yes. Once we run out of single letters, we can start combining them. For example, after 'z', we could use 'aa', 'ab', 'ac' ... 'az', and then 'ba', 'bb', etc., to continue the sequence endlessly.

Figure it Out - 3.1

• 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.
  Addition: Take the two collections of sticks and combine them into a single pile.
  Subtraction: Take the larger collection and remove one stick from it for every stick in the smaller collection. The remaining sticks are the answer.
  Multiplication: For every single stick in the first collection, create a full copy of the second collection. Combine all the copies together.
  Division: Take the first collection of sticks and keep separating them into small bundles, where each bundle has the exact same number of sticks as the second collection. The total number of bundles you create is the answer.
• 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?
  After reaching 'z' (26), we use two letters starting with 'aa' (27), 'ab' (28) ... up to 'zz'. Once two-letter combinations are exhausted, we can move to three-letter combinations 'aaa', 'aab', and so on. This creates an infinite standard sequence to represent all numbers.

• 3. Try making your own number system.
  (Student Activity) Example: Let a circle represent 1, a triangle represent 5, and a square represent 10. The number 17 would be represented as: 1 square, 1 triangle, 2 circles.

Figure it Out - 3.2

• 1. Represent the following numbers in the Roman system.
  (i) 1222    (ii) 2999    (iii) 302    (iv) 715
  (i) 1222 = MCCXXII
  (ii) 2999 = MMCMXCIX
  (iii) 302 = CCCII
  (iv) 715 = DCCXV

• 1. Try adding the following numbers without converting them to Hindu numerals:
  (a) CCXXXII + CCCXIII
  Grouping the C's, X's, and I's together:
  (CC + CCC) + (XXX + X) + (II + III) = CCCCC + XXXX + V.
  Since CCCCC is D, and XXXX is XL, the answer is DCXLV.
• 2. (b) LXXXVII + LXXVIII
  Grouping symbols:
  L + L = C
  XXX + XX = L
  V + V = X
  II + III = V
  Putting it all together: C + L + X + V = CLXV.
• 3. Try to find the product of the following pairs of landmark numbers: V × L, L × D, V × D, VII × IX.
  V × L = CCL (5 × 50 = 250)
  L × D = (50 × 500 = 25,000, which requires special Roman bar notation)
  V × D = MMD (5 × 500 = 2500)
  VII × IX = LXIII (7 × 9 = 63)
• 4. Multiply CCXXXI and MDCCCLII. (Daredevil Contest)
  CCXXXI = 231
  MDCCCLII = 1852
  Product = 231 × 1852 = 427,812. (In Roman numerals, this requires advanced bar notation, showing how difficult multiplication was in this system!)

Figure it Out - 3.3

• 1. Identify the features of the Hindu number system that make it efficient when compared to the Roman number system.
  1. Place Value System: The position of a digit determines its value, eliminating the need to create new symbols for larger numbers.
  2. Use of Zero (0): Zero acts as a placeholder, making it easy to distinguish between numbers like 32 and 302.
  3. Limited Symbols: It uses only 10 symbols (0-9) to represent infinitely large numbers.
  4. Ease of Arithmetic: Addition, subtraction, multiplication, and division are straightforward through standard algorithms, unlike the complex grouping needed in the Roman system.
• 2. Using the ideas discussed in this section, try refining the number system you might have made earlier.
  (Student Activity) Example: If I used shapes, I can now assign a place value rule to them. Instead of repeating a square ten times to make a hundred, I can use a "zero" shape as a placeholder. The rightmost shape represents 1s, the next represents 10s, and so on.

Figure it Out - 3.4

• 1. Represent the following numbers in the Egyptian system: 10458, 1023, 2660, 784, 1111, 70707.
  10458: 1 pointing finger, 4 coiled ropes, 5 heel bones, 8 strokes.
  1023: 1 lotus flower, 2 heel bones, 3 strokes.
  2660: 2 lotus flowers, 6 coiled ropes, 6 heel bones.
  784: 7 coiled ropes, 8 heel bones, 4 strokes.
  1111: 1 lotus flower, 1 coiled rope, 1 heel bone, 1 stroke.
  70707: 7 pointing fingers, 7 coiled ropes, 7 strokes.
• 2. What numbers do these numerals stand for?
  (i) 2 spirals (200) + 7 heel bones (70) + 6 strokes (6) = 276
  (ii) 4 lotus flowers (4000) + 3 spirals (300) + 2 strokes (2) = 4302
• 3. Instead of grouping together 10 collections of size equal to the previous landmark number, can we get a number system by grouping together 5 collections...? Can this 5 be replaced by any positive integer?
  Yes, if we group by 5s, we create a base-5 number system. The landmark numbers become 1, 5, 25, 125, etc. And yes, the number 5 can be replaced by any positive integer greater than 1 (like base-2, base-8, base-12, etc.) to form valid number systems.

• 1. Express the number 143 in this new base-5 system.
  The largest power of 5 in 143 is 125 (5³).
  143 = 125 + 18.
  18 can be grouped into three 5s and three 1s.
  So, 143 = 1(125) + 0(25) + 3(5) + 3(1).
  Using symbols, it would be represented as: 1 Circle, 3 Squares, 3 Triangles.

Figure it Out - 3.5

• 1. Write the following numbers in the above base-5 system using the symbols in Table 2: 15, 50, 137, 293, 651.
  15: 3(5) = 3 Squares.
  50: 2(25) = 2 Hexagons.
  137: 1(125) + 2(5) + 2(1) = 1 Circle, 2 Squares, 2 Triangles.
  293: 2(125) + 1(25) + 3(5) + 3(1) = 2 Circles, 1 Hexagon, 3 Squares, 3 Triangles.
  651: 1(625) + 1(25) + 1(1) = 1 Tilde, 1 Hexagon, 1 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. We can endlessly create higher landmark numbers by multiplying the previous one by 5 (e.g., 3125, 15625, etc.), allowing us to group and represent any infinitely large number.
• 3. Compute the landmark numbers of a base-7 system. In general, what are the landmark numbers of a base-n system?
  The landmark numbers of a base-7 system are powers of 7:
  7⁰ = 1, 7¹ = 7, 7² = 49, 7³ = 343, 7⁴ = 2401, etc.
  In general, the landmark numbers of a base-n system are the powers of n: 1, n, n², n³, n⁴, ...

• 1. To get an idea of how the abacus was used for calculations, let us consider a simple addition problem: 2907 + 43. How would you use this to find the sum?
  First, place counters for 2907 on the abacus: 2 counters on the 1000 line, 9 counters on the 100 line, 0 on the 10 line, and 7 on the 1 line.
  Then, add the counters for 43: 4 counters on the 10 line and 3 counters on the 1 line.
  Combine them together on each line.
• 2. The counters along each line were brought together. What is to be done if the total in a line exceeded 10?
  If the total counters on any line reach 10 or more, you remove 10 counters from that line and add exactly 1 counter to the line immediately above it (representing the next power of 10). This is the process of "carrying over".

Figure it Out - 3.6

• 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, whenever you have 10 of a particular symbol, they are replaced by 1 symbol of the next higher landmark number. For example, 10 strokes are immediately replaced by 1 heel bone.
• 2. Create your own number system of base 4, and represent numbers from 1 to 16.
  In a base-4 system, the allowed digits are 0, 1, 2, and 3. The landmark numbers are 1, 4, 16, 64, etc.
  1 = 1
  2 = 2
  3 = 3
  4 = 10
  5 = 11
  6 = 12
  7 = 13
  8 = 20
  9 = 21
  10 = 22
  11 = 23
  12 = 30
  13 = 31
  14 = 32
  15 = 33
  16 = 100
• 3. Give a simple rule to multiply a given number by 5 in the base-5 system that we created.
  To multiply a number by 5 in a base-5 system, simply shift all the symbols one place value to the left, which is equivalent to adding a zero (the lowest place value placeholder) to the right end of the number.

• 1. Example: Let us represent the number 640 in this system... Can we represent this more compactly?
  Yes. Instead of writing 10 separate symbols for 60 and 40 separate symbols for 1, we can use a positional place value system. We place the symbol for '10' in the 60s position and the symbol for '40' in the 1s position.

Figure it Out - 3.7

• 1. Represent the following numbers in the Mesopotamian system -
  (i) 63 (ii) 132 (iii) 200 (iv) 60 (v) 3605
  (i) 63: 1(60) + 3(1). Represented by a '1' symbol in the 60s place, space, '3' symbol in the 1s place.
  (ii) 132: 2(60) + 12(1). Represented by a '2' symbol in the 60s place, space, '12' symbol in the 1s place.
  (iii) 200: 3(60) + 20(1). Represented by a '3' symbol in the 60s place, space, '20' symbol in the 1s place.
  (iv) 60: 1(60) + 0(1). Represented by a '1' symbol in the 60s place, followed by the placeholder (blank space or zero symbol) in the 1s place.
  (v) 3605: 1(3600) + 0(60) + 5(1). Represented by a '1' symbol in the 3600s place, a placeholder in the 60s place, and a '5' symbol in the 1s place.
• 2. Look at the representation of 60. What will be the representation for 3,600?
  3600 is exactly 1 × 60². It would be represented by the symbol for '1' placed in the 3600s position, followed by two placeholders (blank spaces or zero symbols) representing 0 in the 60s position and 0 in the 1s position.

• 1. Represent the following numbers using the Mayan system:
  (i) 77 (ii) 100 (iii) 361 (iv) 721
  (i) 77: 3(20) + 17(1). Placed vertically: The symbol for 3 (three dots) is written above the symbol for 17 (three bars and two dots).
  (ii) 100: 5(20) + 0(1). Placed vertically: The symbol for 5 (one bar) is written above the shell symbol (representing 0).
  (iii) 361: 1(360) + 0(20) + 1(1). Top level: 1 dot. Middle level: shell (0). Bottom level: 1 dot.
  (iv) 721: 2(360) + 0(20) + 1(1). Top level: 2 dots. Middle level: shell (0). Bottom level: 1 dot.

• 1. Where does the Hindu/Indian number system figure in the evolution of ideas of number representation? What are its landmark numbers? And does it use a place value system?
  The Hindu number system represents the culmination and highest point of the evolution of number systems. Its landmark numbers are powers of 10 (1, 10, 100, 1000, etc.). Yes, it is a fully developed place value system that uses the digit 0 as both a placeholder and an independent number.

Figure it Out - 3.8

• 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 and Heng symbols to clearly distinguish adjacent place values. If only Zong (vertical lines) were used, the number 41 would look like 4 vertical lines next to 1 vertical line. Without significant spacing, this could easily be misinterpreted as a single group of 5 vertical lines, meaning the number 5.
• 2. 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 the number zero are fundamental in almost every aspect of daily life and all professions: computing, engineering, banking, architecture, and medicine. Without zero and the place value system, advanced mathematics (like algebra and calculus) would be nearly impossible, and modern digital technology (which relies on binary 0s and 1s) would not exist.
• 3. 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?
  If we had 8 fingers, we would likely use a base-8 (octal) system. We would only need 8 symbols (0, 1, 2, 3, 4, 5, 6, 7). The landmark numbers would be 1, 8, 64, 512, etc. If we used base-5, we would only use 5 symbols (0, 1, 2, 3, 4).
• 4. Try writing the base-10 Hindu numeral 25 as base-8 and base-5 Hindu numerals, respectively. Can you write it in base-2?
  Base-8: 25 = 3(8) + 1(1). So it is written as 31.
  Base-5: 25 = 1(25) + 0(5) + 0(1). So it is written as 100.
  Base-2: 25 = 16 + 8 + 1. So it is 1(16) + 1(8) + 0(4) + 0(2) + 1(1). Written as 11001.

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