Figure it Out — The Idea of a Base — A Story of Numbers

1. 13 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 63

   Write the following numbers in the base-5 system created in the chapter: 15, 50, 137, 293, 651.

   Hint. The base-5 landmark numbers are 1, 5, 25, 125, 625. Start from the largest landmark that fits and take as many as possible.

   The base-5 landmark numbers are the powers of 5: 1, 5, 25, 125, 625, 3125, … For each number, start from the largest landmark that fits and take as many of each as possible, since taking fewer would leave a remainder too big for the smaller landmarks to absorb without repeating five of them.

   15 = 5 + 5 + 5 → three 5-symbols (no 1-symbols needed)

   50 = 25 + 25 → two 25-symbols

   137 = 125 + 5 + 5 + 1 + 1 → one 125-symbol, two 5-symbols, two 1-symbols Check: 125 + 10 + 2 = 137 ✓

   293 = 125 + 125 + 25 + 5 + 5 + 5 + 1 + 1 + 1 → two 125-symbols, one 25-symbol, three 5-symbols, three 1-symbols Check: 250 + 25 + 15 + 3 = 293 ✓

   651 = 625 + 25 + 1 → one 625-symbol, one 25-symbol, one 1-symbol Check: 625 + 25 + 1 = 651 ✓

   Important rule to watch: no symbol may appear five or more times, because five of any landmark make the next one up. In 293, for instance, three 5s is fine, but a fourth and fifth would have to be regrouped into a 25.

   ✦ 15 = 5+5+5; 50 = 25+25; 137 = 125+5+5+1+1; 293 = 125+125+25+5+5+5+1+1+1; 651 = 625+25+1

2. 22 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 63

   Is there a number that cannot be represented in our base-5 system above? Why or why not?

   Hint. Think about what you would draw to show that a collection is empty.

   Yes — zero cannot be represented.

   The reason is that in this system a number is written by repeating landmark symbols, and the numeral for a number is literally the collection of symbols standing for its parts. Zero has no parts to write down, and the system was given no symbol meaning 'nothing', so there is simply nothing to put on the page.

   Every number from 1 upwards can be written, because we can always take as many of the largest fitting landmark as possible, then move down to the next, and this process always terminates. It is only zero that falls outside.

   This is exactly the gap the chapter is building towards. The Egyptian and early base-5 style systems never needed a zero, because a missing power of the base is shown simply by leaving that symbol out. But the moment a system uses position instead of distinct symbols, an empty position has to be marked somehow — and that is where the placeholder, and eventually the number zero, becomes indispensable.

   ✦ Yes — zero, because the system has no symbol for it and a numeral here is just a collection of landmark symbols, so there is nothing to write.

3. 32 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 63

   Compute the landmark numbers of a base-7 system. In general, what are the landmark numbers of a base-n system?

   Hint. The first landmark is always 1, and each next one is the previous multiplied by the base.

   Step 1 — Build the base-7 landmarks. By definition the first landmark number is 1, and each next landmark is obtained by multiplying the current one by the base, which here is 7: 7⁰ = 1 7¹ = 7 7² = 49 7³ = 343 7⁴ = 2401 and so on without end.

   So the landmark numbers of a base-7 system are 1, 7, 49, 343, 2401, …

   Step 2 — Generalise. Repeating the same reasoning with any base n, the landmark numbers are n⁰ = 1, n¹ = n, n², n³, n⁴, … — that is, all the powers of n starting from n⁰.

   This is precisely why the definition of a base-n system is worth stating: because every landmark is a power of the same number n, multiplying any two landmarks gives another landmark (nᵃ × nᵇ = nᵃ⁺ᵇ). That single fact is what makes arithmetic in a base system so much easier than in the Roman system, where the jumps between landmarks are irregular.

   ✦ Base-7 landmarks are 1, 7, 49, 343, 2401, … In general the landmarks of a base-n system are the powers of n: n⁰, n¹, n², n³, …

Solutions written by the tuition.in editorial team and checked against the NCERT Class 8 Mathematics textbook Ganita Prakash Part 1, Reprint 2026-27 (hegp103.pdf). This chapter is a history of number *systems* — tally marks, Roman, Egyptian, base-5, Mesopotamian, Mayan, Chinese rod and Hindu numerals — not a chapter on rational numbers. Questions appear as numbered 'Figure it Out' blocks plus 'Math Talk'/'Try This' prompts in the running text; every numeric answer here is checked against the book's own printed answer key at the end of the chapter. Four sub-parts whose questions exist only as printed Egyptian/base-5 glyph images (the addition drills on pages 65 and the products on page 68) are deliberately omitted rather than guessed at, since the operands cannot be recovered from the text.. Questions are referenced from the NCERT textbook for identification.