Figure it Out — Shortcomings of the Egyptian System — A Story of Numbers

1. 12 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 69

   Can there be a number whose representation in Egyptian numerals has one of the symbols occurring 10 or more times? Why not?

   Hint. What happens the moment you have ten copies of the same landmark symbol?

   No, there cannot be — not in a properly written numeral.

   The reason is built into how the system is constructed: ten copies of any landmark number make exactly the next landmark number, since each landmark is ten times the one before it. So the moment a symbol appears ten times, those ten can and must be replaced by a single symbol of the next size up.

   For example, ten 100-symbols are worth 10 × 100 = 1000, so they get written as one 1000-symbol instead.

   This means every symbol can appear at most nine times in a correctly written Egyptian numeral — which is precisely why the Hindu system needs digits only from 0 to 9. The digit in each place is nothing other than a count of how many times that landmark occurs, and that count can never reach ten.

   It is worth noticing that this same rule is what makes 'carrying' work in ordinary addition: when a column reaches ten, ten of that unit are exchanged for one of the next.

   ✦ No — ten copies of any landmark make exactly the next landmark, so they would always be exchanged for one symbol of the next size. Every symbol can appear at most nine times.

2. 23 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 70

   Create your own number system of base 4, and represent numbers from 1 to 16.

   Hint. The base-4 landmarks are 1, 4, 16. Invent one symbol for each and repeat them — remembering no symbol may appear four or more times.

   Step 1 — Fix the landmark numbers and give each a symbol. In a base-4 system the landmarks are the powers of 4: 4⁰ = 1 → let this be ∟ 4¹ = 4 → let this be ∆ 4² = 16 → let this be □

   Step 2 — Write each number by grouping into these landmarks, always taking as many of the largest as possible, and never letting a symbol appear four or more times (since four of any landmark make the next one up).

   Number	Grouping	Numeral
   1	1	∟
   2	1+1	∟∟
   3	1+1+1	∟∟∟
   4	4	∆
   5	4+1	∆∟
   6	4+1+1	∆∟∟
   7	4+1+1+1	∆∟∟∟
   8	4+4	∆∆
   9	4+4+1	∆∆∟
   10	4+4+1+1	∆∆∟∟
   11	4+4+1+1+1	∆∆∟∟∟
   12	4+4+4	∆∆∆
   13	4+4+4+1	∆∆∆∟
   14	4+4+4+1+1	∆∆∆∟∟
   15	4+4+4+1+1+1	∆∆∆∟∟∟
   16	16	□

   Notice what happens at 16: rather than write a fourth ∆, the four 4s are exchanged for the single new symbol □, because four of one landmark always make the next. Any other choice of symbols would work equally well — what matters is the grouping rule, not the shapes.

   ✦ With ∟ = 1, ∆ = 4, □ = 16: 1 = ∟, 2 = ∟∟, 3 = ∟∟∟, 4 = ∆, 5 = ∆∟, 6 = ∆∟∟, 7 = ∆∟∟∟, 8 = ∆∆, 9 = ∆∆∟, 10 = ∆∆∟∟, 11 = ∆∆∟∟∟, 12 = ∆∆∆, 13 = ∆∆∆∟, 14 = ∆∆∆∟∟, 15 = ∆∆∆∟∟∟, 16 = □

3. 32 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 70

   Give a simple rule to multiply a given number by 5 in the base-5 system that we created.

   Hint. Five is the base here. What did multiplying by 10 do in the base-10 Egyptian system?

   Rule: replace every symbol in the numeral by the symbol for the next landmark up.

   Here is the reasoning. In the base-5 system the landmarks are 1, 5, 25, 125, 625, … and each one is five times the previous. So multiplying any single landmark by 5 gives exactly the next landmark: 5⁰ × 5 = 5¹, 5¹ × 5 = 5², 5² × 5 = 5³, and so on.

   A number in this system is a sum of landmarks, and multiplication distributes over that sum, so multiplying the whole number by 5 multiplies each landmark in it by 5 — meaning each symbol simply moves up one step.

   Example: 137 = 125 + 5 + 5 + 1 + 1. Multiplying by 5 turns the 125 into a 625, each 5 into a 25, and each 1 into a 5: 625 + 25 + 25 + 5 + 5 = 685 ✓ (and indeed 137 × 5 = 685)

   This is the exact counterpart of the base-10 rule that multiplying by 10 just appends a zero. In general, in any base-n system, multiplying by n shifts every symbol up one landmark — which in a place value system is the familiar 'shift left by one place'.

   ✦ Replace every symbol by the symbol for the next landmark up — the exact counterpart of 'append a zero' when multiplying by 10 in base 10.

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.