Figure it Out — Place Value: Mesopotamian and Mayan Systems — A Story of Numbers

1. 13 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 73

   Represent the following numbers in the Mesopotamian system: (i) 63 (ii) 132 (iii) 200 (iv) 60 (v) 3605

   Hint. This is a base-60 system. Split each number into how many 3600s, how many 60s and how many 1s, then write each of those counts using the symbols for 1 and 10.

   The Mesopotamian system is base-60, so its landmarks are 1, 60, 3600, 216000, … A number is written by saying how many of each landmark it contains, with each count itself written using the symbols for 1 and for 10. No count can reach 60, since sixty of one landmark make the next.

   (i) 63 = (1 × 60) + 3 → one 60, then three 1s

   (ii) 132 = (2 × 60) + 12 = (2 × 60) + 10 + 2 → two 60s, then the count 12 written as one 10-symbol and two 1-symbols Check: 120 + 12 = 132 ✓

   (iii) 200 = (3 × 60) + 20 → three 60s, then the count 20 written as two 10-symbols Check: 180 + 20 = 200 ✓

   (iv) 60 = (1 × 60) + 0 → a single 1-symbol standing in the 60s position, with nothing in the units position This one exposes the system's weakness: the numeral for 60 looks just like the numeral for 1, and only the spacing tells them apart.

   (v) 3605 = (1 × 3600) + (0 × 60) + 5 → one symbol in the 3600s position, an empty 60s position, then five 1s Check: 3600 + 5 = 3605 ✓ The empty middle position had to be shown by a blank space, and inconsistent spacing across manuscripts is exactly what made such numerals ambiguous — the problem that the later placeholder symbol, and ultimately zero, was invented to solve.

   ✦ 63 = 1×60 + 3; 132 = 2×60 + 10 + 2; 200 = 3×60 + 20; 60 = 1×60 (with an empty units place); 3605 = 1×3600 + 5 (with an empty 60s place)

2. 23 marksGanita Prakash Cl-8 Part 1, Math Talk, page 76

   Represent the following numbers using the Mayan system: (i) 77 (ii) 100 (iii) 361 (iv) 721

   Hint. The Mayan landmarks are 1, 20, 360 (not 400). Symbols are written one below the other, lowest row = number of 1s.

   The Mayan landmarks are 1, 20 and then 360 — note that the third is 360, not 20 × 20 = 400, which is why this is not a true base-20 system. Within each row, a dot is 1 and a bar is 5, and the rows are written one below another with the units row at the bottom.

   (i) 77 = (3 × 20) + (17 × 1) → 20s row: 3 · 1s row: 17 Check: 60 + 17 = 77 ✓

   (ii) 100 = (5 × 20) + (0 × 1) → 20s row: 5 · 1s row: 0 (the seashell placeholder) Check: 100 + 0 = 100 ✓

   (iii) 361 = (1 × 360) + (0 × 20) + (1 × 1) → 360s row: 1 · 20s row: 0 · 1s row: 1 Check: 360 + 0 + 1 = 361 ✓

   (iv) 721 = (2 × 360) + (0 × 20) + (1 × 1) → 360s row: 2 · 20s row: 0 · 1s row: 1 Check: 720 + 0 + 1 = 721 ✓

   The last three all need the placeholder in the middle row, and this is the Mayans' real achievement: they arrived independently at both place value and a symbol for zero. What they lacked was a consistent base, since the jump from 20 to 360 is by 18 rather than 20, so their landmarks are not all powers of one number and their arithmetic loses the advantage a true base gives.

   ✦ 77 = 3×20 + 17; 100 = 5×20 + 0; 361 = 1×360 + 0×20 + 1; 721 = 2×360 + 0×20 + 1

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.