NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 — Figure it Out (Page 73)#
The Mesopotamian system is a base-60 system. Its landmark numbers are $1, 60, 60^2 = 3600, \ldots$ Inside each place, a number from 1 to 59 is written with two symbols: a wedge for 1 and a chevron (‹) for 10. For example, 23 is 2 chevrons and 3 wedges.
In the book’s notation, ◇1 marks the 60s and ◇2 marks the 3600s. In the compact (place value) form, these markers are dropped and the position alone tells the place.
1. Represent the following numbers in the Mesopotamian system — (i) 63 (ii) 132 (iii) 200 (iv) 60 (v) 3605 #
Solution
We split each number into 3600s, 60s and 1s, and write each part with wedges and chevrons:
| Number | Split | 3600s place | 60s place | 1s place |
|---|---|---|---|---|
| 63 | $1 \times 60 + 3$ | — | 1 wedge | 3 wedges |
| 132 | $2 \times 60 + 12$ | — | 2 wedges | 1 chevron, 2 wedges |
| 200 | $3 \times 60 + 20$ | — | 3 wedges | 2 chevrons |
| 60 | $1 \times 60 + 0$ | — | 1 wedge | (blank) |
| 3605 | $1 \times 3600 + 0 \times 60 + 5$ | 1 wedge | (blank) | 5 wedges |
In the book’s notation with landmark markers:
- 63 = 1 wedge ◇1 3 wedges
- 132 = 2 wedges ◇1 1 chevron 2 wedges
- 200 = 3 wedges ◇1 2 chevrons
- 60 = 1 wedge ◇1
- 3605 = 1 wedge ◇2 5 wedges
Notice: in the compact form, 60 is just “1 wedge” followed by a blank, which looks exactly like 1. And 3605 needs a blank place in the middle. Such blanks are hard to see, which is why the later Mesopotamians brought in a placeholder symbol, an early form of zero.