NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 — Figure it Out (Pages 69–70)#
1. Can there be a number whose representation in Egyptian numerals has one of the symbols occurring 10 or more times? Why not? #
Solution
No. Each Egyptian landmark number is 10 times the one before it. So 10 copies of any symbol are always replaced by one copy of the next symbol: 10 strokes make a heel bone, 10 heel bones make a coil, and so on. That is why each symbol appears at most 9 times, just like a digit from 0 to 9.
(The only exception would be the largest symbol, 1,000,000, for numbers of 10 million or more, because the Egyptians had no symbol for $10^7$ in everyday use.)
2. Create your own number system of base 4, and represent numbers from 1 to 16. #
Solution
In base 4, the landmark numbers are the powers of 4:
| Landmark | $4^0 = 1$ | $4^1 = 4$ | $4^2 = 16$ |
|---|---|---|---|
| Our symbol | ● | ▲ | ■ |
Each symbol is used at most 3 times, because 4 of them make the next symbol.
| Number | Grouping | Numeral | Number | Grouping | Numeral |
|---|---|---|---|---|---|
| 1 | 1 | ● | 9 | $4 + 4 + 1$ | ▲▲ ● |
| 2 | $1 + 1$ | ●● | 10 | $4 + 4 + 1 + 1$ | ▲▲ ●● |
| 3 | $1 + 1 + 1$ | ●●● | 11 | $4 + 4 + 1 + 1 + 1$ | ▲▲ ●●● |
| 4 | 4 | ▲ | 12 | $4 + 4 + 4$ | ▲▲▲ |
| 5 | $4 + 1$ | ▲ ● | 13 | $4 + 4 + 4 + 1$ | ▲▲▲ ● |
| 6 | $4 + 1 + 1$ | ▲ ●● | 14 | $4 + 4 + 4 + 1 + 1$ | ▲▲▲ ●● |
| 7 | $4 + 1 + 1 + 1$ | ▲ ●●● | 15 | $4 + 4 + 4 + 1 + 1 + 1$ | ▲▲▲ ●●● |
| 8 | $4 + 4$ | ▲▲ | 16 | 16 | ■ |
Any other three symbols would do just as well.
3. Give a simple rule to multiply a given number by 5 in the base-5 system that we created. #
Solution
Each landmark number times 5 is the next landmark number: × 5 = , × 5 = , × 5 = , × 5 = , × 5 = .
Rule: to multiply a numeral by 5, replace every symbol by the next symbol in the sequence → → → → → , keeping the same number of each.
Example: stands for $125 + 10 + 3 = 138$.
Multiplying by 5, each symbol moves up one step:
× 5 =
Check: $625 + 50 + 15 = 690$, and $138 \times 5 = 690$. ✓
This is just like multiplying by 10 in our own system, where each digit moves one place to the left.