NCERT Solutions for Class 8 Maths – Ganita Prakash
Chapter 3: A Story of Numbers
Exercise 3.7
Q1. Can there be a number whose representation in Egyptian numerals has one symbol occurring 10 or more times? Why not?
Answer:
❌ No, this is not possible.
Reason:
- The Egyptian number system follows a grouping rule.
- Whenever a symbol appears 10 times, it is replaced by one symbol of the next higher value.
Example:
- 10 symbols of 1 → replaced by 1 symbol of 10
- 10 symbols of 10 → replaced by 1 symbol of 100
- 10 symbols of 100 → replaced by 1 symbol of 1000
Conclusion:
- In Egyptian numerals, no symbol can appear 10 or more times.
- The maximum number of repetitions for any symbol is 9.
✔️ Therefore, such a number cannot exist in the Egyptian number system.
2. Create your own number system of base 4 and represent numbers from 1 to 16.
Base-4 system uses the digits: 0, 1, 2, 3
Now, the numbers from 1 to 16 in base-4 are:
| Number in Decimal System | Number in Base-4 System |
|---|---|
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 10 |
| 5 | 11 |
| 6 | 12 |
| 7 | 13 |
| 8 | 20 |
| 9 | 21 |
| 10 | 22 |
| 11 | 23 |
| 12 | 30 |
| 13 | 31 |
| 14 | 32 |
| 15 | 33 |
| 16 | 100 |
Conclusion: In base-4, after 3, the next number is 10 because 4 ones make 1 group of 4.
3. Give a simple rule to multiply a given number by 5 in the base-5 system that we created.
In base-5 system, the number 5 is written as 10.
So, multiplying any number by 5 in base-5 is the same as multiplying by 10 in decimal system.
Rule: To multiply a number by 5 in base-5 system, just put one 0 to the right of the number.
Examples:
235 × 5 = 2305
145 × 5 = 1405
Conclusion: In base-5 system, multiplying by 5 means shifting the number one place to the left and writing 0 in the ones place.