NCERT Solutions for Class 8 Maths – Ganita Prakash
Chapter: A Story of Numbers
Exercise 3.9
1. Why did the Chinese alternate between Zong and Heng symbols?
The Chinese alternated between Zong (vertical) and Heng (horizontal) symbols to clearly distinguish place values.
If only Zong symbols were used, numbers could look similar and create confusion.
Example:
41 = 4 tens + 1 ones
Using only Zong symbols → |||| |
This could also be misinterpreted as 5 (|||||), so spacing and alternation help avoid ambiguity.
Conclusion: Alternating symbols ensures clarity and avoids misreading.
2. Form a base-2 system using ‘ukasar’ and ‘urapon’:
Let:
- urapon = 0
- ukasar = 1
Base-2 numbers:
- 1 = ukasar
- 2 = ukasar-urapon
- 3 = ukasar-ukasar
- 4 = ukasar-urapon-urapon
Comparison with Gumulgai system:
- Base-2 system uses place value.
- Gumulgai system is based on grouping (mostly 2s).
- Base-2 is more systematic and efficient.
3. Importance of Hindu numerals and 0:
In daily life:
- Counting money
- Time and measurements
- Banking and transactions
In professions:
- Engineering
- Science and research
- Computer programming
- Business and accounting
Without 0:
- No place value system
- Difficult calculations
- Modern technology would not exist
Conclusion: Hindu numerals and 0 are essential for modern mathematics and life.
4. If we had 8 fingers (Base-8 system):
Base-8 digits: 0–7
Convert 25 (base-10):
25 ÷ 8 = 3 remainder 1
3 ÷ 8 = 0 remainder 3
25 = 318
Convert 25 into base-5:
25 ÷ 5 = 5 remainder 0
5 ÷ 5 = 1 remainder 0
1 ÷ 5 = 0 remainder 1
25 = 1005
Convert 25 into base-2:
25 = 16 + 8 + 1
= 110012
Conclusion:
Different bases change representation but not the value of numbers.