NCERT Class 9 Maths Chapter 1 Number Systems Exercise 1.3 Solutions | Step-by-Step Explanation
Chapter: Number Systems
Class: 9 (CBSE)
Exercise: 1.3
Board: CBSE 2026
Introduction
Here are the complete step-by-step solutions of NCERT Class 9 Maths Chapter 1 – Number Systems Exercise 1.3. This exercise focuses on decimal expansions of rational numbers, recurring decimals, irrational numbers, and classification of numbers.
Exercise 1.3 Solutions
Question 1
Write the following in decimal form and say what kind of decimal expansion each has:
(i) 36/100
36 ÷ 100 = 0.36
Type: Terminating decimal
(ii) 1/11
1 ÷ 11 = 0.090909…
Type: Non-terminating recurring decimal
(iii) 4 1/8
Convert to improper fraction:
4 1/8 = 33/8
33 ÷ 8 = 4.125
Type: Terminating decimal
(iv) 3/13
3 ÷ 13 = 0.230769230769…
Type: Non-terminating recurring decimal
(v) 2/11
2 ÷ 11 = 0.181818…
Type: Non-terminating recurring decimal
(vi) 329/400
329 ÷ 400 = 0.8225
Type: Terminating decimal
Question 2
You know that 1/7 = 0.142857… Predict the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7 without long division.
Since 1/7 = 0.142857 (repeating block: 142857),
2/7 = 0.285714…
3/7 = 0.428571…
4/7 = 0.571428…
5/7 = 0.714285…
6/7 = 0.857142…
Explanation: The digits repeat in cyclic order.
Question 3
Express the following in the form p/q (q ≠ 0):
(i) 0.6̅
Let x = 0.6666…
10x = 6.6666…
10x − x = 6.6666 − 0.6666
9x = 6
x = 6/9 = 2/3
Answer: 2/3
(ii) 0.47̅
Let x = 0.474747…
100x = 47.4747…
100x − x = 47
99x = 47
x = 47/99
Answer: 47/99
(iii) 0.001̅
Let x = 0.001001001…
1000x = 1.001001…
1000x − x = 1
999x = 1
x = 1/999
Question 4
Express 0.99999… in the form p/q.
Let x = 0.9999…
10x = 9.9999…
10x − x = 9
9x = 9
x = 1
Therefore: 0.999… = 1
Question 5
What is the maximum number of digits in the repeating block of 1/17?
When we divide 1 by 17:
1/17 = 0.0588235294117647…
The repeating block contains 16 digits.
Answer: Maximum 16 digits.
Question 6
What property must q satisfy for p/q to have a terminating decimal expansion?
If p/q is in lowest form, then q must be of the form:
2m × 5n
Where m and n are non-negative integers.
Example:
1/8 (8 = 2³) → Terminating
1/25 (25 = 5²) → Terminating
Question 7
Write three numbers whose decimal expansions are non-terminating non-recurring.
Examples:
- √2
- √3
- π
These are irrational numbers.
Question 8
Find three irrational numbers between 5/7 and 9/11.
5/7 ≈ 0.714
9/11 ≈ 0.818
Three irrational numbers between them:
- √0.6
- √0.65
- √0.7
Question 9
Classify the following numbers as rational or irrational:
(i) √23 → Irrational (not a perfect square)
(ii) √225 = 15 → Rational
(iii) 0.3796 → Rational (terminating decimal)
(iv) 7.478478… → Rational (recurring decimal)
(v) 1.101001000100001… → Irrational (non-terminating non-recurring)
Conclusion
- Terminating decimals occur when denominator has only factors 2 and/or 5.
- Recurring decimals are rational numbers.
- Non-terminating non-recurring decimals are irrational numbers.
- 0.999… equals 1.
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