NCERT Class 9 Maths Chapter 1 Number Systems Exercise 1.4 Solutions | Step-by-Step Explanation
Chapter: Number Systems
Class: 9
Exercise: 1.4
Exercise 1.4 Solutions
Question 1
Classify the following numbers as rational or irrational:
(i) 2 − √5
√5 is irrational. Rational − irrational = irrational.
Answer: Irrational
(ii) (3 + √23) − √23
= 3 + √23 − √23
= 3
3 is a rational number.
Answer: Rational
(iii) (2√7) / (7√7)
= 2√7 / 7√7
Cancel √7:
= 2/7
Answer: Rational
(iv) 1 / √2
√2 is irrational. 1 divided by irrational remains irrational.
Answer: Irrational
(v) 2π
π is irrational. Rational × irrational = irrational.
Answer: Irrational
Question 2
Simplify each of the following expressions:
(i) (3 + √3)(2 + √2)
= 3×2 + 3√2 + 2√3 + √6
= 6 + 3√2 + 2√3 + √6
(ii) (3 + √3)(3 − √3)
Using identity: (a + b)(a − b) = a² − b²
= 3² − (√3)²
= 9 − 3
= 6
(iii) (√5 + √2)²
Using identity: (a + b)² = a² + b² + 2ab
= 5 + 2 + 2√10
= 7 + 2√10
(iv) (√5 − √2)(√5 + √2)
Using identity: (a − b)(a + b) = a² − b²
= 5 − 2
= 3
Question 3
π = c/d where c is circumference and d is diameter. How is π irrational?
The ratio c/d is constant for every circle.
However, this constant cannot be expressed exactly in the form p/q.
Its decimal expansion is non-terminating and non-recurring.
Therefore, π is irrational.
Question 4
Represent √9.3 on the number line.
Construction Steps:
- Draw a number line and mark OA = 9.3 units.
- Extend to point B such that AB = 1 unit.
- Find midpoint of OB and draw a semicircle.
- At point A, draw perpendicular meeting semicircle at P.
- AP = √9.3
- With O as centre and radius OP, mark the point on number line.
Question 5
Rationalise the denominators:
(i) 1 / √7
Multiply numerator and denominator by √7:
= √7 / 7
(ii) 1 / (√7 − √6)
Multiply by conjugate (√7 + √6):
= (√7 + √6) / (7 − 6)
= √7 + √6
(iii) 1 / (√5 + √2)
Multiply by conjugate (√5 − √2):
= (√5 − √2) / (5 − 2)
= (√5 − √2) / 3
(iv) 1 / (√7 − 2)
Multiply by conjugate (√7 + 2):
= (√7 + 2) / (7 − 4)
= (√7 + 2) / 3
Conclusion
- Operations involving irrational numbers usually remain irrational.
- Use algebraic identities to simplify surds.
- Use conjugates to rationalise denominators.
- π is irrational because its decimal expansion is non-terminating and non-recurring.
Visit Stepify.in for complete NCERT Class 9 Maths Chapter 1 solutions.