Identify Rational or Irrational Numbers and Give Decimal Representation
Question: Identify whether the following numbers are rational or irrational. Give decimal representation of rational numbers:
- (i) \( \sqrt{4} \)
- (ii) \( 3\sqrt{18} \)
- (iii) \( \sqrt{1.44} \)
- (iv) \( \sqrt{\frac{9}{27}} \)
- (v) \( -\sqrt{64} \)
- (vi) \( \sqrt{100} \)
Solution:
(i) \( \sqrt{4} \)
\[ \sqrt{4} = 2 \]
2 is a rational number (can be written as \( \frac{2}{1} \)).
Decimal form: 2.0
(ii) \( 3\sqrt{18} \)
\[ \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \]
\[ 3\sqrt{18} = 3 \times 3\sqrt{2} = 9\sqrt{2} \]
Since \( \sqrt{2} \) is irrational, \( 9\sqrt{2} \) is irrational.
Conclusion: Irrational number
(iii) \( \sqrt{1.44} \)
\[ \sqrt{1.44} = 1.2 \]
1.2 is a terminating decimal, hence rational.
Decimal form: 1.2
(iv) \( \sqrt{\frac{9}{27}} \)
\[ \frac{9}{27} = \frac{1}{3} \]
\[ \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}} \]
Since \( \sqrt{3} \) is irrational, the number is irrational.
Conclusion: Irrational number
(v) \( -\sqrt{64} \)
\[ \sqrt{64} = 8 \]
\[ -\sqrt{64} = -8 \]
-8 is a rational number.
Decimal form: -8.0
(vi) \( \sqrt{100} \)
\[ \sqrt{100} = 10 \]
10 is a rational number.
Decimal form: 10.0
Final Answers:
- \( \sqrt{4} \) → Rational (2.0)
- \( 3\sqrt{18} \) → Irrational
- \( \sqrt{1.44} \) → Rational (1.2)
- \( \sqrt{\frac{9}{27}} \) → Irrational
- \( -\sqrt{64} \) → Rational (-8.0)
- \( \sqrt{100} \) → Rational (10.0)
Concept Used:
- Square root of a perfect square is rational.
- Square root of a non-perfect square is irrational. :contentReference[oaicite:0]{index=0}
- Rational numbers have terminating or repeating decimals. :contentReference[oaicite:1]{index=1}