Examine Whether the Following Numbers are Rational or Irrational
Question: Examine whether the following numbers are rational or irrational:
- (i) \( \sqrt{7} \)
- (ii) \( \sqrt{4} \)
- (iii) \( 2 + \sqrt{3} \)
Solution:
(i) \( \sqrt{7} \)
7 is not a perfect square. Hence, \( \sqrt{7} \) cannot be expressed in the form \( \frac{p}{q} \). Its decimal expansion is non-terminating and non-repeating.
Conclusion: \( \sqrt{7} \) is an irrational number. :contentReference[oaicite:0]{index=0}
(ii) \( \sqrt{4} \)
\[ \sqrt{4} = 2 \]
Since 2 can be written as \( \frac{2}{1} \), it is a rational number.
Conclusion: \( \sqrt{4} \) is a rational number.
(iii) \( 2 + \sqrt{3} \)
We know that \( \sqrt{3} \) is irrational.
Sum of a rational number (2) and an irrational number (\( \sqrt{3} \)) is always irrational.
Conclusion: \( 2 + \sqrt{3} \) is an irrational number.
Final Answers:
- \( \sqrt{7} \) → Irrational
- \( \sqrt{4} \) → Rational
- \( 2 + \sqrt{3} \) → Irrational
Concept Used:
- Square root of a non-perfect square is irrational.
- Square root of a perfect square is rational.
- Sum of a rational and irrational number is irrational.