Examples of Irrational Numbers Based on Operations
Solution:
(i) Difference is a Rational Number
\[ \sqrt{2} – \sqrt{2} = 0 \ (\text{rational}) \]
(ii) Difference is an Irrational Number
\[ \sqrt{5} – \sqrt{3} \]
This is irrational.
(iii) Sum is a Rational Number
\[ (2 + \sqrt{3}) + (2 – \sqrt{3}) = 4 \]
(iv) Sum is an Irrational Number
\[ \sqrt{2} + \sqrt{3} \]
This is irrational.
(v) Product is a Rational Number
\[ \sqrt{2} \times \sqrt{2} = 2 \]
(vi) Product is an Irrational Number
\[ \sqrt{2} \times \sqrt{3} = \sqrt{6} \]
(vii) Quotient is a Rational Number
\[ \frac{\sqrt{8}}{\sqrt{2}} = \sqrt{4} = 2 \]
(viii) Quotient is an Irrational Number
\[ \frac{\sqrt{5}}{\sqrt{2}} = \sqrt{\frac{5}{2}} \]
Final Answer Summary:
- (i) \( \sqrt{2} – \sqrt{2} = 0 \)
- (ii) \( \sqrt{5} – \sqrt{3} \)
- (iii) \( (2+\sqrt{3}) + (2-\sqrt{3}) = 4 \)
- (iv) \( \sqrt{2} + \sqrt{3} \)
- (v) \( \sqrt{2} \times \sqrt{2} = 2 \)
- (vi) \( \sqrt{2} \times \sqrt{3} = \sqrt{6} \)
- (vii) \( \frac{\sqrt{8}}{\sqrt{2}} = 2 \)
- (viii) \( \frac{\sqrt{5}}{\sqrt{2}} \)