Question

Give an example of two irrational numbers whose
(i) difference is an irrational number.
(ii) difference is a rational number.
(iii) sum is an irrational number.
(iv) sum is a rational number.
(v) product is an irrational number.
(vi) product is a rational number.
(vii) quotient is an irrational number.
(viii) quotient is a rational number.

Answer 1

(i) 2 irrational numbers with difference an irrational number will be $3-\sqrt{5}\mathrm{and}3+\sqrt{5}$.
(ii) 2 irrational numbers with difference is a rational number will be $5+\sqrt{3}\mathrm{and}2+\sqrt{3}$
(iii) 2 irrational numbers with sum an irrational number $7+\sqrt{5}\mathrm{and}\sqrt{6}-8$
(iv) 2 irrational numbers with sum a rational number is $3-\sqrt{2}\mathrm{and}3+\sqrt{2}$
(v) 2 irrational numbers with product an irrational number will be $6+\sqrt{3}\mathrm{and}7-\sqrt{3}$
(vi) 2 irrational numbers with product a rational number will be $\left(5+\sqrt{7}\right)\mathrm{and}\left(5-\sqrt{7}\right)$
(vii) 2 irrational numbers with quotient an irrational number will be $\sqrt{15}\mathrm{and}\sqrt{5}$
(viii) 2 irrational numbers with quotient a rational number will be $\sqrt{63}\mathrm{and}\sqrt{7}$.