Question

Give an example of each, of two irrational numbers whose:

(i) difference is a rational number.

(ii) difference is an irrational number.

(iii) sum is a rational number.

(iv) sum is an irrational number.

(v) product is a rational number.

(vi) product is an irrational number.

(vii) quotient is a rational number.

(viii) quotient is an irrational number.

Answer 1

(i) Two numbers whose difference is also a rational number. e.g. $\sqrt{2},\sqrt{2}$ which are irrational numbers.

$\therefore$ Difference = $\sqrt{2}-\sqrt{2}$ = 0 which is also a rational number.
(ii) Two numbers whose difference is an irrational number.

e.g. $\sqrt{3}and\sqrt{2}$ which are numbers.

Now difference = $\sqrt{3}-\sqrt{2}$ which is also an irrational number.

(iii) Let two irrational numbers be $\sqrt{3}and-\sqrt{3}$ which are irrational numbers.

Now sum = $\sqrt{3}$ + $-\sqrt{3}$ = $\sqrt{3}-\sqrt{3}$ = 0

Which is a rational number.
(iv) Let two numbers be $\sqrt{5},\sqrt{3}$ which are irrational numbers.

Now sum = $\sqrt{5}+\sqrt{3}$ which is an irrational number
(v) Let number be $\sqrt{3}+\sqrt{2}$ and $\sqrt{3}-\sqrt{2}$ which are irrational numbers.

Now product = $(\sqrt{3}+\sqrt{2})$ $(\sqrt{3}-\sqrt{2})$ = 3 - 2 = 1 which is a rational number.
(vi) Let numbers be $\sqrt{3}$ and $\sqrt{5}$, which are irrational number.

Now product = $\sqrt{3}$ $\times$ $\sqrt{5}$ $\sqrt{3\times 5}$ = $\sqrt{15}$

which is an irrational number.
(vii) Let numbers be 6 $\sqrt{2}$ and 2$\sqrt{2}$ which are irrational numbers.

Quotient = $\frac{6\sqrt{2}}{2\sqrt{2}}$ = 3 which is a rational number.
(viii) Let numbers be $\sqrt{3}$ and $\sqrt{5}$ which are irrational numbers.

Now quotient = $\frac{\sqrt{3}}{\sqrt{5}}$ = $\sqrt{\frac{3}{5}}$ which is an irrational number.