Question

3. 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. √2, √2 which are irrational numbers.

∴ Difference = √2 - √2 = 0 which is also a rational number.

(ii) Two numbers whose difference is an irrational number.

e.g. √3 and √2 which are numbers.

Now difference = √3 - √2 which is also an irrational number.

(iii) Let two irrational numbers be √3 and -√3 which are irrational numbers.

Now sum = √3 + (-√3) = √3 - √3 = 0

Which is a rational number.

(iv) Let two numbers be √5, √3 which are irrational numbers.

Now sum = √5 + √3 which is an irrational number.

(v) Let number be √3 + √2 and √3 - √2 which are irrational numbers.

Now product = (√3 + √2) (√3 - √2) = 3 - 2 = 1 which is a rational number.

(vi) Let numbers be √3 and √5, which are irrational number.

Now product = √3 × √5 = \(\sqrt{3 \times 5}\) = √15

which is an irrational number.

(vii) Let numbers be 6√2 and 2√2 which are irrational numbers.

Quotient = \(\frac {6\sqrt 2}{2 \sqrt 2}\) = 3 which is a rational number.

(viii) Let numbers be √3 and √5 which are irrational numbers.

Now quotient = \(\frac{\sqrt 3}{\sqrt 5} = \sqrt {\frac 35}\) which is an irrational number.