Question

5. Give an example to show that (i) the sum of two irrational numbers may be a rational number. (ii) The difference of two irrational numbers may be a rational number. (iii) The product of two irrational numbers may be a rational number. (ii) The quotient of two irrational numbers may be a rational number.

Answer 1

(i) Let n₁ = 1+√3 and n₂ = 1-√3.

It is cleared that n₁ & n₂ both one irrational number (∵ rational + irrational = irrational)

Now, n₁+n₂ = (1+√3)+(1-√3) = 2 which is a rational number 

i.e; if two numbers are irrational then their sum need not be irrational, it may be rational.

(ii) Let n₁ = 1+√3 and n₂ = -1+√3

It is cleared that both n₁ & n₂ are irrational number.

Now, n₁ - n₂ = (1+√3) - (-1+√3)

= 1+√3 + 1-√3

= 2 Which is a rational number.

Hence, the difference of two irrational number may be a rational number.

(iii) Let n₁ = 1+√3 and n₂ = 1-√3

It is cleared that both n₁&n₂ are irrational number

Then n₁n₂ = (1+√3)(1-√3)

= 1-3    (∵(a+b)(a-b) = a² - b²)

= -2 which is a rational number.

Hence, the product of two irrational number may be a rational number.

(iv) Let n₁ = 2+2√3

It is cleared that both n₁&n₂ are irrational number.

Then \(\frac{n_1}{n_2}\) = \(\frac{2+2\sqrt{3}}{1+\sqrt{3}}\) = \(\frac{2(1+\sqrt3)}{1+\sqrt{3}}\) = 2 which is a rational number.

Hence, the quotient of two irrational number may be a rational number.