Question

State whether the given statement is true or false.

(i) The sum of two rationals is always rational.

(ii) The product of two rationals is always rational.

(iii) The sum of two irrationals is always an irrational.

(iv) The product of two irrationals is always an irrational.

(v) The sum of a rational and an irrational is irrational.

(vi) The product of a rational and an irrational is irrational.

Answer 1

(i) True.

(ii) True.

(iii) False.

Reason:

Sum of two irrational can be rational. Let us take an example,

Consider two irrational numbers: (3 + √2) and (3 – √2)

Sum: (3 + √2) + (3 – √2) = 3 + √2 + 3 – √2 = 6 which is rational.

(iv) False.

Reason:

Product of two irrational numbers can be rational. Let us take an example,

Consider two irrational numbers: (3 + √2) and (3 – √2)

Product: (3 + √2)(3 – √2 ) 

= (3)² – (√2 )² 

= 9 – 2 = 7; which is rational

(v) True.

(vi) True.