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 an irrational 

(iv) The product of two irrationals is 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) The sum of two rationals is always rational - True 

(ii) The product of two rationals is always rational - True 

(iii) The sum of two irrationals is an irrational - False 

Counter example: 2 + √3 and 2 - √3 are two irrational numbers. But their sum is 4, which is a rational number. 

(iv) The product of two irrationals is an irrational – False 

Counter example: 

2 √3 and 4 √3 are two irrational numbers. But their product is 24, which is a rational number. 

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

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