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 rational and an irrational is irrational.
vi. The product of a rational and an irrational is irrational.
i. True ii. True
iii. False iv. False
v. True vi. True
(iii) Let’s take (2 + 
) and (2 ‒ 
).
These are irrational numbers.
Their sum = (2 + 
) + (2 ‒ 
) = 4 {which is rational}.
∴ it is false.
(iv) Let’s take (3 + 
) and (3 ‒ 
).
These are irrational numbers.
Their product = (3 + 
) × (3 ‒ 
) = 9 + 3√2 – 3√2 – (√2)2
= 9 – 4 = 5 {which is rational}
∴ it is false.
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