Question

State whether the given statement is true of 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 and irrational is irrational.
(vi) The product of a rational and an irrational is irrational.

Answer 1

(i) True
(ii) True
(iii) False
Counter example: $2+\sqrt{3}\mathrm{and}2-\sqrt{3}$ are two irrational numbers. But their sum is 4, which is a rational number.
(iv) False
Counter example: $2\sqrt{3}\mathrm{and}4\sqrt{3}\mathrm{are}\mathrm{two}\mathrm{irrational}\mathrm{number}s.\text{B}\mathrm{ut}\mathrm{their}\mathrm{product}\mathrm{is}24,\mathrm{which}\mathrm{is}\text{a}\mathrm{rational}\mathrm{number}.$
(v) True
(vi) True