State whether the given statement is ture 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 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.

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Solution

Sol:

(i) True

(ii) True

(iii) False

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

(iv) False

Counter example: 23–√3 and 43–√3 are two irrational numbers. But their product is 24, which is a rational number.

(v) True

(vi) True

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Similar questions

State, in each case, whether the given statement is true or false.

(i) The sum of two rational numbers is rational.

(ii) The sum of two irrational numbers is irrational.

(iii) The product of two rational numbers is rational.

(iv) The product of two irrational numbers is irrational.

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

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

(vii) Every real number is rational.

(viii) Every real number is either rational or irrational.

(ix) $π$ is irrational and $\frac{22}{7}$ is rational.

Which of the following is a true statement ?

(a) The sum of two irrational numbers is an irrational number.

(b) The product of two irrational numbers is an irrational number.

(d) Every real number is either rational or irrational.

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