Question

State in each case, whether the given statement is true of 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 number 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) $\mathrm{\pi }$is irrational and$\frac{22}{7}$is rational.

Answer 1

(i) True

(ii) False
Example: $$\begin{gathered} \left(2+\sqrt{3}\right)+\left(2-\sqrt{3}\right)=4 \\ \mathrm{Here},4\mathrm{is}\mathrm{a}\mathrm{rational}\mathrm{number}. \end{gathered}$$

(iii) True

(iv) False
$$\begin{gathered} \mathrm{Example}:\sqrt{3}\times \sqrt{3}=3 \\ \mathrm{Here},3\mathrm{is}\mathrm{a}\mathrm{rational}\mathrm{number}. \end{gathered}$$

(v) True

(vi) False
Example: $$\begin{gathered} \left(4\right)\times \sqrt{5}=4\sqrt{5} \\ \mathrm{Here},4\sqrt{5}\mathrm{is}\mathrm{an}\mathrm{irrational}\mathrm{number}. \end{gathered}$$

(vii) False
Real numbers can be divided into rational and irrational numbers.

(viii) True

(ix) True