Question 14 Marks
Ravish and Aarushi dedcided to visit world book fair which is organised every year. During their visit Aarushi was fascinated by the cover page of a book with $\pi / e$ written on it. $\pi$ and e are mathematical constants. In Euclidean geometry $\pi$ is defined as the ratio of a circle's circumference to its diameter. It is also referred to as Archimede's constant. The constant e is known as Euler's number and it is the limiting value of $\left(1+\frac{1}{n}\right)^n$ as $n$ approches infinity. Using the knowledge of rational and irrational numbers answer the following questions.
(i) $\pi$ represents
(a) an integer
(b) a rational number
(c) an irrational number
(d) a natural number
(ii) e represents
(a) a natural number
(b) an integer
(c) a rational number
(d) an irrational number
(iii) The product of any two irrational numbers is
(a) always an irrational number $\quad$(b) not necessarily an irrational number $\quad$
(c) never an irrational number $\quad$ (d) always an integer $\quad$
(iv) A rational number between $\sqrt{2}$ and $\sqrt{3}$ is
(a) $\frac{\sqrt{3}-\sqrt{2}}{2}$$\quad$(b) $\frac{\sqrt{3}+\sqrt{2}}{2}$$\quad$(c) $1 . \overline{6}$ $\quad$(d) $0 . \overline{2}+0 . \overline{3}$$\quad$
(i) $\pi$ represents
(a) an integer
(b) a rational number
(c) an irrational number
(d) a natural number
(ii) e represents
(a) a natural number
(b) an integer
(c) a rational number
(d) an irrational number
(iii) The product of any two irrational numbers is
(a) always an irrational number $\quad$(b) not necessarily an irrational number $\quad$
(c) never an irrational number $\quad$ (d) always an integer $\quad$
(iv) A rational number between $\sqrt{2}$ and $\sqrt{3}$ is
(a) $\frac{\sqrt{3}-\sqrt{2}}{2}$$\quad$(b) $\frac{\sqrt{3}+\sqrt{2}}{2}$$\quad$(c) $1 . \overline{6}$ $\quad$(d) $0 . \overline{2}+0 . \overline{3}$$\quad$
Answer
View full question & answer→(i) (c): $\pi$ is an irrational number its value is $3.1415926538 \ldots$
(ii) (d): $e$ is an irrational number and its value is $2.71828182845 \ldots$
(iii) (b): The product of irrational numbers $2+\sqrt{3}$ and $2-\sqrt{3}$ is 1 , which is rational. However, the product of rational numbers $(2+\sqrt{3})$ and $\sqrt{3}$ is $2 \sqrt{3}+3$ which is an irrational number. Hence, the product of two irrational numbers is sometimes rational and sometimes irrational.
(iv) (c): $\frac{\sqrt{3}-\sqrt{2}}{2}$ and $\frac{\sqrt{3}+\sqrt{2}}{2}$ are irrational numbers. So, options (a) and (b) are not correct.
We find that $0 . \overline{2}+0 . \overline{3}=0 . \overline{5}$, which does not lie between $\sqrt{2}$ and $\sqrt{3}$.
We have $\sqrt{2}=1.412 \ldots$ and $\sqrt{3}=1.736 \ldots$
Clearly, $\sqrt{2}<1 . \overline{6}<\sqrt{3}$. Hence, option (c) is correct.
(ii) (d): $e$ is an irrational number and its value is $2.71828182845 \ldots$
(iii) (b): The product of irrational numbers $2+\sqrt{3}$ and $2-\sqrt{3}$ is 1 , which is rational. However, the product of rational numbers $(2+\sqrt{3})$ and $\sqrt{3}$ is $2 \sqrt{3}+3$ which is an irrational number. Hence, the product of two irrational numbers is sometimes rational and sometimes irrational.
(iv) (c): $\frac{\sqrt{3}-\sqrt{2}}{2}$ and $\frac{\sqrt{3}+\sqrt{2}}{2}$ are irrational numbers. So, options (a) and (b) are not correct.
We find that $0 . \overline{2}+0 . \overline{3}=0 . \overline{5}$, which does not lie between $\sqrt{2}$ and $\sqrt{3}$.
We have $\sqrt{2}=1.412 \ldots$ and $\sqrt{3}=1.736 \ldots$
Clearly, $\sqrt{2}<1 . \overline{6}<\sqrt{3}$. Hence, option (c) is correct.
