Rational & Irrational Numbers — Case-Study Based Question

(i) Real numbers are numbers which include both rational and irrational numbers.

(ii) Every rational number can be expressed as $\dfrac{p}{q}$, where p and q are integers and q ≠ 0.

(iii) Irrational numbers cannot be expressed as $\dfrac{p}{q}$, where p and q are integers and q ≠ 0.

Answer the following questions :

(a) Is the difference between a rational number and an irrational number always rational?

(b) Is 0.5555 ....... a rational number ?

(c) Is 0.505005 ...... a rational number ?

(d) Is the product of irrational number (3 - $\sqrt{7}$) and its rationalising factor a rational number?

Answer

(a) No, because rational - irrational = irrational

Example : 2 - $\sqrt{3}$ is an irrational number as $\sqrt{3}$ is an irrational number.

Hence, required answer is no.

(b) As 0.5555 is a repeating decimal number, so it is a rational number.

Hence, required answer is Yes.

(c) 0.505005 is non-terminating and non-repeating number.

So it is an irrational number, not a rational number.

Hence, required answer is No.

(d) Rationalizing factor of 3 - $\sqrt{7}$ is 3 + $\sqrt{7}$

= (3 - $\sqrt{7}$) × (3 - $\sqrt{7}$)

= (3)² - $(\sqrt{7})^2$

= 9 - 7

= 2.

2 is a rational number.

Hence, required answer is Yes.

According to Taruner Swapna Scheme of West Bengal Government, backward / economically disadvantaged students of West Bengal can receive financial aid and devices like Smartphones / Tablets for digital learning, bridging the digital gap and supporting online studies.

Rohan who resides in Purulia district of West Bengal received a tablet and he decided to complete his education through e-learning. One day he was studying number system, where he learnt about rational and irrational numbers, etc. He was particularly interested in irrational numbers like 7 + $4\sqrt{3}$.

Based on the above information, answer the following :

(i) What is the rationalising factor of 7 + $4\sqrt{3}$ ?

(ii) What is the reciprocal of 7 + $4\sqrt{3}$ ? Are rationalising factor and reciprocal same?

(iii) If x = 7 + $4\sqrt{3}$, then find the value of $\Big(x + \dfrac{1}{x}\Big)$.

Answer

(i) For a number of the form a + $b\sqrt{c}$, the rationalizing factor is a - $b\sqrt{c}$.

So, for 7 + $4\sqrt{3}$, the rationalizing factor is 7 - $4\sqrt{3}$.

Hence rationalizing factor = 7 - $4\sqrt{3}$.

(ii) Reciprocal of 7 + $4\sqrt{3}$ is :

$\dfrac{1}{7 + 4 \sqrt{3}}$

Multiplying numerator and denominator by the rationalising factor :

$\dfrac{1}{7 + 4 \sqrt{3}} × \Big(\dfrac{7 - 4 \sqrt{3}}{7 - 4 \sqrt{3}} \Big)$

= $\dfrac{7 - 4 \sqrt{3}}{7^2 - (4 \sqrt{3})^2}$

= $\dfrac{7 - 4 \sqrt{3}}{49 - 48}$

= 7 - $4\sqrt{3}$.

Hence, reciprocal of 7 + $4\sqrt{3}$ = 7 - $4\sqrt{3}$. Thus, the reciprocal and the rationalising factor both are same.

(iii) Given,

x = 7 + $4\sqrt{3}$,

$\dfrac{1}{x}$ = 7 - $4\sqrt{3}$

$x + \dfrac{1}{x} = (7 + 4 \sqrt{3}) + (7 - 4\sqrt{3})$ = 14.

Hence, value = 14.