Rational & Irrational Numbers — Case-Study Based Questions

Case Study

The union of the set of rational numbers (Q) and irrational numbers (P) form the set of real numbers (R). Rational numbers are the set of numbers which can be written in the form of $\dfrac{a}{b}$, where a and b are integers and b is not equal to zero. The decimal expansion of a rational number is either terminating or non-terminating repeating. The number which cannot be expressed in the form $\dfrac{a}{b}$ are called irrational numbers. The decimal expansion of irrational numbers is non-terminating non-repeating.

Based on this information, answer the following questions:

1. Every rational number is:
   (a) a natural number
   (b) a whole number
   (c) an integer
   (d) a real number

2. Every real number is:
   (a) an integer
   (b) a rational number
   (c) an irrational number
   (d) either a rational number or an irrational number.

3. The sum of two irrationals is:
   (a) irrational
   (b) rational
   (c) either rational or irrational
   (d) neither rational or irrational.

4. The product of a rational and irrational number is:
   (a) an irrational number
   (b) a rational number
   (c) either a rational number or an irrational number
   (d) neither a rational number nor an irrational number.

5. The number of irrational number is:
   (a) finite
   (b) infinite
   (c) neither finite or infinite
   (d) none of these.

Answer

1. Real numbers include all rational and irrational numbers.

Rational numbers are a subset of real numbers.

Hence, Option (d) is the correct option.

2. Real numbers include all rational and irrational numbers.

Hence, Option (d) is the correct option.

3. The sum of two irrational numbers can be either rational or irrational.

Hence, Option (c) is the correct option.

4. If the rational number is zero, the product is rational.

If the rational number is non-zero, the product is irrational.

Hence, Option (c) is the correct option.

5. The number of irrational number is infinite as the number of real numbers is also infinite.

Hence, Option (b) is the correct option.

Case Study

Ms Mehta teaches maths in a school. One day after teaching the lesson of number system, she wanted to check the understanding of the students of her class. So, she wrote two numbers, $\dfrac{3}{11}\text{ and } 0.\overline{52}$ on the blackboard and asked few questions based on them. You please try to answer the following questions asked by Ms Mehta.

1. The decimal expansion of $\dfrac{3}{11}$ is:
   (a) terminating
   (b) non-terminating
   (c) non-terminating non-repeating
   (d) non-terminating repeating

2. $0.\overline{52}$ is:
   (a) non-terminating non-repeating
   (b) non-terminating repeating
   (c) non-terminating
   (d) terminating

3. The decimal form of $\dfrac{3}{11}$:
   (a) 0.27
   (b) 0.2727
   (c) $0.\overline{27}$
   (d) 0.3

4. $0.\overline{52}$ as vulgar fraction becomes:
   
   (a) $\dfrac{52}{99}$
   
   (b) $\dfrac{52}{100}$
   
   (c) $\dfrac{26}{25}$
   
   (d) $\dfrac{13}{25}$

5. The sum of $0.\overline{52} \text{ and } \dfrac{3}{11}$ is :
   (a) $\dfrac{79}{99}$
   
   (b) $\dfrac{70}{99}$
   
   (c) $\dfrac{52}{99}$
   
   (d) $\dfrac{40}{99}$

Answer

1. On dividing,

$\dfrac{3}{11} = 0.\overline{27}$

Hence, Option (d) is the correct option.

2. $0.\overline{52}$ = 0.525252...

Hence, Option (b) is the correct option.

3. $\dfrac{3}{11}$ = 0.272727.... = $0.\overline{27}$

Hence, Option (c) is the correct option.

4. Given,

Let x = 0.525252..     .........(1)

Multiply both side by 100(as two digits are recurring)

⇒ 100x = 52.5252..     .........(2)

Subtracting eqn (1) from eqn (2), we get :

⇒ 100x - x = 52.5252.. - 0.525252

⇒ 99x = 52

⇒ x = $\dfrac{52}{99}$.

Hence, Option (a) is the correct option.

5. From part 4,

$0.\overline{52} = \dfrac{52}{99}$

Adding $\dfrac{52}{99}$ and $\dfrac{3}{11}$

$\Rightarrow \dfrac{52}{99} + \dfrac{3}{11} \\[1em] \Rightarrow \dfrac{52 + 27}{99} \\[1em] \Rightarrow \dfrac{79}{99}.$

Hence, Option (a) is the correct option.