Question

Classify the following numbers as rational or irrational:
(i) \[2 - \sqrt 5 \]
(ii) \[\left( {3 + \sqrt {23} } \right) - \sqrt {23} \]
 (iii) \[\dfrac{{2\sqrt 7 }}{{7\sqrt 7 }}\]

Answer 1

Hint: Here, we need to classify the given numbers as rational or irrational. We will simplify the given expressions. We will use the fact that the sum or difference of a rational and irrational number is always irrational. If the final expression includes an irrational number, then the number is irrational.

Complete step-by-step answer:
(i) The given number is \[2 - \sqrt 5 \].
The number 2 is a rational number because it can be written as \[\dfrac{2}{1}\].
Since 5 is not a perfect square, we cannot simplify \[\sqrt 5 \] and write it in the form \[\dfrac{p}{q}\].
Therefore, \[\sqrt 5 \] is an irrational number.
Now, we know that the sum or the difference between a rational and irrational number is always irrational. Therefore, the difference of the rational number 2 and the irrational number \[\sqrt 5 \] is irrational.
Thus, \[2 - \sqrt 5 \] is an irrational number.

(ii) The given number is \[\left( {3 + \sqrt {23} } \right) - \sqrt {23} \].
Simplifying the expression, we get
\[ \Rightarrow \left( {3 + \sqrt {23} } \right) - \sqrt {23} = 3 + \sqrt {23} - \sqrt {23} \]
The two irrational numbers \[\sqrt {23} \] and \[\sqrt {23} \] can be subtracted from each other.
Therefore, we get
\[ \Rightarrow \left( {3 + \sqrt {23} } \right) - \sqrt {23} = 3\]
The number 3 is a rational number because it can be written as \[\dfrac{3}{1}\].
\[\therefore \] \[\left( {3 + \sqrt {23} } \right) - \sqrt {23} \] is a rational number.

(iii) The given number is \[\dfrac{{2\sqrt 7 }}{{7\sqrt 7 }}\].
Rewriting the expression, we get
\[ \Rightarrow \dfrac{{2\sqrt 7 }}{{7\sqrt 7 }} = \dfrac{2}{7} \times \dfrac{{\sqrt 7 }}{{\sqrt 7 }}\]
The two irrational numbers \[\sqrt 7 \] and \[\sqrt 7 \] can be divided by each other.
Therefore, we get
\[ \Rightarrow \dfrac{{2\sqrt 7 }}{{7\sqrt 7 }} = \dfrac{2}{7} \times 1 = \dfrac{2}{7}\]
The number \[\dfrac{2}{7}\] is a rational number because it is in the form \[\dfrac{p}{q}\], where \[q \ne 0\].
\[\therefore \] \[\dfrac{{2\sqrt 7 }}{{7\sqrt 7 }}\] is a rational number.

Note: A rational number is a number which can be written in the form \[\dfrac{p}{q}\], where the denominator \[q \ne 0\]. For example, \[5,\dfrac{7}{2}, - \dfrac{{15}}{7},5.6\], etc. are rational numbers. Rational numbers include every integer, fraction, decimal.
An irrational number is a number that is opposite to a rational number. They cannot be written in the form \[\dfrac{p}{q}\]. For example, \[\sqrt 2 ,\sqrt 5 ,\sqrt 6 \], etc. are irrational numbers.