Question

Classify the following number as rational or irrational-
A) \[2 - \sqrt 5 \]
B) \[\left( {3 + \sqrt {23} } \right) - \sqrt {23} \]
C) \[\dfrac{{2\sqrt 7 }}{{7\sqrt 7 }}\]
D) \[\dfrac{1}{{\sqrt 2 }}\]
E) \[2\pi \]

Answer 1

Hint:
Here we have to find the following numbers as rational numbers or irrational numbers. A rational number is a number that can be expressed as the fraction of two integers. The numbers, which cannot be expressed in terms of fraction, or the numbers, which are not rational, are called irrational numbers. We have to classify each option based on the properties of rational numbers and irrational numbers.

Complete step by step solution:
A) \[2 - \sqrt 5 \] is an irrational number because it is the difference between a rational number and an irrational number which is also called a surd. Every surd is an irrational number. Here 2 is the rational number and \[\sqrt 5 \] is an irrational number.
 Therefore, \[2 - \sqrt 5 \] is an irrational number.
B) We will first simplify the given expression. We will subtract the like terms.
\[\left( {3 + \sqrt {23} } \right) - \sqrt {23} = 3 + \sqrt {23} - \sqrt {23} = 3\]
We get an answer as 3. It is a rational number since it can be expressed in the form of \[\dfrac{p}{q}\] as \[\dfrac{3}{1}\] .
Therefore, \[(3 + \sqrt {23} ) - \sqrt {23} \] is a rational number.
C) We will first simplify the given expression. We will divide the like terms
\[\dfrac{{2\sqrt 7 }}{{7\sqrt 7 }}\] = \[\dfrac{2}{7}\]
The obtained fraction is a rational number since it can be expressed in the form of \[\dfrac{p}{q}\] as \[\dfrac{2}{7}\] .
Therefore, \[\dfrac{{2\sqrt 7 }}{{7\sqrt 7 }}\] is a rational number.
D) \[\dfrac{1}{{\sqrt 2 }}\]
Multiplying by \[\sqrt 2 \] on the numerator and the denominator, we get
\[\dfrac{1}{{\sqrt 2 }} \times \dfrac{{\sqrt 2 }}{{\sqrt 2 }} = \dfrac{{\sqrt 2 }}{2}\]
The obtained fraction is an irrational number because it is the division of a rational number by a surd.
Therefore, \[\dfrac{1}{{\sqrt 2 }}\] is an irrational number.
E) \[2\pi \] is an irrational number. Since, \[\pi = 3.14159\]……… is an irrational number which is multiplied by a constant, it also becomes an irrational number.
Therefore, \[2\pi \] is an irrational number.

Note:
Another method to classify the numbers as rational or irrational is based on the terminating and recurring digits. Recurring of digits is the repetition of digits. If the integer is in decimal then a non-terminating and non-recurring decimal is called an irrational number and a terminating and recurring decimal is called a rational number. The same method can be used when it is in fraction too.