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Classify the following number as rational or irrational and choose the correct option.

(1) $2 - \sqrt{5}$

(2) $(3 + \sqrt{23}) - \sqrt{23}$

(3) $\frac{2 \sqrt{7}}{7 \sqrt{7}}$

(4) $\frac{1}{\sqrt{2}}$

(5) $2 π$

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2 and 4 are rational

1, 4, 5 are irrational

1, 2 are rational

1, 3, 2 are irrational

answer is B.

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Detailed Solution

Rational numbers are the nubers which can be written in the form of m/n where both m and n are integers while n is not equal to zero.
We can not represent the irrational numbers as a simple fraction.
(1) The number given in first part is,

2 - \sqrt{5}

Here,

2

is a rational number which can be represent as 2/1 while

\sqrt{5}

is the irrational number.
So, the given number

2 - \sqrt{5}

is an irrational number.

2)

The number given in second part is,

(3 + \sqrt{23}) - \sqrt{23}

= 3 + \sqrt{23} - \sqrt{23}

= 3

Here, 3 is the rational number.
Thus

(3 + \sqrt{23}) - \sqrt{23}

is also a rational number.

3)

The number given in Third part is,

\frac{2 \sqrt{7}}{7 \sqrt{7}}

Cancel out the same term

\sqrt{7}

to get,

= \frac{2}{7}

Thus

\frac{2 \sqrt{7}}{7 \sqrt{7}}

is also a rational number.

4)

The number given in forth part is,

\frac{1}{\sqrt{2}} :

As, we know that

\sqrt{2}

is an irrational number,
So, the term

\frac{1}{\sqrt{2}},

is also an irrational number.

5)

The number given in fifth part is,

2 π

Here,

π

is considered as an irrational number,
So, the number 2

π

is an irrational number. Correct option is 2.

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