Classify the following numbers as rational or irrational .
(i) `2-sqrt(5)`
(ii) `(3+sqrt(23))-sqrt(23)`
(iii) `(2sqrt(7))/(7sqrt(7))`
(iv) `(1)/(sqrt(2))`
(v) `2pi`

Answer

To classify the given numbers as rational or irrational, we will analyze each number step by step.

### Step-by-Step Solution:

**(i) \(2 - \sqrt{5}\)**
- The term \(\sqrt{5}\) is an irrational number because it cannot be expressed as a fraction of two integers.
- Therefore, \(2 - \sqrt{5}\) is also irrational, as the subtraction of a rational number (2) and an irrational number (\(\sqrt{5}\)) results in an irrational number.
**Classification:** Irrational

**(ii) \((3 + \sqrt{23}) - \sqrt{23}\)**
- Simplifying this expression:
\((3 + \sqrt{23}) - \sqrt{23} = 3\)
- The number 3 is a whole number and can be expressed as \(\frac{3}{1}\), which is in the form of \(\frac{P}{Q}\) where \(Q \neq 0\).
**Classification:** Rational

**(iii) \(\frac{2\sqrt{7}}{7\sqrt{7}}\)**
- We can simplify this expression:
\(\frac{2\sqrt{7}}{7\sqrt{7}} = \frac{2}{7}\) (since \(\sqrt{7}\) cancels out)
- The number \(\frac{2}{7}\) is in the form of \(\frac{P}{Q}\) where \(Q \neq 0\).
**Classification:** Rational

**(iv) \(\frac{1}{\sqrt{2}}\)**
- The term \(\sqrt{2}\) is irrational.
- Therefore, \(\frac{1}{\sqrt{2}}\) is also irrational because it cannot be expressed as a fraction of two integers.
**Classification:** Irrational

**(v) \(2\pi\)**
- The number \(\pi\) is known to be irrational.
- Therefore, multiplying an irrational number (\(\pi\)) by a rational number (2) results in an irrational number.
**Classification:** Irrational

### Summary of Classifications:
- (i) \(2 - \sqrt{5}\) : Irrational
- (ii) \((3 + \sqrt{23}) - \sqrt{23}\) : Rational
- (iii) \(\frac{2\sqrt{7}}{7\sqrt{7}}\) : Rational
- (iv) \(\frac{1}{\sqrt{2}}\) : Irrational
- (v) \(2\pi\) : Irrational