Classify the following numbers as rational or irrational :
(i) ` 27/7` (ii) 3.1416 (iii) ` pi` (iv) `3. overline(142857)`
(v) 5.636363… (vi) 2.040040004…. (vii) 1.535335333…..

Answer

To classify the given numbers as rational or irrational, we need to understand the definitions of both types of numbers:

- **Rational Numbers**: These are numbers that can be expressed in the form \( \frac{P}{Q} \), where \( P \) and \( Q \) are integers, and \( Q \neq 0 \). Rational numbers can be terminating or repeating decimals.

- **Irrational Numbers**: These are numbers that cannot be expressed as a fraction of two integers. They are non-terminating and non-repeating decimals.

Now, let's classify each of the given numbers step by step.

### Step 1: Classify \( \frac{27}{7} \)
- This number is in the form \( \frac{P}{Q} \) where \( P = 27 \) and \( Q = 7 \).
- Since both are integers and \( Q \neq 0 \), \( \frac{27}{7} \) is a **Rational Number**.

### Step 2: Classify \( 3.1416 \)
- This number is a terminating decimal.
- Since it can be expressed as a fraction (for example, \( \frac{31416}{10000} \)), it is a **Rational Number**.

### Step 3: Classify \( \pi \)
- The value of \( \pi \) is approximately \( 3.14159... \) and it is known to be non-terminating and non-repeating.
- Therefore, \( \pi \) is an **Irrational Number**.

### Step 4: Classify \( 3.\overline{142857} \)
- The notation \( 3.\overline{142857} \) indicates that the digits \( 142857 \) repeat indefinitely.
- Since it can be expressed as a fraction, it is a **Rational Number**.

### Step 5: Classify \( 5.636363... \)
- This number is a repeating decimal (the digits \( 63 \) repeat).
- It can be expressed as a fraction, so it is a **Rational Number**.

### Step 6: Classify \( 2.040040004... \)
- This number is non-terminating and does not have a repeating pattern.
- Therefore, it is an **Irrational Number**.

### Step 7: Classify \( 1.535335333... \)
- Similar to the previous number, this is also non-terminating and does not have a repeating pattern.
- Thus, it is an **Irrational Number**.

### Summary of Classifications:
1. \( \frac{27}{7} \) - Rational
2. \( 3.1416 \) - Rational
3. \( \pi \) - Irrational
4. \( 3.\overline{142857} \) - Rational
5. \( 5.636363... \) - Rational
6. \( 2.040040004... \) - Irrational
7. \( 1.535335333... \) - Irrational