Classify the following numbers as terminating decimal or non-terminating recurring decimal or non-terminating non-recurring decimal:
(i) 0.1666… (ii) 0.250 (iii) 1.01001000100001…
(iv) 0.27696 (v) 2.142857142857… (vi) 0.3
(vii) 0.2359872785.... (viii) 0.484848848... (ix) 2.502500250002....
(x) 4.123456789
Also classify these given numbers as Rational and irrational numbers.

Answer

To classify the given numbers as terminating decimal, non-terminating recurring decimal, or non-terminating non-recurring decimal, and also to classify them as rational or irrational numbers, we will analyze each number step by step.

### Step-by-Step Solution:

1. **(i) 0.1666…**
- This number has a repeating decimal (the digit 6 repeats).
- Classification: Non-terminating recurring decimal.
- Rational/Irrational: Rational number.

2. **(ii) 0.250**
- This number has a finite number of decimal places.
- Classification: Terminating decimal.
- Rational/Irrational: Rational number.

3. **(iii) 1.01001000100001…**
- The decimal does not repeat; it keeps changing.
- Classification: Non-terminating non-recurring decimal.
- Rational/Irrational: Irrational number.

4. **(iv) 0.27696**
- This number has a finite number of decimal places.
- Classification: Terminating decimal.
- Rational/Irrational: Rational number.

5. **(v) 2.142857142857…**
- This number has a repeating decimal (the sequence 142857 repeats).
- Classification: Non-terminating recurring decimal.
- Rational/Irrational: Rational number.

6. **(vi) 0.3**
- This number has a finite number of decimal places.
- Classification: Terminating decimal.
- Rational/Irrational: Rational number.

7. **(vii) 0.2359872785…**
- The decimal does not repeat; it keeps changing.
- Classification: Non-terminating non-recurring decimal.
- Rational/Irrational: Irrational number.

8. **(viii) 0.484848848…**
- This number has a repeating decimal (the sequence 48 repeats).
- Classification: Non-terminating recurring decimal.
- Rational/Irrational: Rational number.

9. **(ix) 2.502500250002…**
- The decimal does not have a repeating pattern; it keeps changing.
- Classification: Non-terminating non-recurring decimal.
- Rational/Irrational: Irrational number.

10. **(x) 4.123456789**
- This number has a finite number of decimal places.
- Classification: Terminating decimal.
- Rational/Irrational: Rational number.

### Summary of Classifications:
- **Non-terminating recurring decimals**: (i) 0.1666…, (v) 2.142857142857…, (viii) 0.484848848…
- **Terminating decimals**: (ii) 0.250, (iv) 0.27696, (vi) 0.3, (x) 4.123456789
- **Non-terminating non-recurring decimals**: (iii) 1.01001000100001…, (vii) 0.2359872785…, (ix) 2.502500250002….

### Rational and Irrational Classifications:
- **Rational Numbers**: (i), (ii), (iv), (v), (vi), (viii), (x)
- **Irrational Numbers**: (iii), (vii), (ix)