Which of the following is irrational?

Options

A. `0.14`
B. `0.14bar(16)`
C. `0.bar(1416)`
D. `0.4014001400014.........`

Explanation

To determine which of the following numbers is irrational, we will analyze each option based on the definitions of rational and irrational numbers.

### Step-by-Step Solution:

1. **Understanding Rational and Irrational Numbers**:
- Rational numbers can be expressed in the form \( \frac{P}{Q} \), where \( P \) and \( Q \) are integers and \( Q \neq 0 \). They can be either terminating or repeating decimals.
- Irrational numbers cannot be expressed in this form. They are non-terminating and non-repeating decimals.

2. **Analyzing Each Option**:
- **Option A: 0.14**
- This is a terminating decimal (it stops after two decimal places).
- Therefore, it can be expressed as \( \frac{14}{100} \) or \( \frac{7}{50} \).
- **Conclusion**: 0.14 is a rational number.

- **Option B: 0.1416 (with a bar over 16)**
- This means 0.1416161616..., which is a repeating decimal.
- It can be expressed as a fraction.
- **Conclusion**: 0.1416 (with a bar) is a rational number.

- **Option C: 0.1416 (with a bar over 6)**
- This means 0.1416161616..., which is also a repeating decimal.
- It can be expressed as a fraction.
- **Conclusion**: 0.1416 (with a bar over 6) is a rational number.

- **Option D: 0.4014001400014...**
- This number does not have a repeating pattern and continues indefinitely.
- Since it is non-terminating and non-repeating, it cannot be expressed as a fraction.
- **Conclusion**: 0.4014001400014... is an irrational number.

3. **Final Answer**:
- The irrational number among the options is **Option D: 0.4014001400014...**.