Which of the following is the decimal expansion of an irrational number

Options

A. `4.561`
B. `0.bar(12)`
C. `5.010010001`
D. `6.03`

Answer

To determine which of the given decimal expansions represents an irrational number, we need to understand the characteristics of irrational numbers. Irrational numbers are defined as numbers that cannot be expressed as a fraction of two integers. Their decimal expansions are non-terminating and non-repeating.

Let's analyze each option step by step:

### Step 1: Identify the characteristics of irrational numbers
- **Irrational numbers**: Non-terminating and non-repeating decimal expansions.

### Step 2: Examine each option
1. **Option 1**: A terminating decimal (e.g., 0.5)
- **Analysis**: Since this decimal terminates, it can be expressed as a fraction (e.g., 0.5 = 5/10 = 1/2). Therefore, it is a rational number.

2. **Option 2**: A repeating decimal (e.g., 0.121212...)
- **Analysis**: This decimal has a repeating pattern (1, 2), which means it can also be expressed as a fraction. Hence, it is a rational number.

3. **Option 3**: A non-terminating and non-repeating decimal (e.g., 0.1010010001...)
- **Analysis**: This decimal does not terminate and does not have a repeating pattern. Therefore, it cannot be expressed as a fraction and is classified as an irrational number.

4. **Option 4**: Another terminating decimal (e.g., 0.75)
- **Analysis**: Similar to option 1, this decimal terminates and can be expressed as a fraction (0.75 = 75/100 = 3/4). Thus, it is a rational number.

### Conclusion
After analyzing all options, we conclude that **Option 3** is the only decimal expansion that represents an irrational number.

### Final Answer
**Option 3** is the decimal expansion of an irrational number.

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