Which one of the following is a correct statement?

Options

A. Decimal expansion of a rational number is terminating
B. Decimal expansion of a rational number is non-terminating
C. Decimal expansion of an irrational number is terminating
D. Decimal expansion of an irrational number is non-terminating and non-repeating

Answer

To solve the problem, we need to analyze each of the four statements regarding decimal expansions and determine which one is correct. Let's break it down step by step.

### Step 1: Understand Decimal Expansions
Decimal expansions can be classified into two main categories:
1. **Terminating Decimal**: A decimal that ends after a finite number of digits (e.g., 2.34).
2. **Non-Terminating Decimal**: A decimal that continues indefinitely. This can be further divided into:
- **Recurring Decimal**: A decimal that has a repeating pattern (e.g., 2.333...).
- **Non-Recurring Decimal**: A decimal that does not have any repeating pattern (e.g., π).

### Step 2: Analyze Each Statement
Now, let's analyze the four statements given in the question.

1. **Statement 1**: "The decimal expansion of a rational number is terminated."
- This statement is **false** because rational numbers can have both terminating (e.g., 0.5) and non-terminating recurring decimal expansions (e.g., 1/3 = 0.333...).

2. **Statement 2**: "The decimal expansion of a rational number is non-terminated."
- This statement is also **false** because, as mentioned, rational numbers can be either terminating or non-terminating.

3. **Statement 3**: "The decimal expansion of an irrational number is terminated."
- This statement is **false** because irrational numbers cannot have terminating decimal expansions; they are always non-terminating.

4. **Statement 4**: "The decimal expansion of an irrational number is non-terminating and non-repeating."
- This statement is **true**. Irrational numbers have decimal expansions that are non-terminating and do not repeat.

### Conclusion
After analyzing all four statements, we conclude that the correct statement is:
- **Option 4**: "The decimal expansion of an irrational number is non-terminating and non-repeating."