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 determine which statement is correct regarding rational and irrational numbers, we will analyze each statement step by step.

### Step 1: Understanding Rational Numbers
Rational numbers are defined as numbers that can be expressed in the form of a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).

**Hint:** Rational numbers can be identified by their decimal expansions, which can either terminate (like 1.5) or repeat (like 0.333...).

### Step 2: Understanding Irrational Numbers
Irrational numbers cannot be expressed as a fraction of two integers. Their decimal expansions are non-terminating and non-repeating. Examples include numbers like \( \sqrt{2} \) or \( \pi \).

**Hint:** If a number's decimal goes on forever without repeating a pattern, it is likely irrational.

### Step 3: Analyzing the Statements
Now, let's evaluate each of the given statements:

1. **Decimal expansion of a rational number is terminating.**
- This statement is partially true; rational numbers can have terminating decimal expansions, but they can also have non-terminating repeating decimal expansions. Thus, this statement is not universally correct.

2. **Decimal expansion of a rational number is non-terminating.**
- This statement is incorrect. While some rational numbers have non-terminating decimal expansions, they must be repeating. Therefore, this statement is false.

3. **Decimal expansion of an irrational number is terminating.**
- This statement is false. By definition, irrational numbers cannot have terminating decimal expansions.

4. **Decimal expansion of an irrational number is non-terminating and non-repeating.**
- This statement is true. Irrational numbers have decimal expansions that do not terminate and do not repeat.

### Conclusion
The correct statement is: **Decimal expansion of an irrational number is non-terminating and non-repeating.**

**Final Answer:** Option 4 is the correct statement.