Decimal representation of a rational cannot be

Options

A. terminating
B. non-terminating non- repeating
C. non-terminating repeating
D. none of these

Answer

To solve the question regarding the decimal representation of a rational number, we need to understand the properties of rational numbers and their decimal forms.

### Step-by-Step Solution:

1. **Definition of Rational Numbers**:
A rational number is defined as any number that can be expressed in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).

**Hint**: Remember that rational numbers can be fractions where the denominator is not zero.

2. **Decimal Representation of Rational Numbers**:
The decimal representation of a rational number can either terminate or repeat.

**Hint**: Think about how fractions can be converted to decimals; some will end (terminate) while others will go on (repeat).

3. **Terminating Decimals**:
A decimal is said to be terminating if it has a finite number of digits after the decimal point. For example, \( 0.75 \) is a terminating decimal.

**Hint**: Identify examples of fractions that result in finite decimal expansions.

4. **Repeating Decimals**:
A decimal is repeating if it has one or more digits that repeat indefinitely. For example, \( 0.333... \) (which represents \( \frac{1}{3} \)) is a repeating decimal.

**Hint**: Consider how some fractions lead to decimals that never end but keep repeating a certain pattern.

5. **Conclusion**:
Therefore, the decimal representation of a rational number cannot be non-terminating and non-repeating. Such decimals are characteristic of irrational numbers.

**Hint**: Recall that irrational numbers cannot be expressed as a fraction of two integers and have decimal forms that neither terminate nor repeat.

### Final Answer:
The decimal representation of a rational number cannot be non-terminating and non-repeating.