Product of two rational numbers is

Options

A. always rational
B. rational or irrational
C. always irrational
D. None of these

Answer

To solve the question regarding the product of two rational numbers, let's break it down step by step.

### Step-by-Step Solution:

1. **Understanding Rational Numbers**:
- A rational number is defined as a number that can be expressed in the form \( \frac{P}{Q} \) where \( P \) and \( Q \) are integers and \( Q \neq 0 \).
- Let's denote two rational numbers as \( A \) and \( B \).
- We can express them as:
\[ A = \frac{P_1}{Q_1} \quad \text{and} \quad B = \frac{P_2}{Q_2} \]
- Here, \( P_1, P_2 \) are the numerators and \( Q_1, Q_2 \) are the denominators of the rational numbers.

2. **Calculating the Product**:
- The product of the two rational numbers \( A \) and \( B \) is given by:
\[ A \times B = \frac{P_1}{Q_1} \times \frac{P_2}{Q_2} \]
- To multiply these fractions, we multiply the numerators together and the denominators together:
\[ A \times B = \frac{P_1 \times P_2}{Q_1 \times Q_2} \]

3. **Defining New Variables**:
- Let \( P = P_1 \times P_2 \) and \( Q = Q_1 \times Q_2 \).
- Thus, we can rewrite the product as:
\[ A \times B = \frac{P}{Q} \]

4. **Conclusion**:
- Since \( P \) is the product of two integers and \( Q \) is the product of two non-zero integers (because \( Q_1 \) and \( Q_2 \) cannot be zero), \( Q \) is also non-zero.
- Therefore, \( \frac{P}{Q} \) is in the form of a rational number.
- This shows that the product of two rational numbers is also a rational number.

### Final Answer:
The product of two rational numbers is always a rational number.

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