Is zero a rational number? Can it be written in the form `(p)/(q)` where p and q are integers and q`ne`0?

Answer

To determine if zero is a rational number, we need to check if it can be expressed in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).

### Step-by-Step Solution:

1. **Understanding Rational Numbers**:
- A rational number is defined as any number that can be expressed as the quotient of two integers \( p \) and \( q \), where \( q \) is not equal to zero.

2. **Expressing Zero in the Form \( \frac{p}{q} \)**:
- We need to find integers \( p \) and \( q \) such that \( 0 = \frac{p}{q} \).
- We can choose \( p = 0 \) (since zero is an integer) and any non-zero integer for \( q \).

3. **Choosing Values for \( p \) and \( q \)**:
- Let’s take \( p = 0 \) and \( q = 1 \). This gives us:
\[ 0 = \frac{0}{1} \]
- Here, \( q \) is not zero, satisfying the condition \( q \neq 0 \).

4. **Generalizing the Expression**:
- We can also express zero as \( \frac{0}{2} \), \( \frac{0}{3} \), \( \frac{0}{4} \), etc., where \( q \) can be any non-zero integer.
- Therefore, zero can be expressed in the form \( \frac{p}{q} \) for multiple values of \( q \).

5. **Conclusion**:
- Since we can express zero as \( \frac{p}{q} \) with \( p = 0 \) and \( q \) being any non-zero integer, we conclude that zero is indeed a rational number.

### Final Answer:
Yes, zero is a rational number because it can be written in the form \( \frac{p}{q} \) where \( p = 0 \) and \( q \) is any non-zero integer.

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