To prove that the sum of two rational numbers is rational, we start with the definition of rational numbers. A rational number can be expressed as the quotient of two integers, where the denominator is not zero.
If we denote the two rational numbers as \( \frac{a}{b} \) and \( \frac{c}{d} \), where \( a, b, c, \) and \( d \) are integers and \( b \neq 0 \), \( d \neq 0 \), then the sum of these two rational numbers can be expressed as:
\[ \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \]
In this expression:
- \( a \) can be any integer since it represents the numerator of the first rational number.
- \( b \) is the denominator of the first rational number, where \( b \neq 0 \).
- \( c \) is the numerator of the second rational number.
- \( d \) is the denominator of the second rational number, where \( d \neq 0 \).
Given the context of your question, to use an appropriate value for \( a \) in the proof, you could select any integer, such as \( a = 0 \), which is also a rational number (\( \frac{0}{1} \)). This will allow you to express the sum of two rational numbers \( \frac{a}{b} \) and \( \frac{c}{d} \) in the required form.
In summary, you could use \( a = 0 \) or any integer value for \( a \) in your proof to show that the sum of the two rational numbers remains rational.