Final answer:
Bayani should use the definition of rational numbers to prove that the sum of any two rational numbers is also a rational number by showing that the sum can be expressed as a fraction with an integer numerator and a non-zero integer denominator.
Step-by-step explanation:
The best method Bayani could use to continue proving that the sum of any two rational numbers is a rational number is by using the definition of rational numbers. Specifically, he should demonstrate that the sum of two rational numbers, which by definition can be written as fractions with integer numerators and non-zero integer denominators, can be expressed as another fraction with the same properties.
Bayani can demonstrate this by taking any two rational numbers, \(\frac{a}{b}\) and \(\frac{c}{d}\), where a, b, c, and d are integers with b and d non-zero. He can then find a common denominator and add the two numerators. The result is again an integer numerator over an integer denominator, thus proving that the sum is rational:
\(\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}\)
Here, assuming b and d are non-zero, bd is also non-zero. Since the addition and multiplication of integers yield integers, ad+bc and bd are integers, confirming the sum is a rational number.