Final answer:
Bayani should use the definition of rational numbers to prove that the sum of any two rational numbers is also rational by expressing the sum as a fraction with integer numerator and denominator, and providing an example to demonstrate the process.
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 to use the definition of rational numbers. Rational numbers are any numbers that can be expressed as a fraction ⅛, where ⅛ and ⅝ are integers, and ⅝ is not zero. To prove that the sum of these is also rational, consider two rational numbers ⅟⁄⅝ and ⅚⁄⅜. Their sum is ⅟⅝⁄⅝⅜ + ⅚⅜⁄⅝⅜, which simplifies to (⅟⅜ + ⅚⅝)⁄(⅝⅜), still in the form of a fraction of integers where the denominator is not zero, thus showing that the sum is rational.
Intuition can be a helpful guide in figuring out the rules of addition and subtraction, particularly with fractions. We know that the result should make sense and be reasonable, such as understanding that ½ + ¼ must be less than 1. This understanding helps verify that we need a common denominator, and it should be logical and fit within the bounds of what we know.
Remember some key points when working with whole numbers: the commutative property of addition (A + B = B + A), and to check that the answer is reasonable. These foundational principles apply to fractions as well. By using intuition and logical reasoning, we can also apply these principles successfully when adding or subtracting fractions.
It's important to eliminate terms wherever possible to simplify the algebra, and always check the answer to see if it is reasonable, confirming that the mathematical rules have been applied correctly.