Addition and Subtraction of Rational Numbers: Like vs Unlike Denominators and Properties

Adding and subtracting rational numbers are critical algebraic operations with systematic procedural rules. When adding or subtracting rational numbers with identical denominators, we simply combine the numerators over the shared denominator: (a/c) ± (b/c) = (a ± b)/c. When denominators differ, we must find the least common denominator (LCD), rewrite each fraction as an equivalent fraction, and then execute the arithmetic. We examine the algebraic properties of addition in Q: Closure (the sum of any two rational numbers is always rational), Commutative property (a + b = b + a), Associative property ((a + b) + c = a + (b + c)), the Additive Identity (0), and the Additive Inverse property (a + (-a) = 0). Subtraction is conceptualized as adding the additive inverse: a - b = a + (-b).