The discussion clarifies that the equation 3 x 2 = 2 x 3 illustrates the commutative law of multiplication, not the distributive law, which is defined as a x (b + c) = a x b + a x c. Proving these laws varies based on the number system used, such as natural numbers, integers, rational numbers, and real numbers, with different complexities involved in each. The most challenging proof involves constructing real numbers from rational numbers, as detailed in Rudin's "Principles of Mathematical Analysis." Some prefer an axiomatic approach to real numbers, where the distributive law is accepted as an axiom rather than proven. Understanding these concepts often requires a foundational knowledge of set theory and the definitions of real numbers.
