Part 3/9:
Negative times a negative should be positive because flipping twice cancels out, leaving you directionally unchanged.
While intuitive, this idea isn't a proof. To understand why it must be true, we need to delve into the formal logic and rules that govern numbers.
Building a Rigorous Foundation Using Algebra
Mathematicians base their reasoning on a set of agreed-upon rules called axioms. These axioms include properties like associativity, distributivity, and the existence of additive inverses.
One foundational fact is that -2 + 2 = 0. This expresses that negative two and positive two cancel each other out, leaving zero.
Now, to analyze multiplication involving negatives, consider the rule of distributivity:
[A \times (B + C) = A \times B + A \times C]