Proving |ab| = |a||b|

Intro to Proofs · Axiom Academy

Systematically prove an identity using proof by cases on the signs of the variables. Excellent work! You've constructed a systematic proof by cases. Here's what we covered: Proof by cases: When a definition depends on conditions (like sign), split into all possibilities so nothing is missed. Exhaustive & mutually exclusive: The four cases covered every sign combination of a and b, with no overlap. Recognizing symmetry: Cases 2 and 3 share the same structure, so we referenced earlier work instead of repeating it. Each case separately: We proved the identity in each individual case before concluding it holds universally. Synthesis: The final step combined all cases into the general conclusion for all real numbers. This identify-the-cases, prove-each, synthesize approach is fundamental to mathematical proof — you'll use it throughout analysis, algebra, and beyond.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.