A visual proof that a^2 – b^2 = (a + b)(a – b)

Original Article ↗ Hacker News Discussion ↗

Scope and validity of the visual proof

  • Many note the diagram only obviously covers (a > 0, b > 0, a > b).
  • Critics argue this makes it at best a partial proof, since the algebraic identity holds more generally (e.g., for all reals, or even any commutative ring).
  • Others respond that the goal is to convey the core idea, not to cover every case or abstract algebraic setting.

Handling negative, zero, and swapped values

  • Debate over whether one can assume (b < a) “without loss of generality”:
    • One side: you can handle (b > a) by swapping labels and adjusting signs.
    • Other side: that step is itself algebraic work and not present in the picture, so the visual argument is incomplete.
  • Some attempt to extend the picture using signed/negative areas or “oriented area”; others find negative area visually unintuitive.
  • Edge cases like (a = 0), (b = 0), (a = b), and negative inputs are discussed; consensus that the picture doesn’t transparently handle them.

Visual proofs vs algebraic proofs

  • Several comments stress that visual arguments can be deceptive (e.g., “missing square” puzzles, bogus “(\pi = 4)” constructions).
  • Others emphasize that this diagram is best seen as an illustration or intuition pump, not a fully formal proof.
  • Some argue that once you rely on algebraic clean-up for edge cases, you might as well just do the full algebraic proof via the distributive law.

Teaching, intuition, and cognition

  • Many wish they had seen such diagrams in school; they help connect algebra and geometry and make memorized identities feel meaningful.
  • Others report the opposite: algebra feels natural, while geometric reasoning does not.
  • Teachers’ practices vary: some avoid visual proofs to maintain rigor, others think multiple representations (symbolic and visual) deepen understanding.

Related concepts and resources

  • Discussion touches on area-as-multiplication, integration as “area under a curve,” and signed/oriented areas simplifying geometric reasoning.
  • Links and references are shared to “proofs without words,” visual math sites, YouTube channels, and Pythagorean theorem visual proofs.