Algebraic Manipulation in ℝ

Real Analysis · Axiom Academy

EXAMPLE Algebraic Manipulation in ℝ Proving the zero-product property — if ab = 0 then a = 0 or b = 0 — from the field axioms alone In we take it for granted that a product is zero only when one of the factors is zero — that is what lets us solve x(x-3)=0 by reading off x=0 or x=3 . But this is a theorem , not an assumption. Prove it directly from the field axioms: if ab = 0 , then a = 0 or b = 0 . Every line of the proof was justified by one of these axioms of : Every nonzero element has an inverse that multiplies with it to give 1 . The grouping of a product can be rearranged freely. Multiplying by 1 leaves any element unchanged. A consequence of distributivity: any element times 0 is 0 . You proved the zero-product property using nothing but the field axioms of — no appeal to "obviousness," every step justified. Split into cases: if a = 0 the conclusion holds immediately, so the real work is the case , where we showed b = 0 . A nonzero hypothesis unlocks the inverse: the existence of a^ -1 — available only because — is the lever that drives the whole argument. Cite an axiom at every step: a rigorous proof names the inverse, associativity, identity, and zero properties as it uses them. This is why factoring works: reading roots off x(x-3)=0 is exactly this theorem in action. It needs a field: in a ring with zero divisors (e.g. , where ) the property fails — it is the inverses of that rule zero divisors out.

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