Proving √2 Irrational via Well-Ordering

Intro to Proofs · Axiom Academy

EXAMPLE Proving is Irrational via Well-Ordering An elegant alternative to the standard parity proof, using the principle of minimal counterexamples Show that is irrational. Instead of the usual even/odd argument, assume where q is the smallest positive denominator that works, then build a representation with an even smaller denominator — an impossibility. From a denominator q we construct q' = p - q . Because forces q < p < 2q , the new denominator lands strictly between 0 and q — a smaller positive integer giving the same value . The two brackets have equal length p-q : copying that distance back from 0 pins q' strictly inside . Since q' is a positive integer denominator for smaller than q , the assumption that q was smallest collapses. You worked through an elegant alternative to the parity proof. Here is what makes the well-ordering approach powerful: Well-Ordering Principle: every non-empty set of positive integers has a smallest element — the foundation this proof leans on. Minimal Counterexample: by assuming a counterexample with the smallest possible denominator, we set up a target to contradict. Infinite Descent: producing a smaller denominator would force an impossible infinite descending chain of positive integers. The Key Identity: comes from a clean algebraic rearrangement, not the even/odd trick. Generalizes: the same descent proves is irrational for any non-perfect-square n .

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