Constructing √2

Real Analysis · Axiom Academy

Use the Completeness Axiom to build a real number whose square is exactly 2. The rational numbers have a "hole" where should be — no fraction squares to 2 . Working entirely inside the real numbers, prove that the set has a least upper bound , and that . This is . is impossible — couldn't be an upper bound. is impossible — couldn't be the least upper bound. You just built from the ground up. Here is what made it work: Existence from completeness: the Completeness Axiom turned " S is bounded above" into " exists" — the step cannot make. Trichotomy + contradiction: ruling out and forces . Explicit witnesses: the carefully chosen and made each contradiction concrete, not hand-waved. Result: is a real number, not just a symbol. The same supremum technique constructs for any — and more broadly, it is how the completeness of manufactures the limits, roots, and suprema that analysis relies on.

This is the written version of the interactive lesson above. See the full Real Analysis course.