Properties and Axioms

Intro to Proofs · Axiom Academy

The field and order axioms that govern real numbers — and knowing when a proof step must cite them. When you write a proof, every step must be justified — but justified with what ? You can't prove everything, or you'd fall into infinite regress. Instead, mathematics rests on a small base of axioms we simply accept as true, and everything else is built on top of them. The real numbers form a field : addition and multiplication obey a fixed set of rules. These are the properties you'll cite most often when justifying algebra. Beyond arithmetic, the reals carry an order . The order axioms govern how inequalities behave, and they're essential for any proof involving < and >. Trichotomy: for any , exactly one holds: Multiplying by a positive preserves order: 4. When to Cite: The Art of Proof Writing Here proof writing becomes an art. Not every tiny step needs explicit justification — your proof would be unreadable. The skill is knowing your audience and what they'll accept as "obvious." Combining like terms (2x + 3x = 5x) Well-known facts (√2 is irrational) Direct definition applications 5. Example: A Properly Justified Proof Let's put it together with a short proof that cites an axiom at every non-obvious step. (multiply both sides by 2; since 2 > 0, order is preserved) (add 3 to both sides; addition preserves order) (rewrite each side; distributive property)

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