Field and Order Axioms

Real Analysis · Axiom Academy

Two short lists of rules — the field axioms and the order axioms — pin down exactly what it means to be a real number, before we ever mention limits. 1. The Field Axioms — Arithmetic That Never Leaves the Set A set F with two operations + and is a field when both operations are commutative , associative , have an identity ( 0 for + , 1 for ), and have inverses (every a has -a ; every has a^ -1 ) — and multiplication distributes over addition. Watch the consequence below: starting from two numbers inside the field, every one of lands you right back inside it. Identities + inverses for both operations a+b and are always elements of F — no operation escapes the set. a+0=a and for every a . The two identities are distinct: . Every a has an opposite -a (so subtraction works); every has a reciprocal a^ -1 (so division works). a(b+c) = ab + ac — the one axiom that ties addition and multiplication together. 2. The Order Axioms — Putting the Numbers in a Line An order on a field is a relation obeying four rules. The first two arrange every number on a single line; the last two say the order respects the arithmetic — you can add to both sides freely, and multiply both sides by anything positive. The animation runs all four on the number line . For any a,b , exactly one holds: , a = b , or . If and , then . (A positive scaling keeps the order.)

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