The field axioms
An ordered field is the starting point of real analysis. A field is a set with two operations, addition and multiplication, satisfying the field axioms: both operations are commutative and associative, multiplication distributes over addition, there are distinct identities 0 and 1, every element has an additive inverse, and every nonzero element has a multiplicative inverse. The rationals Q, the reals R, and the complex numbers C are all fields. The integers Z are not — 2 has no multiplicative inverse inside Z.
Everything you learned in school algebra — cancelling, moving terms across an equals sign, that a product is zero only if a factor is zero — is a *theorem* provable from these axioms, not an extra rule. That is the spirit of rigor: a short, fixed list of assumptions, and every later fact earned by proof.
Claim: in any field, x*0 = 0 for every x.
Proof.
x*0 = x*(0+0) (0 is the additive identity)
= x*0 + x*0 (distributive law)
Add the inverse -(x*0) to both sides:
x*0 + (-(x*0)) = (x*0 + x*0) + (-(x*0))
0 = x*0 + (x*0 + -(x*0)) (associativity)
0 = x*0 + 0
0 = x*0. QED
Nothing about 'numbers' was used — only the axioms.Adding an order
A field becomes an ordered field when we single out a set P of *positive* elements that is closed under addition and multiplication, and such that for every x exactly one of these holds: x is in P, x = 0, or -x is in P. We then *define* x < y to mean y - x is in P. From this, all of the inequality rules follow: you may add the same quantity to both sides, multiply both sides by a positive number, and the direction flips when you multiply by a negative.
One pretty consequence: squares are never negative. If x is not 0, then either x or -x is positive, and a positive times itself is positive, so x*x > 0. In particular 1 = 1*1 > 0. This already rules out the complex numbers as an ordered field, since there i*i = -1 < 0.