The nine axioms of algebra are:
- Associativity (distributive law)
- Identity element for addition (e.g. zero)
- Identity element for multiplication (e.g. 1)
- Additive inverses (i.e. negatives)
- Multiplicative inverses (i.e. reciprocals)
- Commutative law for addition
- Commutative law for multiplication
- Total ordering
- (Least) upper bounds for sets.
Axioms 1 to 5 define groups. If we add 6 and 7, we define fields. If we also add 8, we define an algebra. If we also add 9, we get completely ordered fields.
Regarding numbers: 1 to 9 defines real numbers, 1 to 8 defines rational numbers, 6 and 7 transform groups to Abelian groups, 5 introduces rational numbers (fractions), 4 introduces negative integers, 2 introduces zero.