The power and reliability of algebra come from its unwavering consistency. These are the definitive rules that govern the system.

1. The Axioms of Equality: These include the reflexive (a=a), symmetric (if a=b, then b=a), and transitive (if a=b and b=c, then a=c) properties.

2. The Commutative Rules: These state that the order doesn't matter for addition (a+b=b+a) and multiplication (a*b=b*a).

3. The Associative Rules: These state that the grouping doesn't matter for addition ( (a+b)+c=a+(b+c) ) and multiplication.

4. The Distributive Rule: This is the vital link between the two operations, stating that a(b+c) = ab + ac.

5. The Identity Rules: Adding 0 (the additive identity) or multiplying by 1 (the multiplicative identity) changes nothing.

6. The Inverse Rules: For every number 'a', there is an additive inverse '-a' that sums to zero, and a multiplicative inverse '1/a' that multiplies to 1.

7. The Order of Operations (PEMDAS): This is the strict, hierarchical order in which all calculations must be performed.

8. The Rules of Exponents: The six key rules for handling powers when multiplying, dividing, or raising a power to another power.

9. The Properties of Inequalities: The rules for how an inequality symbol behaves, especially the critical 'flip' rule when multiplying or dividing by a negative.