Properties of Real Numbers

These properties name the moves you already make.

Commutative: order does not matter for + and ×. 3 + 5 = 5 + 3.
Associative: grouping does not matter for + and ×. (2 + 3) + 4 = 2 + (3 + 4).
Distributive: a(b + c) = ab + ac.
Identity: adding 0 or multiplying by 1 changes nothing.
Inverse: a + (−a) = 0 and a × 1/a = 1.

Two warnings. Subtraction and division are neither commutative nor associative: 5 − 3 is not 3 − 5.

These matter because when a proof or a teacher asks "why are you allowed to do that," these are the answers.

The reals form a field: an abelian group under addition, the nonzero reals an abelian group under multiplication, and multiplication distributes over addition. Every algebraic manipulation you will ever perform is a finite chain of these axioms plus the properties of equality.

Distributivity is the only axiom linking the two operations, which is why it does so much work — factoring, expanding, combining like terms and the FOIL pattern are all it.

Subtraction and division are not separate operations but shorthand for adding an inverse and multiplying by an inverse. This explains their failure to commute directly: a − b = a + (−b) and b + (−a) are genuinely different numbers.

The additive identity has no multiplicative inverse, which is the structural reason division by zero is undefined rather than merely inconvenient: no real number x satisfies 0·x = 1.