The properties of real numbers are the bedrock axioms upon which all of algebra is built. They are the ultimate 'rules of the game.'

1. The Commutative Properties: These say the order doesn't matter for addition or multiplication. (a+b = b+a) and (a*b = b*a).

2. The Associative Properties: These say the grouping doesn't matter for addition or multiplication. (a+b)+c = a+(b+c).

3. The Distributive Property: This is the crucial link between addition and multiplication. a(b+c) = ab + ac.

4. The Identity Properties: Adding 0 does nothing (Additive Identity). Multiplying by 1 does nothing (Multiplicative Identity).

5. The Inverse Properties: Every number 'a' has an opposite '-a' that adds to zero. Every non-zero 'a' has a reciprocal '1/a' that multiplies to 1.

You use these properties in every single step of every single problem, whether you realize it or not.

When you 'combine like terms,' you are using the distributive property.

When you 'cancel something out,' you are using the inverse property.

Knowing these properties by name isn't just for tests; it's about understanding the deep, fundamental structure of how numbers work.