Properties of real numbers
Which rules about adding and multiplying hold for every real number at once, and why can algebra rely on them?
Where you are 3 inside
What this area is about
Algebra is mostly rewriting: an expression or an equation turned into a handier one that is worth the same. Trying numbers can never show a rewrite is safe for every value of the letters. A short list of properties true of all real numbers can, and this area names them so that each step of a simplification or a solution has a reason.
How these fit together
Laws for rewriting. Commutative, associative and distributive properties: order and grouping do not change a sum or a product, and a(b + c) = ab + ac for all real a, b and c.
Numbers that do nothing, and numbers that undo. Identity and inverse elements: adding 0 and multiplying by 1 change nothing; every real a has an opposite with a + (-a) = 0, and every a \ne 0 has a reciprocal with \displaystyle a \cdot \frac{1}{a} = 1. Zero has no reciprocal.
Staying inside a set. Closure asks whether an operation on two members of a set always gives a member: the whole numbers are closed under addition but not under subtraction.
A path through it
There is no forced order. The rewriting laws come first naturally, then identities and inverses, which explain each step of solving an equation. Closure comes last, since it uses sorting numbers into sets and subtracting integers.
Finish here and every rewrite in later algebra can name the property that allows it.