Properties of Real Numbers

The real numbers obey a standard list of algebraic and order properties -- these are the rules that justify every later algebraic manipulation in the chapter. For all $a,b,c\in R$:

Closure and associativity. $a+b\in R$, $ab\in R$; $(a+b)+c=a+(b+c)$ and $a(bc)=(ab)c$ (grouping does not matter for a finite sum or product).

Identities and inverses. $a+0=a$, $a(1)=a$; for every $a$, $a+(-a)=0$; for every $b\ne0$, $b\left(\dfrac1b\right)=1$.

Commutativity and distributivity. $a+b=b+a$, $ab=ba$; $a(b+c)=ab+ac$. …