This is an old revision of the document!

## Arithmetic structure of real numbers

### Definition

 $\langle \mathbb R,+_\mathbb{R},\cdot_\mathbb{R} \rangle$
 $r +_\mathbb{R} s = \{q+_\mathbb{Q}p\ |\ q\in r\land p\in s\}$ $r -_\mathbb{R} s = \{q-_\mathbb{Q}p\ |\ q\in r\land p\in \mathbb Q\setminus s\}$ $-_{\mathbb R}r = \{q-_\mathbb{Q}p\ |\ q<0\land p\in \mathbb Q\setminus r\}$
 $r\ge 0\land s\ge 0\implies r\cdot_\mathbb{R}s = \{q\cdot_\mathbb{Q}p\ |\ (q\in r)\land (p\in s)\land (q,p\ge 0)\}\cup\{q\ |\ (q\in\mathbb Q)\land (q<0)\}$ $r\ge 0\land s < 0\implies r\cdot_\mathbb{R}s = -(r\cdot_\mathbb{R}(-s))$ $r < 0\land s\ge 0\implies r\cdot_\mathbb{R}s = -((-r)\cdot_\mathbb{R}s)$ $r < 0\land s < 0\implies r\cdot_\mathbb{R}s = (-r)\cdot_\mathbb{R}(-s)$
 $r\ge 0\land s > 0\implies r/_\mathbb{R}s = \{q/_\mathbb{Q}p\ |\ (q\in r)\land (p\in \mathbb Q\setminus s)\}$ $r\ge 0\land s < 0\implies r/_\mathbb{R}s = -(r/_\mathbb{R}(-s))$ $r < 0\land s > 0\implies r/_\mathbb{R}s = -((-r)/_\mathbb{R}s)$ $r < 0\land s < 0\implies r/_\mathbb{R}s = (-r)/_\mathbb{R}(-s)$

The operations $+_\mathbb{Q}$ and $\cdot_\mathbb{Q}$ on the right hand sides are these of arithmetic structure of rational numbers.