Order Properties of Real Numbers

Definition

There exists a nonempty subset $\mathbb{P}$ of $\mathbb{R}$, called the set of positive real numbers, that satisfies the following properties:

1. $a,b\in \mathbb{P}⟹a+b\in \mathbb{P}$
2. $a,b\in \mathbb{P}⟹ab\in \mathbb{P}$
3. If $a\in \mathbb{R}$, then exactly one of the following holds:

$$a\in \mathbb{P},a=0,-a\in \mathbb{P}$$

The first two properties ensure compatibility with addition and multiplication operations.

The last one is often called the tricothomy property

• If $a\in \mathbb{P}$, we write $a>0$ and say that $a$ is a real positive number.
• If $a\in \mathbb{P}\cup \{0\}$, we write $a\ge 0$ and say that $a$ is a real nonnegative number.
• If $-a\in \mathbb{P}$, we write $a<0$ and say that $a$ is a real negative number.
• If $-a\in \mathbb{P}\cup \{0\}$, we write $a\le 0$ and say that $a$ is a real nonpositive number.