The Order Properties of Real Numbers

# The Order Properties of Real Numbers

We will now take a look at some more axioms regarding the field of real numbers $\mathbb{R}$. Be sure to first review the The Axioms of the Field of Real Numbers page first since we will still use these properties in proving subsequent theorems.

# The Order Axioms of the Field of Real Numbers

Let $P$ be a nonempty subset of $\mathbb{R}$ which we define as **the set of positive real numbers**, that satisfies the following properties:

**Axiom O1:**If $a, b \in P$ then $(a + b) \in P$.

**Axiom O2:**If $a, b \in P$ then $(ab) \in P$.

**Axiom O3:**If $a \in \mathbb{R}$ then only one of the following holds: $a \in P$ or $-a \in P$ or $a = 0$. (The Trichotomy Property)

Furthermore, we will also use the following notation to denote this order of the real number system.

- If $a \in P$ then we can write $a > 0$ (or $0 < a$).

- Furthermore, we write $a > b$ (or $b < a$) whenever $(a - b) \in P$, and we write $a ≥ b$ if $a > b$ or $a - b = 0$.

- If $-a \in P$ then $a < 0$ (or $0 > a$).

- Furthermore, we write $b > a$ (or $a < b$) whenever $(b - a) \in P$, and we write $b ≥ a$ if $b > a$ or $b - a = 0$.

Note: Some authors use the notation $\mathbb{R}^+$ to denote the set of positive real numbers as opposed to $P$. We will not use this notation in order to prevent confusion. |