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.
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