Properties of Ordered Field

Theorem
Let $\struct {k, +, \cdot}$ be a totally ordered field with unity $1$, zero $0$.

Denote the strict order by $<$ and the weak order by $\leq$.

Let $\Char k$ denote the characteristic of $k$.

Then the following hold for all $x, y, z \in k$:


 * $(1): \quad x < 0 \iff -x > 0$
 * $(2): \quad x > y \iff x-y > 0$
 * $(3): \quad x < y \iff -x > -y$
 * $(4): \quad \paren {z < 0} \land \paren {x < y} \implies xz > yz$
 * $(5): \quad x \neq 0 \implies x^2 > 0$
 * $(6): \quad 1 > 0$
 * $(7): \quad \Char k = 0$
 * $(8): \quad x > y > 0 \iff y^{-1} > x^{-1} > 0$

Proof
By definition of ordering, the relation $\le$ is:
 * reflexive
 * transitive
 * antisymmetric

and furthermore, every pair of elements is comparable.

The order is compatible with $k$ in the sense that, for all $x, y, z, c \in k$:
 * $x < y \implies x + z < y + z$
 * $c > 0,\ x < y \implies c x < c y$

The proof is by repeated deduction from these properties.


 * $(1): \quad x < 0 \iff -x > 0$:

Conversely:


 * $(2): \quad x > y \iff x-y > 0$:

Conversely:


 * $(3): \quad x < y \iff -x > -y$:

Conversely:


 * $(4): \quad (z < 0) \land (x < y) \implies xz > yz$:

By parts 1 and 3 above, if $z < 0$, $x < y$ then $-z > 0$ and $-x > -y$.

Then:
 * $x z = \paren {-x} \paren {-z} > \paren {-y} \paren {-z} = y z$

If $x > 0$, then:
 * $(5): \quad x \neq 0 \implies x^2 > 0$:
 * $x^2 = x \cdot x > x \cdot 0 = 0$

If $x < 0$, then by 1, $-x > 0$, so:
 * $x^2 = \paren {-x} \cdot \paren {-x} > \paren {-x} \cdot 0 = 0$


 * $(6): \quad 1 > 0$:

This is immediate from $(5)$, noting that $1 = 1^2$ is a square.


 * $(7): \quad \Char k = 0$:

By $(6)$, we have:
 * $0 < 1 < 1 + 1 < 1 + 1 + 1 < \cdots$

so $n \cdot 1 \neq 0$ for all $n \in \N$.


 * $(8): \quad x > y > 0 \iff y^{-1} > x^{-1} > 0$:

First let $x > 0$, and suppose that $x^{-1} < 0$.

Then by $(4)$:
 * $0 = 0 \cdot x^{-1} > x \cdot x^{-1} = 1$

which contradicts $(6)$, so $x^{-1} > 0$.

Now let $x > y > 0$.

Then:

The converse follows upon interchanging $x^{-1} \leftrightarrow x$ and $y^{-1} \leftrightarrow y$.