Ordered fields

An ordered field is a field $\mathbb{F}$ which is also an ordered set, such that for all $x,y,z\in \mathbb{F}$,

• If $y<z$, then $x+y<x+z$, and
• If $x>0,y>0$ then $xy>0$.

Note that a field which is ordered is NOT an ordered field.

If $x>0$, we call $x$ positive. If $x<0$, we call $x$ negative.

The following statements are true in every ordered field.

• If $x>0$, then $-x<0$ and vice versa.
• if $x>0$ and $y<z$, then $xy<xz$.
• If $x<0$ and $y<z$ then $xy>xz$.
• If $x\ne 0$ then $x^2>0$. In particular, $1>0$.
• If $0<x<y$, then $0<\frac{1}{y}<\frac{1}{x}$.