ordered field

Ordered field

A field $(F,+,\cdot )$ with a linear order $\le$ such that:

1. $a\le b⟹a+c\le b+c$ (compatible with addition)
2. $0\le a$ and $0\le b⟹0\le a\cdot b$ (compatible with multiplication)

is called an ordered field.

The natural numberss $\mathbb{N}$ form neither an ordered ring nor an ordered field (no additive inverses).
The integers $\mathbb{Z}$ form an ordered ring with 1, but not an ordered field (not every element has a multiplicative inverse in $\mathbb{Z}$).
The rational numbers $\mathbb{Q}$ form an ordered field with the usual order $\le$.
The real numbers $\mathbb{R}$ form an ordered field with the usual order $\le$.
The complex numbers $\mathbb{C}$ do not form an ordered field:

$\mathbb{C}$ is not an ordered field

$\mathbb{C}$ can be totally ordered (e.g., compare real parts first, then imaginary parts if tied). But no ordering satisfies both axioms above.

Multiplication breaks it: Consider $i$. Either $i>0$ or $i<0$:

• If $i>0$: then $i\cdot i>0$, so $-1>0$.
• If $i<0$: then $-i>0$, so $(-i)(-i)>0$, meaning $-1>0$.

Addition breaks it: If $-1>0$, adding $1$ to both sides gives $0>1$. But $1>0$. Contradiction.

Any geometric ordering (by angle, radius, etc.) fails similarly.