Properties of the Absolute Value in an Ordered Field

Let $F$ be an ordered field and let $x,y,c\in F$. Absolute values are as in that definition; $x-y$ abbreviates $x+(-y)$ and $x^{2}$ abbreviates $x\cdot x$. For $s,t\in F$ we write $s<t$ to mean that $s\le t$ and $s\ne t$. Then the following hold.

1. (Nonnegativity)(i) $|x|$ equals $x$ or $-x$; moreover $0\le|x|$, and $|x|=0$ if and only if $x=0$.

2. (Symmetry)(ii) $|-x|=|x|$.

3. (Bounds by the absolute value)(iii) $-|x|\le x$ and $x\le|x|$.

4. (Multiplicativity)(iv) $|xy|=|x|\,|y|$.

5. (Triangle inequality)(v) $|x+y|\le|x|+|y|$.

6. (Two-sided bound)(vi) $|x|\le c$ holds if and only if both $-c\le x$ and $x\le c$ hold.

7. (Reverse triangle inequality)(vii) $\bigl||x|-|y|\bigr|\le|x-y|$.

8. (Agreement with the complex modulus)(viii) If $F$ is the field of real numbers, then for every $x\in F$ the absolute value $|x|$ equals the modulus of $x$ regarded as a complex number.

9. (Strict two-sided bound)(ix) $|x|<c$ holds if and only if both $-c<x$ and $x<c$ hold.