Let $F$ be an \reftext{def:ordered-field-c54-2026b}{ordered field} and let $x,y,c\in F$. Absolute values are as in \reftext{def:absolute-value-ordered-field-2026a}{that definition}; $x-y$ abbreviates $x+(-y)$ and $x^{2}$ abbreviates $x\cdot x$. Then the following hold.

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

\textbf{2. (Symmetry)} $|-x|=|x|$.

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

\textbf{4. (Multiplicativity)} $|xy|=|x|\,|y|$.

\textbf{5. (Triangle inequality)} $|x+y|\le|x|+|y|$.

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

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

\textbf{8. (Agreement with the complex modulus)} If $F$ is the field of \reftext{def:real-numbers-c54-2026c}{real numbers}, then for every $x\in F$ the absolute value $|x|$ equals the \reftext{def:complex-modulus-2026a}{modulus} of $x$ regarded as a \reftext{def:complex-numbers-2026a}{complex number}.