Definition:Absolute Value/Ordered Integral Domain
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Definition
Let $\struct {D, +, \times, \le}$ be an ordered integral domain whose zero is $0_D$.
Then for all $a \in D$, the absolute value of $a$ is defined as:
- $\size a = \begin{cases} a & : 0_D \le a \\ -a & : a < 0_D \end{cases}$
where $a > 0_D$ denotes that $\neg \paren {a \le 0_D}$.
Also see
- Integers form Ordered Integral Domain
- Rational Numbers form Ordered Integral Domain
- Real Numbers form Ordered Integral Domain
from which it follows that the definition for numbers is compatible with this.
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Ordered and Well-Ordered Integral Domains: $\S 7$. Order