Definition:Strictly Negative
Definition
Let $\struct {R, +, \circ, \le}$ be an ordered ring whose zero is $0_R$.
Then $x \in R$ is strictly negative if and only if:
- $x \le 0_R$ and $x \ne 0_R$
That $x$ is strictly negative may be (more conveniently) denoted $0_R < x$ or $x > 0_R$.
Thus, the set of all strictly negative elements of $R$ is denoted:
- $R_{<0_R} := \set {x \in R: x < 0_R}$
Numbers
The concept of strictly negative can be applied to the following sets of numbers:
- $(1): \quad$ The integers $\Z$
- $(2): \quad$ The rational numbers $\Q$
- $(3): \quad$ The real numbers $\R$
Integers
The strictly negative integers are the set defined as:
| \(\ds \Z_{< 0}\) | \(:=\) | \(\ds \set {x \in \Z: x < 0}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \set {-1, -2, -3, \ldots}\) |
That is, all the integers that are strictly less than zero.
Rational Numbers
The strictly negative rational numbers are the set defined as:
- $\Q_{< 0} := \set {x \in \Q: x < 0}$
That is, all the rational numbers that are strictly less than zero.
Real Numbers
The strictly negative real numbers are the set defined as:
- $\R_{<0} := \set {x \in \R: x < 0}$
That is, all the real numbers that are strictly less than zero.
Also known as
The notation $R_-^*$ is frequently seen for $R_{< 0_R}$, that is for $\set {x \in R: 0_R > x}$.
However, the notations $R_-$ and $R^-$ are also frequently seen for both $\set {x \in R: 0_R \ge x}$ and $\set {x \in R: 0_R > x}$, and so suffer badly from ambiguity.
Some treatments of this subject reserve the term negative to define $x \in R$ where $0_R > x$, using the term non-positive for what is defined on this website as negative.
With the conveniently unambiguous notation that has been adopted on this site, the distinction between the terms loses its importance, as the symbology removes the confusion.
Also see
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Ordered and Well-Ordered Integral Domains: $\S 7$. Order