Definition:Negative
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Definition
Let $\left({R, +, \circ, \le}\right)$ be an ordered ring whose zero is $0_R$.
Then $x \in R$ is negative iff $x \le 0_R$.
The set of all negative elements of $R$ is denoted:
- $R_{\le 0_R} := \left\{{x \in R: x \le 0_R}\right\}$
Numbers
The concept of 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 $\R$
The complex numbers can not be ordered, so there is no such concept as a negative complex number.
Integers
As the set of integers is the Inverse Completion of Natural Numbers, it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the Unique Minus between the two elements of each tuple is the same.
Thus negative can be formally defined on $\Z$ as a relation induced on those equivalence classes as specified in the definition of integers.
That is, the integers being defined as all the Unique Minus congruence classes, negative can be defined directly as the relation specified as follows:
The integer $z \in \Z: z = \left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$ is negative iff $b > a$.
The set of negative integers is denoted $\Z_{\le 0}$.
An element of $\Z$ can be specifically indicated as being negative by prepending a $-$ sign:
- $-x \in \Z \iff x \in \Z_{\le 0}$
Rational Numbers
The negative rational numbers are the set defined as:
- $\Q_{\le 0} := \left\{{x \in \Q: x \le 0}\right\}$
That is, all the rational numbers that are less than or equal to zero.
Real Numbers
The negative real numbers are the set defined as:
- $\R_{\le 0} := \left\{{x \in \R: x \le 0}\right\}$
That is, all the real numbers that are less than or equal to zero.
Also known as
The notations $R_-$ and $R^-$ are also frequently seen for $\left\{{x \in R: x \le 0_R}\right\}$.
However, these notations are also used to denote $\left\{{x \in R: x < 0_R}\right\}$, i.e. $R_{< 0_R}$, and hence are ambiguous.
Some treatments of this subject use the term define non-positive to define $x \in R$ where $0_R \le x$, reserving the term negative for what is defined on this website as strictly 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
- C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 2.7$
