Definition:Positive

Definition
Let $\left({R, +, \circ, \le}\right)$ be an ordered ring whose zero is $0_R$.

Then $x \in R$ is positive iff $0_R \le x$.

The set of all positive elements of $R$ is denoted:


 * $R_{\ge 0_R} := \left\{{x \in R: 0_R \le x}\right\}$

Also known as
The notations $R_+$ and $R^+$ are frequently seen for $\left\{{x \in R: 0_R \le x}\right\}$.

However, these notations are also used for $\left\{{x \in R: 0_R < x}\right\}$, that is, $R_{> 0_R}$, and so suffer from being ambiguous.

Also defined as
Some treatments of this subject use the term define non-negative to define $x \in R$ where $0_R \le x$, reserving the term positive for what is defined on this website as strictly positive.

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

 * Definition:Strictly Positive
 * Definition:Negative
 * Definition:Strictly Negative