Positive numbers

# Positive numbers

## Definition

Typically, a positive number is a real number $p$ such that, using the usual strict total order on the real line,

(1)$p \gt 0$

For emphasis, such a number may be called strictly positive or positive definite.

A real number $p$ is nonnegative if

(2)$p \geq 0 ,$

using the usual total order on the real line. The only nonnegative number that is not positive is zero. Such a number may also be called weakly positive or positive semidefinite; sometimes (especially in French), it is called simply ‘positive’.

The formulas (1) and (2) may be used as well for $p$ taken from various extensions and restrictions of the real line. For example:

• If $p$ is a rational number, then we may use the usual orders on the set of rational numbers, or equivalently interpret $p$ as a special real number and use the definitions for a real number. Thus positive rational numbers specialise positive real numbers. (The same thing goes for other subalgebras? of the real line.)
• If $p$ is a complex number, then we may use the usual strict preorder and partial order on the complex plane, or equivalently require first that $p$ is real and then use the definitions for a real number. Thus positive complex numbers are the same as positive real numbers. However, the term ‘nonnegative’ should not be used here. (The same thing goes for other hypercomplex extensions of the real line.)
• If $p$ is a surreal number, then we use the usual orders on the surreal line; we may not assume that $p$ is real this time. Thus positive surreal numbers generalise positive real numbers. (The same thing goes for other hyperreal? extensions of the real line.)

The set of (say real) positive numbers may be denoted $\dot{\mathbb{R}}^+$, $]0,\infty[$, $\mathbb{R}_{\gt0}$, or in many other ways; the corresponding notations for the set of nonnegative numbers are $\bar{\mathbb{R}}^+$, $[0,\infty[$, and $\mathbb{R}_{\geq0}$, while unadorned $\mathbb{R}^+$ is ambiguous.

## Applications

Karl Weierstraß's epsilontic analysis relies on repeatedly quantifying over the (strictly) positive real numbers. These positive numbers are typically denoted $\epsilon$ and $\delta$, hence the term ‘epsilontics’. This quantification replaces the use of (individual) positive infinitesimal numbers in intuitive pre-epsilontic analysis (or in rigorous post-epsilontic nonstandard analysis). In predicative mathematics, one may prefer to quantify over positive rational numbers, or even over the numbers $1/n$ for $n$ a positive integer, instead. (The important thing is to have a small set of positive numbers whose topological closure includes zero.)

Last revised on December 26, 2023 at 00:44:45. See the history of this page for a list of all contributions to it.