transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
Typically, a positive number is a real number $p$ such that, using the usual strict total order on the real line,
For emphasis, such a number may be called strictly positive or positive definite.
A real number $p$ is nonnegative if
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:
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.
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.