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 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 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 taken from various extensions and restrictions of the real line. For example:
The set of (say real) positive numbers may be denoted , , , or in many other ways; the corresponding notations for the set of nonnegative numbers are , , and , while unadorned is ambiguous.
Karl Weierstraß's epsilontic analysis relies on repeatedly quantifying over the (strictly) positive real numbers. These positive numbers are typically denoted and , 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 for 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.