In
mathematics, the set of positive real numbers,
is the subset of those
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s that are greater than zero. The non-negative real numbers,
also include zero. Although the symbols
and
are ambiguously used for either of these, the notation
or
for
and
or
for
has also been widely employed, is aligned with the practice in algebra of denoting the exclusion of the zero element with a star, and should be understandable to most practicing mathematicians.
In a
complex plane,
is identified with the positive real axis, and is usually drawn as a horizontal
ray. This ray is used as reference in the
polar form of a complex number. The real positive axis corresponds to
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s
with
argument
Properties
The set
is
closed under addition, multiplication, and division. It inherits a
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
from the
real line and, thus, has the structure of a multiplicative
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
or of an additive
topological semigroup.
For a given positive real number
the
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of its integral powers has three different fates: When
the
limit is zero; when
the sequence is constant; and when
the sequence is
unbounded.
and the
multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/ ...
function exchanges the intervals. The functions
floor,