Positive real numbers

In mathematics, the set of positive real numbers, $\mathbb{R}_{>0} = \left\{ x \in \mathbb{R} \mid x > 0 \right\}$, is the subset of those real numbers that are greater than zero.

In a complex plane, $\mathbb{R}_{>0}$ 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 numbers $z = |z| \mathrm{e}^{\mathrm{i}\varphi}$ with argument $\varphi = 0$.

Notation

Alternative to $\mathbb{R}_{>0}$, the non-standard symbols $\mathbb{R}_{+}$ and $\mathbb{R}^{+}$ are often used. However, this may lead to confusion, as some authors use them to denote the set of non-negative real numbers, $\mathbb{R}_{\geq 0} = \left\{ x \in \mathbb{R} \mid x \geq 0 \right\}$, which explicitly includes zero. The non-negative reals serve as the range for metrics, norms, and measures in mathematics.

Properties

The set $\mathbb{R}_{>0}$ is closed under addition, multiplication, and division. It inherits a topology from the real line and, thus, has the structure of a multiplicative topological group or of an additive topological semigroup.

For a given positive real number x, the sequence {xⁿ} of its integral powers has three different fates: When x ∈ (0, 1) the limit is zero and when x ∈ (1, ∞) the limit is infinity, while the sequence is constant for x = 1. The case x > 1 thus leads to an unbounded sequence.

$\mathbb{R}_{>0} = (0,1) \cup \{ 1 \} \cup (1,\infty)$ and the multiplicative inverse function exchanges the intervals. The functions floor, $\operatorname{floor} : [ 1 , \infty ) \to \mathbb{N},\, x \mapsto \lfloor x \rfloor$, and excess, $\operatorname{excess} : [ 1 , \infty ) \to (0,1),\, x \mapsto x - \lfloor x \rfloor$, have been used to describe an element $x \in \mathbb{R}_{>0}$ as a continued fraction $[ n_0; n_1, n_2, \ldots]$ which is a sequence of integers obtained from the floor function after the excess has been reciprocated. For $x \in \mathbb{Q}$ the sequence terminates with an exact fractional expression of $x$, and for quadratic irrational $x$ the sequence becomes a periodic continued fraction.

In the study of classical groups, for every $n \in \mathbb{N}$, there is a normal subgroup relation SL(n,ℝ) ◁ GL(n,ℝ), the general linear group, such that the quotient group is the positive real numbers. In this context $\mathbb{R}_{>0}$ is considered to be a Lie group.

Logarithmic measure

If $[a,b] \subseteq \mathrm{R}_{>0}$ is an interval, then $\mu([a,b]) = \log(b / a)$ determines a measure on certain subsets of $\mathbb{R}_{>0}$. In fact, it is an invariant measure with respect to multiplication $[a,b] \to [az,bz]$ by a $z \in \mathbb{R}_{>0}$. In the context of topological groups, this measure is an example of a Haar measure.

The utility of this measure is shown in its use for describing stellar magnitudes and noise levels in decibels, among other applications of the logarithmic scale. For purposes of international standards ISO 80000-3 the dimensionless quantities are referred to as level (logarithmic quantity).

See also

• Bohr-Mollerup theorem
• Dagum distribution
• Fransén-Robinson constant
• Fréchet mean
• Generalized mean
• Meijer G-function
• Nesbitt's inequality
• Semifield
• Silverman's game
• Stolarsky mean

References

• Joseph Kist and Sanford Leetsma (1970) "Additive semigroups of positive real numbers", Mathematische Annalen 188:214 to 18.