Positive real numbers

In mathematics, the set of positive real numbers, ${\displaystyle \mathbb {R} _{>0}=\left\{x\in \mathbb {R} \mid x>0\right\},}$ is the subset of those real numbers that are greater than zero. The non-negative real numbers, ${\displaystyle \mathbb {R} _{\geq 0}=\left\{x\in \mathbb {R} \mid x\geq 0\right\},}$ also include zero. Although the symbols ${\displaystyle \mathbb {R} _{+}}$ and ${\displaystyle \mathbb {R} ^{+}}$ are ambiguously used for either of these, the notation ${\displaystyle \mathbb {R} _{+}}$ or ${\displaystyle \mathbb {R} ^{+}}$ for ${\displaystyle \left\{x\in \mathbb {R} \mid x\geq 0\right\}}$ and ${\displaystyle \mathbb {R} _{+}^{*}}$ or ${\displaystyle \mathbb {R} _{*}^{+}}$ for ${\displaystyle \left\{x\in \mathbb {R} \mid x>0\right\}}$ 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.^[1]

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