: Mathématiques: positive real number

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positive real number

In mathematics, the set of positive real numbers, ${\\displaystyle \\mathbb {R} _{>0}=\\left\\{x\\in \\mathbb {R} \\mid x>0\ ight\\},}$ is the subset of those real numbers that are greater than zero. The non-negative real numbers, ${\\displaystyle \\mathbb {R} _{\\geq 0}=\\left\\{x\\in \\mathbb {R} \\mid x\\geq 0\ ight\\},}$ 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 ${\\displaystyle \\left\\{x\\in \\mathbb {R} \\mid x\\geq 0\ ight\\}}$ and ${\\displaystyle \\mathbb {R} _{+}^{*}}$ or ${\\displaystyle \\mathbb {R} _{*}^{+}}$ for ${\\displaystyle \\left\\{x\\in \\mathbb {R} \\mid x>0\ ight\\}}$ 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, ${\\displaystyle \\mathbb {R} _{>0}}$ 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 ${\\displaystyle \\varphi =0.}$