How does one denote the set of all positive real numbers?

What is the “standard” way to denote all positive (or non-negative) real numbers? I’d think

\mathbb R^+

but I believe that that is usually used to denote “all real numbers including infinity”.

So is there a standard way to denote the set

\{x \in \mathbb R : x \geq 0\} \; ?

Answer

The unambiguous notations are: for the positive-real numbers

\mathbb{R}_{>0} = \left\{ x \in \mathbb{R} \mid x > 0 \right\} \,,

and for the non-negative-real numbers

\mathbb{R}_{\geq 0} = \left\{ x \in \mathbb{R} \mid x \geq 0 \right\} \,.

Notations such as \mathbb{R}_{+} or \mathbb{R}^{+} are non-standard and should be avoided, becuase it is not clear whether zero is included. Furthermore, the subscripted version has the advantage, that n-dimensional spaces can be properly expressed. For example, \mathbb{R}_{>0}^{3} denotes the positive-real three-space, which would read \mathbb{R}^{+,3} in non-standard notation.


Addendum:

In Algebra one may come across the symbol \mathbb{R}^\ast, which refers to the multiplicative units of the field \big( \mathbb{R}, +, \cdot \big). Since all real numbers except 0 are multiplicative units, we have

\mathbb{R}^\ast = \mathbb{R}_{\neq 0} = \left\{ x \in \mathbb{R} \mid x \neq 0 \right\} \,.

But caution! The positive-real numbers can also form a field, \big( \mathbb{R}_{>0}, \cdot, \star \big), with the operation x \star y = \mathrm{e}^{ \ln(x) \cdot \ln(y) } for all x,y \in \mathbb{R}_{>0}. Here, all positive-real numbers except 1 are the “multiplicative” units, and thus

\mathbb{R}_{>0}^\ast = \mathbb{R}_{\neq 1} = \left\{ x \in \mathbb{R}_{>0} \mid x \neq 1 \right\} \,.

Attribution
Source : Link , Question Author : dtech , Answer Author : Björn Friedrich

Leave a Comment