section_path: Vocabulary and notation
section_name: Vocabulary and notation
paragraph_text: Vocabulary and notation
  
  The set of all real numbers is denoted R {\displaystyle \mathbb {R} } (blackboard bold) or R (upright bold). As it is naturally endowed with the structure of a field, the expression field of real numbers is frequently used when its algebraic properties are under consideration.
  
  The sets of positive real numbers and negative real numbers are often noted R + {\displaystyle \mathbb {R} ^{+}} and R − {\displaystyle \mathbb {R} ^{-}}, respectively; R + {\displaystyle \mathbb {R} _{+}} and R − {\displaystyle \mathbb {R} _{-}} are also used. The non-negative real numbers can be noted R ≥ 0 {\displaystyle \mathbb {R} _{\geq 0}} but one often sees this set noted R + ∪ { 0 }. {\displaystyle \mathbb {R} ^{+}\cup \{0\}.} In French mathematics, the positive real numbers and negative real numbers commonly include zero, and these sets are noted respectively R + {\displaystyle \mathbb {R_{+}} } and R −. {\displaystyle \mathbb {R} _{-}.} In this understanding, the respective sets without zero are called strictly positive real numbers and strictly negative real numbers, and are noted R + ∗ {\displaystyle \mathbb {R} _{+}^{*}} and R − ∗. {\displaystyle \mathbb {R} _{-}^{*}.}
  
  The notation R n {\displaystyle \mathbb {R} ^{n}} refers to the set of the n -tuples of elements of R {\displaystyle \mathbb {R} } (real coordinate space), which can be identified to the Cartesian product of n copies of R. {\displaystyle \mathbb {R}.} It is an n - dimensional vector space over the field of the real numbers, often called the coordinate space of dimension n; this space may be identified to the n - dimensional Euclidean space as soon as a Cartesian coordinate system has been chosen in the latter. In this identification, a point of the Euclidean space is identified with the tuple of its Cartesian coordinates.
  
  In mathematics, real is used as an adjective, meaning that the underlying field is the field of the real numbers (or the real field). For example, real matrix, real polynomial and real Lie algebra. The word is also used as a noun, meaning a real number (as in "the set of all reals").