Real coordinate space

In mathematics, real coordinate space of n dimensions, written Rn (/ɑːrˈɛn/ ar-EN) (also written ℝn with blackboard bold) is a coordinate space that allows several (n) real variables to be treated as a single variable. With various numbers of dimensions (sometimes unspecified), Rn is used in many areas of pure and applied mathematics, as well as in physics. With component-wise addition and scalar multiplication, it is the prototypical real vector space and is a frequently used representation of Euclidean n-space. Due to the latter fact, geometric metaphors are widely used for Rn, namely a plane for R2 and three-dimensional space for R3. In mathematics, real coordinate space of n dimensions, written Rn (/ɑːrˈɛn/ ar-EN) (also written ℝn with blackboard bold) is a coordinate space that allows several (n) real variables to be treated as a single variable. With various numbers of dimensions (sometimes unspecified), Rn is used in many areas of pure and applied mathematics, as well as in physics. With component-wise addition and scalar multiplication, it is the prototypical real vector space and is a frequently used representation of Euclidean n-space. Due to the latter fact, geometric metaphors are widely used for Rn, namely a plane for R2 and three-dimensional space for R3. For any natural number n, the set Rn consists of all n-tuples of real numbers (R). It is called (the) 'n-dimensional real space'. Depending on its construction from n instances of the set R, it inherits some of the latter's structure, notably: An element of Rn is written where each xi is a real number. For each n there exists only one Rn, the real n-space. Purely mathematical uses of Rn can be roughly classified as follows, although these uses overlap. First, linear algebra studies its own properties under vector addition and linear transformations and uses it as a model of any n-dimensional real vector space. Second, it is used in mathematical analysis to represent the domain of a function of n real variables in a uniform way, as well as a space to which the graph of a real-valued function of n − 1 real variables is a subset. The third use parametrizes geometric points with elements of Rn; it is common in analytic, differential and algebraic geometries. Rn, together with supplemental structures on it, is also extensively used in mathematical physics, dynamical systems theory, mathematical statistics and probability theory. In applied mathematics, numerical analysis, and so on, arrays, sequences, and other collections of numbers in applications can be seen as the use of Rn too. Any function f(x1, x2, … , xn) of n real variables can be considered as a function on Rn (that is, with Rn as its domain). The use of the real n-space, instead of several variables considered separately, can simplify notation and suggest reasonable definitions. Consider, for n = 2, a function composition of the following form: