In mathematics, the real coordinate space or real coordinate n-space, of dimension n, denoted Rⁿ or ${\displaystyle \mathbb {R} ^{n}}$, is the set of all ordered n-tuples of real numbers, that is the set of all sequences of n real numbers, also known as coordinate vectors. Special cases are called the real line R¹, the real coordinate plane R², and the real coordinate three-dimensional space R³. With component-wise addition and scalar multiplication, it is a real vector space.

The coordinates over any basis of the elements of a real vector space form a real coordinate space of the same dimension as that of the vector space. Similarly, the Cartesian coordinates of the points of a Euclidean space of dimension n, Eⁿ (Euclidean line, E; Euclidean plane, E²; Euclidean three-dimensional space, E³) form a real coordinate space of dimension n.

These one to one correspondences between vectors, points and coordinate vectors explain the names of coordinate space and coordinate vector. It allows using geometric terms and methods for studying real coordinate spaces, and, conversely, to use methods of calculus in geometry. This approach of geometry was introduced by René Descartes in the 17th century. It is widely used, as it allows locating points in Euclidean spaces, and computing with them.

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" or the "real n-space".

An element of Rⁿ is thus a n-tuple, and is written $${\displaystyle (x_{1},x_{2},\ldots ,x_{n})}$$ where each x_i is a real number. So, in multivariable calculus, the domain of a function of several real variables and the codomain of a real vector valued function are subsets of Rⁿ for some n.

The real n-space has several further properties, notably:

These properties and structures of Rⁿ make it fundamental in almost all areas of mathematics and their application domains, such as statistics, probability theory, and many parts of physics.

Any function f(x₁, x₂, ..., x_n) 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: $${\displaystyle F(t)=f(g_{1}(t),g_{2}(t)),}$$ where functions g₁ and g₂ are continuous. If