# Definition A point in $\mathbb{R}^n$ is a sequence of $n$ real numbers $(x^1, x^2, \ldots, x^n)$ called an n-tuples of real numbers. We have an intuitive notion of [[Continuous Group|continuity]] in this space, there are points of $\mathbb{R}^n$ arbitrarily close to any given point, a line joining any two points can be subdivided into arbitrarily many pieces that also join points of $\mathbb{R}^n$. See the [[Hausdorff Space]]. # Vector Space $\mathbb{R}^n$ forms a [[vector space]] with addition and multiplication defined the usual way. Given $\pmb{x} = (x^1, x^2, \ldots x^n)$ and $\pmb{y} = (x^1, x^2, \ldots x^n)$, we have $ \pmb{x} + \pmb{y} = (x^1 + y^1, x^2 + y^2, \ldots, x^n + y^n) $ It is a real vector space, i.e., the field $\mathbb{F}$ over which it is defined must be $\mathbb{R}$, otherwise, we do not have closure. ## Applications in Physics $\mathbb{R}^n$ occurs in many applications in physics, especially with $n=3$ and $n =4$. Higher $\mathbb{R}^n$ $n$ form the configuration manifold in classical mechanics.