One-dimensional, two-dimensional, three-dimensional, four-dimensional space, and, in general, $\Bbb R^n$ Often with high school students I find myself confronted with the concept of one-dimensional ($\Bbb R$, where $(x)$ it is a coordinate of a straight line), two-dimensional $\Bbb R^2=\Bbb R\times \Bbb R$ (cartesian plane, $(x,y)\in\Bbb R^2$), three-dimensional $\Bbb R^3$ (like triple coordinates in the space $(x,y,z)$), four-dimensional (our dimension) $\Bbb R^4$ with $(x,y,z,t)$ where $t$ is the time, and $t>0$, and $\Bbb R^n$ spaces.
What idea could be valid to better motivate the meaning of $\Bbb R^n$?

Suppose you have $N$ particles, and to describe each of their location, you need a point in $\Bbb{R}^3$. So, $\xi_1= (x_1, y_1, z_1)\in\Bbb{R}^3$ is the point where the first particle is situated, $\xi_2 = (x_2,y_2,z_2)\in \Bbb{R}^3$ is where the second is situated and so on up to $\xi_N$. If you want to talk about the whole system simultaneously, you need to keep track of everything! You need to know $(\xi_1, \dots, \xi_N)= ((x_1, y_1, z_1), \dots, (x_N,y_N,z_N))$. How much information is there here? Just count: we have $x_1,y_1,z_1, \dots, x_N,y_N,z_N$, which is a total of $3N$ numbers (i.e we're not dealing with $\Bbb{R}$, or $\Bbb{R}^2$ or $\Bbb{R}^3$, but we're dealing with $\Bbb{R}^{3N}$).
So, depending on how much "information" you want to keep track of, you need a higher dimensional space. So, even though we are accustomed to a 3-dimensional world, it makes a lot of sense (to me anyway) to start dealing with $\Bbb{R}^n$ generally for any integer $n\geq 1$.
