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N-Dimensional Space


If $\R $ corresponds to a line, and $\R ^2$ to a plane, and $\R ^3$ to space, does $\R ^4$ correspond to anything? What of $\R ^5$?


Let $n$ be a natural number. We call the set $\R ^n$ $n$-dimensional space (or Euclidean $n$-space, real coordinate space, real Euclidean space). We call elements of $\R ^n$ points. We identify $\R ^1$ with $\R $ in the obvious way.

We call the point associated with $x = (x_1, x_2, \dots , x_n) \in \R ^n$ with $x_i = 0$ for $1 \leqq i \leqq n$ the origin. We denote the origin by $0$. Similarly, we denote the point $x$ with $x_i = 1$ for all $i = 1, \dots , n$ by $1$.


We can not visualize $n$-dimensional space. Thus, our intuition for it comes from real space (see Real Space).


A natural notion of distance for $\R ^n$ generalizes that in $\R ^2$ and $\R ^3$. We define the distance (or Euclidean distance) between $(x_1, x_2, \dots , x_n)$, $(y_1, y_2, \dots , y_n) \in \R ^n$ as

\[ \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + \cdots + (x_n - y_n)^2}. \]

Does this have the properties that distance has in the plane and in space? We discussed these properties It does. Denote the function which associates to $x, y \in \R ^n$ their distance $d: \R ^n \times \R ^n \to \R $. So $d(x, y)$ is the distance between the points corresponding to $x$ and $y$.

$d$ is non-negative, symmetric, and the distance between two points is no larger than the sum of the distances with any third object.1


Let $x, y \in \R ^n$. If $x_i < y_i$ for all $i = 1, \dots , n$ then we say $x$ is less than $y$. Likewise, if $x_i \leq y_i$ for all $i = 1, \dots , n$ then we say $x \leq y$. Likewise for $>$ and $\geq$.


If $x \in \R ^n$ is less than $y \in \R ^n$ then we write $x < y$. Similarly for $x \leq y$, $x > y$ and $x \geq y$. Other notation in the literature for $\R ^n$ includes $E^n$, which is a mnemonic for “euclidean.”

  1. Future editions will include an account. ↩︎
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