The analytic geometry is based on the modern definition of a point in the Euclidean geometry. Recall that a point $P$ in an $n$-dimensional Euclidean vector space $\mathbb R^n$ is represented by $n$ real-numbered coordinates:
$$P=(x_1,\ldots,x_n).$$
In analytic geometry deals with the most common, real-valued vector spaces, i.e. the vector spaces over the field of real numbers $\mathbb R^n$. In general, vector spaces can be defined on other types of fields (e.g. rational numbers $Q^n$). In order to learn more about general vector spaces and their properties, please refer to its formal definition and theory established in the linear algebra branch of BookofProofs.
In the following, we will use the term number spaces as a synonym to the vector space over the field of real numbers $\mathbb R^n$, and we will be using the terms number line, number plane and number space depending on the dimension $n=1,2,3,\ldots$.
In the following example, we see the number line $\mathbb R^1$ with some points on it. Each point on the number line can be represented by exactly one coordinate. Actually, the points can be identified by the real numbers being their coordinates, e.g. $P_0=(-1),$ $P_1=(0),$ $P_2=(\sqrt 2),$, $P_3=(\Pi),$ etc.
In the two-dimensional case, $\mathbb R^2$ consists of all points, which can be represented by exactly two coordinates. This space is called the number plane. In the below example, the same points as in example 1 have now two coordinates: $P_0=(-1,0),$ $P_1=(0,0),$ $P_2=(\sqrt 2,0)$, $P_3=(\Pi,0)$, in which the first coordinate is the same as in example 1, and the second coordinate is zero. Of course, we can designate also any points having a non-zero second coordinate:
In the three-dimensional number space $\mathbb R^3$ each point is represented by three coordinates. Thus, the above four example points would be $P_0=(-1,0,0),$ $P_1=(0,0,0),$ $P_2=(\sqrt 2,0,0)$, $P_3=(\Pi,0,0).$ This demonstrates the following diagram:
In order to get a sense of the three-dimensional number space, you can click on the following "Evaluate" button and animate 10 randomly generated points with your mouse:
In an analogous manner, a point $P$ in an $n$-dimensional number space $\mathbb R^n$, $n > 3$ is represented by exactly $n$ coordinates. While it is perfectly possible to write down all $n$ coordinates of a given point and calculate with them, it is unfortunately not possible to visualize a space with more than $3$ dimensions.
Definitions: 1 2 3
Lemmas: 4
Proofs: 5 6 7
Propositions: 8
Theorems: 9
