平面几何：在$\mathbb{R}^1$中描述一个平面

We'll be a bit pedantic throughout this section/chapter, to make sure our definitions are airtight (assuming only properties of the real numbers).

A point in the plane (or simply point, when the context is clear) is an ordered pair $(a,b)$ of real numbers (called the coordinates of the point). We call the first coordinate the $x$-coordinate, and the second the y-coordinate.

The coordinate plane (or simply plane, when the context is clear) is the set of points in the plane, i.e. the set of ordered pairs $(a,b)$.

The $x$-axis is the set of points with $y$-coordinate equal to $0$, i.e. the set of points $(a,0)$. Similarly, the $y$-axis is the set of points with coordinates of the form $(0,b)$. (Although we haven't defined a line yet, you can intuitively think of the $x$- and $y$-axes as lines.)

We will postpone fully defining a distance metric, to maximize the intuition behind our definitions. In other words, even though we could write out a totally correct formula for the "distance", it would feel like cheating. The approach we'll take is by treating "distance" as a function $d(P,Q)$ of points $P,Q$ in the plane satisfying some simple and intuitive properties, and "derive" a formula for the distance only using these properties. Here's the properties we'll start with:

1. Borrowing from one-dimensional geometry, we set $d((a,b),(a+t,b)) = t$ and $d((a,b),(a,b+t)) = t$ for any reals $a,b,t$.

(Intuition: along lines parallel to the $x$- and $y$-axes, distance corresponds to one-dimensional length.)

1. As a corollary of (1), $d(P,P) = 0$ for any point $P$ in the plane.