AG 1: The Cartesian plane

In analytic geometry, we study geometry using algebra. The first step is to find a way to translates points into numbers, called coordinates.

The number line

You are probably familiar with the idea of the number line. Each point of the number line corresponds to a unique number. Remember that in geometry, lines are in principle infinitely long; no matter how big or small a number is, it will fit on the number line.

The number belonging to each point is called the coordinate of that point. Conversely, if x is a number then (x) stands for the corresponding point on the number line. For instance, (0) represents the “zero point”, or origin of the number line.

Using these coordinates, we can translate the geometry of the points on the line into facts of algebra. The table below shows some basic examples.

geometryalgebra
(x) lies to the left of the originx < 0
(x) lies to the left of (y)x < y
the distance between (x) and (y)
= the length of the line segment with endpoints (x) and (y)
|xy|
the line segment with endpoints (x) and (y) is longer than the line segment with endpoints (a) and (b)$ latex |x – y| < |a – b|

Translate into an algebra statement: (x) lies between points (a) and (b).

If (p) and (q) are points, which point (m) is described by the equation m = (p + q)/2 ?