You are actually familiar with Cartesian Coordinates, they are used
to express addresses in Salt Lake City.
The Cartesian Coordinate System (also called **Rectangular Coordinate System **) is named after
its inventor **Renee Descartes** (1596-1650). It greatly amplifies
our problem solving ability by letting us think about algebraic
problems geometrically, and about geometric problems algebraically.

Consider the Figure above. The Cartesian Coordinate System consists of a
vertical and a horizontal number line that intersect perpendicularly
at their origins.
The intersection is also called the **origin** of the coordinate system.
The number lines are called the **axes** of the system.
The word *axes* is the plural of the word *axis*.
Thus there is a
**horizontal axis**, and a **vertical axis**. The variable along the
horizontal axes is commonly called , and that along the vertical axis is
called . Other choices are possible, but we will use this particular
choice almost universally in this class. Accordingly the horizontal
axis is also called the -axis, and the vertical axis the
-axis. Collectively the axes are also called the **coordinate
axes.** By convention the horizontal axis points to the right, and the
vertical axis points up.

The location of a point can be described by two numbers that are
obtained by projecting the point perpendicularly onto each of the
coordinate axes. The projection onto the horizontal axis is the
-**coordinate,** and the projection onto the vertical axis the
-**coordinate.** A point is described by the two coordinates,
separated by a comma and enclosed by parentheses, with the
-coordinate coming first. The Figure above shows four points
--specifically the points , , , and
--
and their representation as pairs of numbers, all drawn in blue.

Conversely, given two numbers, one being the -coordinate and the other the -coordinate, there is a unique point in the Cartesian Coordinates System that corresponds to that pair of numbers.

Obviously one can draw only a finite part of the infinite coordinate system. Which part to draw depends on the problem at hand. In a certain context your figure may or may not contain the origin, and it may have different scales along the two axes.

The two coordinate axes divide the plane into four parts. These
are called **quadrants,** and they are numbered I, II, III, IV, starting
with the Northeast quadrant and moving counterclockwise. The
quadrants are indicated in red in the above Figure.
We talk about the first, second, etc., quadrant.

Of major importance is the concept of the **graph** of an equation.
Given an equation involving the two variables and , the
graph of an equation is the set of all points whose
coordinates satisfy the equation.

The Figure on this page contains the graph of the equation

The graph of that equation is a straight line extending to infinity in both directions. In this class, we will study equations defining straight lines in great detail.

If the scales on the vertical and horizontal axes are the same then the distance between two points is just what you would measure with a ruler. This fact is illustrated in the above Figure by the magenta triangle. Let be the distance between the points and . The horizontal side of the triangle has a length of , and the vertical side a length of . By the Pythagorean Theorem,

Drawing a similar picture involving two general points
and you can see that the
**distance** between these two points is