As you enter the classroom on the first day, you see a set of desks neatly organized in rows. There are 7 rows of 5 chairs each. How could you create a system so that each desk would be uniquely defined? You could certainly number each desk from 1 to 35. That may work, but there is an easier way. You could also label each of the seven rows A, B, C, D, E, F, G and each of the columns 1,2,3,4,5. The 4th chair in the 3rd row would be labeled C4. Each chair would have a unique label. This is the basis of a coordinate system.

Coordinate systems are set up to help us organize. A supermarket labels its aisles so that you can easily be steered to the location of the tuna fish or the potato chips or the soda. A theater also labels its rows and seats so that you can find your seat with ease. Coordinate systems which have rows and columns (or x-axes and y-axes) are called Cartesian coordinate systems after René Descartes, a 17th-century philosopher and mathematician and all-around bright guy.

If we agree on a way to label the axes, then an ordered pair of numbers can simplify our lives even further. The x-axis is usually the horizontal axis and the y-axis is typically the vertical axis. This is a convention that most people adopt. The positive x-axis is to the right and the positive y-axis is up. After drawing the axes, we define the four quadrants as 1, 2, 3, 4 as shown in the diagram.

The distance from the origin to any point can be easily found on a graph. Drawing a line from the point down to the x-axis and a line from the point to the origin creates a right triangle. The length of the sides of the right triangle are identical to the coordinates of the point. The distance R can then be found using the Pythagorean theorem:

Finding the distance between any two points requires you to draw a line between the two points and forming a right triangle with this line is the hypotenuse. As you can see, the length of one side of the triangle is a distance in the x-direction. This distance can be found by calculating the difference between the x-coordinates of the two points. Similarly, the other side can be found by calculating the difference between the y-coordinates. The resulting equation, using the Pythagorean theorem is:

A Cartesian coordinate system with x-axis and y-axis works well for desks that are aligned in rows and columns. In some classes, the teacher arranges the desks in a large circle. Describe how you can most efficiently describe the position of each chair. Extend your description to a class which has multiple concentric circles of seats (i.e., a theater in the round).