Back to Analytic Geometry
Analytic Geometry:
AG 002 Rectangular coordinates:
By William E. Steinman:
Let us begin with simple things that we already know and progress from there.
We will be concerned at first with the system of real numbers. These are the positive integers, the number zero, and the corresponding negative integers.
We can write the sequence as:
. . . , -4, -3, -2 -1, 0 1, 2 3, 4, . . .
Now, these are the real numbers, but we want also to deal with rational numbers. The rational numbers are all the integers and the ratio of sets of real numbers represented by p/q. The integers are then all of the numbers we get when q = 1 and p can be all the real numbers.
Isn’t that tedious?
So lets define a line hat extends from minus infinity on the left to infinity on the right.
Then we can slap some integers down on that line like this:
Cool. So what?
If we now draw a perpendicular through that line at zero we have created what is called a two dimensional Cartesian coordinate system. That is a coordinate system wherein points on a plane can be represented and plotted. We can then define a point by its distance from the horizontal line (the x axis) and the vertical line (the y axis). The coordinate system will look something like this:
You might recall that we have used such a system before in our geometry studies, but I paid little attention to the formal structure of this coordinate system. We can remedy that now.
Looking at the area between +X and +Y, we call that area the first quadrant or quadrant I. This is the area where we prefer to work because both X a Y will have positive values.
The area between +Y and –X is Quadrant II and you will notice that all values of X will be negative in this quadrant.
The area between –X and –Y is called quadrant III and both X and Y will have negative values.
The area between –Y and +X is quadrant IV and all values of Y will be negative while all values of X will be positive.
We can select a point in this coordinate system and it will acquire values for X and Y depending on where we place it, like this:
If we add a second point we can call it p2 and you may notice that we can calculate the distance between p1 and p2.
Take a look:
Notice that we can draw lines parallel to our axes to create a right triangle.
Notice that the distance between the points is the hypotenuse of that triangle.
Now all we need do is recall Pythagoras theorem.
The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two sides.
