The XY Coordinate System

---

I first started our discussion with the introduction of the number line where all real numbers are located. (Rational & Irrational numbers)(we will call this line the X-axis). Imagine yourself at the zero point. This I will call the origin. You have the freedom to travel to the right (greater than 0 and beyond) or to the left (less than 0 in the negative direction). For each position you located, there will be one number on the X-axis associated with it.

Now, imagine another number line (exactly the same as the first at a 90 degree intersection of our previous line (X-axis) where the zero of that line is exactly the same as the zero of the X-axis. We will call this line the Y-axis. So, if you are positioned at the point where the lines cross, there will be a 0 for the X-axis and a 0 for the Y-axis. We will designate this location using the following symbol (0,0) where the first zero MUST come from the X-axis always. So now your freedom of travel has increased from left or right (on the X-axis) along with up or down (in the Y-axis direction). This way, you can travel anywhere in the plane determend by X & Y axes. But always remember, the first number in the ordered pair is always from the X-axis direction.

Let's take the ordered pair (3,2). To get there from your starting point (0,0)(the origin), you must travel 3 units to the right, then go up 2 units in the Y-axis direction.
Let's take the ordered pair (-5,7). To get to that point, we must move 5 units in the negative direction on the X-axis, then up 7 in the Y-axis direction.
Let's take the ordered pair (-3,-5). To get there, we must travel 3 units in the negative direction on the X-axis, then move down 5 units in the Y-axis direction.
Let's take the ordered pair (0,-10). To get to this point we do not move at all on the X-axis and move directly down 10 units in the Y-axis direction. You will be a point on the Y-axis.
Every point in the plane determined by these two axes can be located. This is called The XY coordinate plane. This plane will serve as visual for many of the important concepts of the calculus. Later, for the serious math student, it will be expanded to 3-D by introducing another axis (Z-axis) with the development of Multivariable Calculus.