What does “x” and “y” even mean?

so after quite a break, I am returning with math post. It is going to be simple analytical geometry or whatever this field is called.

After teaching some people about lines, circles, circles in a hydraulic press (read: ellipse) also vectors and all sorts of shapes that you can think of[1] I bumped into a fundamental issue that my “students” missed out something important even after using it (without knowing what it really was) for a long time.

First, let’s talk points. A point on a number line is quite straightforward. Why? Because you do not need to have a lot of information to describe it. Image result for number line

Any point on the number line can be represented using only one number, one value. If you wanted to represent a line on the number line you would just need to set boundaries for the highest and lowest number, something like 3<x<5 [2]. This line would have the length of 2 and you would find it by just drawing all the numbers like 3,3.1,3.2,3.5,4,5 and all the others in between, those are all the possibilities for “x”.

Image result for grid line

Number lines start to get boring on this level so we have to step up on a grid. Here you need more information. As you can see on the map. Your city/town has two coordinates, latitude and longitude which is just a fancy name for the commonly used “x” and “y”. Here the grid is limited for x which needs to be between +180 and -180 and y between +90 and -90. With these two coordinates, you can locate any point on the Earth. The similar thing is with the commonly used grid but there, “x” and “y” do not have any boundaries[3]. But how do we write a line on the grid? Well, we could just ignore the fact that it is a grid and think about it as a number line. We just need to set “x” or “y” to some number and let it be fixed and unchanging while the other one changes and describes our infinite line (without boundaries) or just a segment (with boundaries). If “y” is constant that means that we can move only left and right and the line will be leveled. If “x” is constant we can only move and find the points on the line by moving up and down and the line will be vertical.

I will continue probably next time with lines that go different directions.


[1] Don’t think of too random shapes, they would be too difficult, you still have to think about “nice” shapes.

[2] Any math is really difficult to write in WordPress so what I actually meant was 3<=x<=5 which means that numbers 3 and 5 are also a possibility.

[3] Of course, Earth is a sphere and that is different geometry but it serves as a good example.