Geometry and Trigonometry

Picture from here (5/13/2015)

What math is really about is this: demonstration of reasoning. You say something and reason for it, but you use geometry (geo: earth + metry: measurement) to demonstrate it. The very first thing about geometry is a line! How to show a line in mathematical language. First we need to have a coordinate system, say x-y coordinate, and this is in a two-dimensional space. If we deal with 3D coordinate then we need another axis, call it z-axis. Any two points in our spaces either 2D or 3D can make a line, so it is fair to say that we need at least two points to make a line. Well, it is good to know that when we talk about the line connecting two points to each other, it's better to call it a line segment, than a line as a line is infinite and it happens to pass through those two points of interest. Also, if similar to a line segment we start from a point and pass by another point and keep going in that direction, we have a ray with a starting point. Now, what does it mean to have points? It's not a painting, it's mathematics, so we should be able to say where that point is located exactly in the space. 

For now, let's talk 2D. A point in a 2D space can be specified by two coordinates, namely the x- and y-coordinates, for example point A at (1, 2) and point B at (-3, 4); point A is 1 step to the positive direction of x-axis from the origin (an assumed point in space), and from there, 2 steps in the positive direction of y-axis; voila, there is our point A. Talking about these concept reminds me the Plato's world of ideas where everything is perfect and represented by mathematics. I strongly suggest that to whoever is interested in philosophy.

When we have the two points located in space, we can draw a line through them which also has a mathematical representation. Each line has a slope (m) in the coordinate system, and an intercept (b) where it intersects with the y-coordinate. The intercept is the distance from the intersecting point of line and y-axis from the origin. If we have these two parameters, it would be easy to show the line. There are different ways to show a line in mathematical language. I prefer to escape from representing all the equations and formulas of the geometrical forms I go over but I try to refer to online sources for further illustrations. I would rather write about what the gist of the story is than to describe the whole story. Also, when we have the coordinates of the two points, we can determine the distance between them and the slope (m) of the line made by them. 

Sometimes we would like to know the angle (α) between two intersecting lines with slopes m1 and m2. It is good to know that if the two lines are perpendicular, slope of one line is the opposite inverse of the slope of the other line. Here are some nice online pages about the story of points and lines in 2D space (accessed 5/13/2015):

• • Points, Line equation forms, and Line slopes: Click Here 

• Distance between two points in space: Here: Click Here

  • Angle between two lines: Click Here

I will have to say that in order to pass the FE exam, going through some concepts of geometry won't suffice at all and it takes hours of practice on solving sample problems and implementing the actual related equations. I will have to do that all too. 

Well we never talked about the shape of the line passing through two points! How about a curved line? what if our line is not a straight line? It can be a curve of different forms but how can we present that with an equation. One of the forms it can take can be presented by a quadratic function. It is called quadratic because the name is coming from the Latin word Quadratus meaning square. Right there, we know that there is going to be a squared form in the equation. 

Imagine a quadratic line in a 2D space! There are two ways to place such a curve in our plane (2D space), either it will intersect with the x-axis at two points or not. If it intersects, the intersecting points are called the roots, and they are real numbers which can be determined through a mathematical calculation using the discriminant of the equation. If the line does not intersect with the x-axis, then the roots are imaginary numbers which require some knowledge of the complex numbers theory to comprehend. This Wikipedia page has a lot of detailed information on quadratic functions, but if one is only concerned about the general format of the equation and the roots, click here (accessed 5/13/2015).

There is a whole big room in Geometry that contains different types of curved lines called Conic Sections. Why they are called conic is because they can be produced by cutting a cone using a flat plane. That is, the intersection of the plane and a cone can take different forms that turns out to be a curve. What's amazing about these curves is that each one of them has a name of its own and obviously an associated mathematical equation describing its geometrical shape. Well, I don't think I am going to go into that area of writing about each one of them, but only to point at their names and maybe some nice facts about them. I, however, refer to easy going good looking pages online for each topic. Different types of conic sections include cone, circle, ellipse, parabola, and hyperbola. This cool page has a lot of nice features explaining the properties of each one of these sections.