This blog is a first post from the series of posts about revisiting the foundations of geometry and mathematics as a whole. We humans are visual animals and we have build theories, understandings about our world through pictures, words and drawing. I would like to invite you to learn mathematics in a more visual way. Let’s begin our journey.
Math has a bad reputation
In schools, mathematics starts with counting, drawing figures and gets into algebra, trigonometry and what not. Students tend to be excited with their early years in maths and already lost interest during their high schools.
This means we constantly face people who think there is no use of mathematics in their life and they have ruined their youth studying shit. I have even met programmers, engineers and computer science students who don’t have firm grip of mathematics and speculate on most mathematical theories and thinking.
Math course should start with geometry
I sincerely think maths must start with geometry as it provides us grounds why maths is necessary. Human mind is always looking for patterns. We are so accustomed to thinking visually that non-intuitive and hard to visualize theories are confusing for us to grasp. In our ever lasting search for patterns, we look for summarizing complex topics like texts, data and information in manner that is visually intuitive.
So, most of our findings are illustrative, graphical and have a lot of charts in them. We have tried to find shapes in data and information all the time. Our brain has been using geometry and comparing shapes forever.
How to think like a mathematician
When you think like a mathematician, first of all, you start to establish some facts. Then you move along by stepping on the facts and establishing more facts.
Here, let’s start with a single point in space. To establish a fact, you must first ascertain what a point is and how is a space defined. A point would be something with zero size. Unlike physics, mathematics allows zero sizes. The space we will be using is a two dimensional Euclidean space i.e. an infinite length and height that doesn’t meet or curve.
If you could draw another point in the same space, in Euclidean space, the shortest path to the point would be a straight line. A line can be considered a path joining two points in space, straight line being the shortest. But you can also think a line as an extension of a point in one dimension. It’s like you have stretched a point in one direction and made a line out of it.
With the introduction of the shortest path, we came across Arithmetic. Our first encounter with numbers will be for comparing sizes. As happened to us in our childhood. Childhood brain compares all figures with sizes before interpreting what it is. Only later it develops complex understanding of shapes.
Now let’s move the line we draw. In mathematics this movement is called translation. In an infinite space, the translation mean nothing. What if we rotate the line from one of the points. For example, in the line above what if we hold one point with a thumb and rotate another point in a circular manner. We will end up creating a circle.
I always wanted to learn things in intuitive way. Because I am very poor in remembering, I hate to memorize and I hate when i can’t visualize what i am learning.
Using the same process of creating a circle, if we pull both points with equal effort vertically, we can create a rectangle and square. So, it’s like we are pulling and stretching points to create different shapes of geometry.
With these basic shapes the circle, square, rectangle and lines, we can create almost all geometric figures.
When we are building intuition about shapes, we have seen that we are stretching (scaling), rotating and moving (translating) points. To create a definite shape and to replicate the process, we need to scale it in a definite fashion. Like if we want to draw two equal rectangles, we need to know what is the height and width of the two lines used to draw one.
Some shapes like circles are rotations. And some shapes like arcs, are circles which start at a certain angle and stops at a certain angle.
Types of numbers
Natural numbers are those who are either equal to or more than zero. Like 0,1,2,3,4 … These are numbers we use to count objects in whole, also called whole numbers.
Integers are numbers which is everything whole numbers have plus all the negative numbers. Like … -4,-3,-2,-1,0,1,2,3 …
Rational numbers are those which can be written in the form of a fraction. Actually all numbers can be written in a fraction form if they are divided by 1. Like 1/1,2/1 … so, all numbers that are countable are rational numbers.
Irrational numbers are those that doesn’t end in a decimal number. Like pi, whose value is 3.14156… and is never is ending.
I will slowly move forward in this series. Please wait for more.