This is not a proof. There are many proofs about irrational vs rational numbers. We know it is connected to odd numbers and is easy to prove algebraically, but can we understand what is going on from a different perspective?
This is one of those mathematical truths that is absolute. It is as foundational as it gets. For the purpose of this post I am going to skip proving this and move on to some constructions which will provide a framework for a different understanding rational and irrational numbers.
First we must talk about circles. When we think of a circle we typically think about it from the center out in terms of it’s radius. This seems as fundamental as the Pythagoras Theorem.
If we change our perspective slightly and instead look at a circle from the bottom corner we find that 2 angles are in fact constant and perfectly bisect any sized circle into the 4 quadrants. Lets keep that in mind moving forward.
How does this relate to irrational numbers? Lets starts with the square root of 2. We would typically look at a problem like this as our first hint about irrational numbers. Figure out the length of c and call it a day.
What is not so obvious is that if we apply the 2 angles above and a circle we find something interesting
On the left you can see the “square” in the square root where the perimeter of the inner square is exactly the square root of 2. When related to a circle on the right side of the image we find what I call the “Bubble Core”. I will explain that name later. Lets stay on track and note that this pattern holds for any size with equal length. Like a Pythagoras Theorem for circles.
So while this “Bubble Core” is connected to Pythagoras Theorem , something else interesting happens. Lets look at anything length 2 or greater a slightly different way. Using circles as units for defining the length, you find a pattern emerges with the square root of 8.
What you find is that what becomes the square root of 8 is in fact the unit circle with the curves chopped off. This lines up perfectly with the gaps between the circles. This pattern again holds true for any size of equal length.
It is pretty clear the relationship between square roots and what we call irrational numbers. That said there is clearly something very rational going on to create these infinite decimals we call irrational numbers.
The Bubble Core is the invisible in actual soap bubbles. It is exactly half the size of the bubble itself and is directly related to how bubbles connect to each other as shown below for context only. I will have more to post on bubbles at a later time. It should also be noted that bubbles are also involved in the SyPi Equation.
It would seem more than anything, that what makes a number rational or irrational is more of a question of “What is and what is not a natural square number?” By this I mean numbers that when square rooted produce a whole number would be a natural square aka rational numbers. While unnatural squares would be numbers that do nor produce a whole number when squared. This is not a proof for irrational numbers, it is just my thoughts leading into writing a proof for irrational numbers.