Finding the Square Roots

Finding the Square Root is a set of infinite geometric series. This idea first came about when dealing with how to represent the idea of the square root geometrically. 

The first concept was a circle with the radius of one unit, or the unit circle. 

I asked, 

"How can I find a squared value of this circle." 


"The distance between the points (0,1) and (1,0) is the square root of 2."

This is a very interesting phenomenon. I began to ask other questions like,

"How many square roots are there with the radius as an element in the set of positive integers?"


"Is it possible to derive all of the square roots through geometry, whose 'rooted' element is an integer? (i.e. n^(1/2); where n is an element of the set of positive integers.)"

So, there are two questions that I'll call questions a and b. 

Question A

(Question B will be discussed in a future blog)

For question a, this could be solved with a logical approach. For instance, the input of the question is infinite; the radius as an element in the set of positive integers. Therefore, the output of that function would be infinite as well. 

r|Z+|; and the diameter is equal to 2*r. 

r=x and r=y; therefore, (x, ø) and (ø, y) would be the corresponding coordinates for the two points on the circumference of the circle where the distance formula would be used.

if r=x v y, then one could find the distance between the points (x, ø) and (ø, y),

therefore, the distance formula ((x2-x1)^2+(y2-y1)^2))^(1/2) can be used,

thus, when r is assigned a value from the set of positive integers, then the resulting square root is equal to (2r^2)^(1/2).

or, r*(2)^(1/2). 

Since the value of r is squared, both negative and positive integers can be used. Additionally, zero can be included since the answer will be zero. To update the theorem, r is an element in the set of integers, |Z|. To gather a squared value of r, one can take the distance from the circumference at 90 degrees at the points (r, 0), and (0, r) as points in an (x, y) coordinate plane; to gain the (2r^2)^(1/2), or r*(2)^(1/2). 

Here's a geometric proof for the previously mentioned concept:

1. Draw 5 concentric circles with the origin (0, 0) and the radius value as an element of this set: (1, 2, 3, 4, 5). 

2. For each circle, find the distance between the coordinates: (r, 0) and (0, r).

All art pieces can be found on Etsy (link below). The following are some derived artworks from this practice of finding the square roots:

The following is the interactive version of the proof mentioned above: