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  1. Why π is an irrational number This is a question I've had for over 5 months and recently I think I've found the perfect solution for it. This is the first draft of my thesis. Be in mind that I'm only a 17-year-old engineering student and that English is my second language. Also, it is possible (and probable) that many other people have written about this, but I've come up with what I've written by myself. According to the Euclidean geometry, a circle is a two-dimensional figure formed by the points equidistant from a center. This set of points is called a circumference and the distance between them and the center is called radio. The ratio between the circumference and the radius of any circle, according to Euclid, is 2π, π being a number close to 3.14. Throughout history, people tried to figure out the exact value of π, until Lambert proved using tangent theory that π was irrational number, meaning that it can not be defined as a fraction of a whole and, more importantly, does not correspond to anything that exists (hence the term irrational). But how is this true? How can a figure that we supposedly see every day have an irrational measurement, i.e., a nonexistent one? The answer to this intriguing question is that Euclidean circles do not exist. It is impossible to find more than four points that have the same physical distance from a central point. You can see it when trying to find points equidistant from another point on a Cartesian plane in R2 (x, y). This is true regardless of the size of the plane. (Note: only natural numbers can be used in this plan because there are only natural numbers in the universe. There aren't two atoms and a half, and even without knowing what is the basic unit of the universe, we know it has a specific x, y, z dimension.) For example, the point (10,10) only has four points which are 5 away from it: (5,10), (15,10), (10,5) and (10,15). There are, however, points which have a distance close to 5, as the point 14,11) which has a distance of square root of 17. It is easy to see on paper the difference between 5 and 17 ^ 0.5 cm, but not between 5 x 10 ^ -10 and 0.5 x 17 ^ 10 ^ -10, something closer to reality when you look at a circle drawn with a modern computer. Or between 5 x 10^-googolplex and 17^0.5 x 10^-googolplex, closer to the circle used when super-computer software tries to estimate estimate π. For this reason, we are faced with (approximate) "circles" in our day to day lives, but in reality there are no perfect circles as Euclid described them.
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