Out of Pod Experience

Kessiaan, thx for the Euler's identity thing.

Awww... you seem so angry. Don't be angry be happy!

Well said.

All of these things we're talking about are a human's way of drawing math on a chalk board and do not hold sway over the universe. Pi, like any repeating decimal, still means exactly what it means in the real world even if our ability to draw it's meaning with decimals is inadequate.

It's silly to write it as a decimal in the first place because it's a relationship and is better shown as a fraction. We just use a symbol for shorthand because the fraction is common and frustrating, then we circumvent it by cutting Pi short to the degree demanded by the accuracy of the task.

We're getting beyond what we can really discuss here without going in-depth into the calculus of it, but we can still demonstrate this.

The ancient Pythagoreans (who came up with that famous triangle theorem) believed that all values could be expressed as a ratio of two whole numbers (aka a fraction). But, they were also the first ones that I know of who came up with a proof that irrational numbers do exist, with this proof of the irrationality of the square root of 2.

http://tinyurl.com/869xu8v

Pi is defined as the ratio of a circle's diameter to it's circumference, that is, πd = c or 2πr = c where d is the diameter, c is the circumference, and r is the radius. Even though π can't be represented as a fraction or a decimal, we can still reason about it.

Take a circle with a radius of 1 centered on the origin of a 2-dimensional coordinate plane. You'll get a figure like this:

http://tinyurl.com/6wg252t

The entirety of trigonometry is based on this figure. Pi appears very frequently in trigonometric relationships because it's fundamental to the unit circle. We know π is an exact number, so π (like all irrational constants) is generally carried through calculations as π rather than a decimal approximation.

If you take the fundamental trigonometric functions (sin, cos) and graph them as f(x) against x, you get the familiar sine wave graphs.

http://tinyurl.com/6vtqkcx

Note that wave goes on forever in both directions and is continuous. That will be important later.

The derivative of a function is the slope of a function at a particular value of the independent variable (x usually), and (in an extreme nutshell) is determined by calculating the slope of a secant line at the limit where the length of the secant segment between the two points of the function where the secant intersects approaches zero. This derivative tells you how the value of the function is changing at whatever X you plug into the derivative equation.

You can calculate the derivative of that derivative (this new function being called the second derivative) in the same manner. Like the first derivative told you how the function was changing, the second derivative tells how the first derivative is changing at whatever X you plug into the new derivative.

Every successive derivative reduces the overall degree of the derived function by one. You can see this with a simple example of the function y = x^2, which is a parabola. At any point on the parabola, the first derivative is the line tangent to the parabola at point x with slope 2x (because the derivative of x^2 is 2x), which means the function describing the slope of all of the lines tangent to the parabola is itself a line. The degree of the original function is 2 (x^2), and the degree of the derivative is 1 (x).

If you take the derivative of the line, you get y = 2. This function has degree 0 (x^0, which equals 1, therefore y = 1*2 = 2) and tells you that the slope of the first derivative is constant, which you already knew.

If you take the derivative of that derivative, you get zero because the value isn't changing.

Let us return to the sine function. We can approximate the sine function around a single point by taking the first derivative and then constructing an approximation function which intersects with the original function at the point and both functions have the same first derivative at that point. In this case we'll just get a tangent line to the function, but by continuing to differentiate the original function and compute approximation functions that agree with successively higher derivatives we will obtain a better approximation. Since the derivative of sin(x) is x'cos(x) and the derivative of cos(x) is -x'sin(x), deriving the full function with a finite number of terms is impossible, but the derivatives do follow a pattern. This is the basis of Taylor's Theorem, which allows us to compute power series for a number of irrational functions.

http://tinyurl.com/qmvkk

Using Taylor's Theorem (and the related Maclaurin theorem, which is just the Taylor Theorem centered at 0), a number of power series for π have been derived. None of them are particularly trivial, but the calculation of e (the exponential identity, another important irrational number) is fairly straightforward.

http://tinyurl.com/8ynuly8 (approximations of π)
http://tinyurl.com/4da742 (more than you ever cared to know about e)

^

After going through the trolls and everything, one problem I think is using infinity. No definition was set and its hard to arrive on how one is using infinity or how they think it should work.

Also when a water droplet falls in a vacuum it forms a sphere. As in the perfect shape of mass with respect to area and volume and handling it all is a sphere or a circle. Only when mass and area are used perfectly is the PI symbol found.

With how things have to be perfect for PI to show up, that might be the reason it is so hard to figure PI out.

You seem to be hitting a conundrum, with the ever increasing small and how it seems big as well.

Using relativity to irrational numbers is smart, since its hard to study irrational numbers, but not being able to study it does lead to confusion and excessive trolling. Good luck anyhow on your thoughts of PI.