Most everyone can see why pi is so significant, but what's so special about e? It looks like someone's got too much time on their hands by the above equation. Well, for starters e is the constant that has several important properties in differential and integral calculus. Namely, the derivative of e raised to the x, is simply e^x. The derivative of all other numbers raised to the x fails this qualification. In fact using the natural logarithm (log base e), the process of derivation of a number to the x power becomes much simpler.
2 million digits of e
[In case you're curious]
2.7182818284590452 ... e starts off appearing to repeat, unfortunately it is irrational.
ok, e makes sense, now why the hell do we wish to waste our time looking at the square root of 2?
Well it all started with the Pythagorean theorem, and with the associated philosophies that suggested all geometric magnitudes could be expressed with rational numbers. Well if you take the diagnol of a unit square (sides equalling one unit) you would end up with the square root of two, and according to the pythagorean philosophies, that should be a rational number... Well, you take a look and decide if it looks rational to you.
5 million digits of the square root of 2
Well, if that isn't enough fun for you, how about the first 250,000 digits of the square root of 9. it's exciting!
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...
Well, if you can see the sequence you'd realize it looks like anyother lame brain teaser sequence. Just add two consecutive numbers to get the next number.
id est, 0+1=1, 1+1=2, 1+2=3, 2+3=5 ...
Now, the only significant prupose to this sequence can be seen in the following diagram This shell represents a graphical illustration of the fibonacci sequence (the fibonacci spiral) and is also a cross sectional view of a nautilus shell. The fibonacci spiral is also seen in galaxy clusters such as the Milky Way. In addition pine cones, flower petal arrangements and seed heads also demonstrate similar graphical depictions of the fibonacci sequence.
The first 500 fibonacci numbers and their respective factors.
The Golden Ratio
Suppose you were to take the ratio of any two numbers of the fibonacci sequence, you would get a series looking something like:
1, 2, 1.5, 1.66..., 1.6, 1.625, 1.61538...
These would appear to be converging by oscillation. What are these number converging to?
They are converging to phi, the Golden Mean.
This number is considered to be the most significant constant in the universe, ratio's of the human body, ripples in a pond, all support the Golden Mean. It is the one number that when squared equals itself plus one, and whose reciprocal is equal to itself minus one. Phi can be derived by adding one to the qauntity (-1+sqrt 5) / 2
Phi: a page full of decimals.