The "Golden Ratio" is what I think a very fun number to play around with. It's symbol is φ, or Φ, each one represents either the ratio or it's inverse, I forget which. I like to use Φ.

Anyway, it's an irrational number. Unlike pi or e, it has a simple equation: (1+√(5))/2

The number is special because Φ - 1 = 1/Φ. Specifically, 1/1.618033989... = .618033989...

So, who cares? Well, there's some interesting stuff about this number. It represents the dimensions of a rectangle that, if you cut a square out of one side, the remaining piece would have the same porportions as the origional rectangle. It relates to the Fibonacci Sequence, Pascal's Triangle, and can be found many places in nature (along with the Fibonacci numbers).

But maybe now you want something you don't already know?

I was experimenting with the Golden Ratio and related numbers. For example, what if I wanted a number n where x - n = 1/x? I devised a formula and called it the "Golden Function:" f(x) = (x+√(x²+4))/2. (Later I found out someone already discovered it, and also already named it the golden function. Odd.)

With this, f(1) = Φ, and all the other numbers in the function have the property f(x) - x = 1/f(x)

I then began experimenting with other things, and looking online for more information/to see if I found anything new.

Here are some random things:

-The Fibonacci Sequence, given by {F₁ = 1, F₂ = 1, F_n = F_n-1 + F_n-2; nЄZ}, (1,1,2,3,5,8,13,21,34,55,89,...) has a closed-form solution:

(Φ^X-(1-Φ)^X)/√(5)

And another relation with the Fibonacci sequence:

lim F_n/F_n-1 = Φ
^x→∞

And yet another:

Φⁿ = F_nΦ + F_n-1

 

 

 

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I remember learning about this in math class. The Golden Ratio is often reffered to as Phi (pronoucned fly). Google it, there is a very interesting site on it, and its uses in mathematics. Also phi ^ 2 = phi + 1. That's so cool .

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As it should, in coordinance with the last bit I mentioned. Φⁿ = F_nΦ + F_n-1, in this case n=2, so Φ² = F₂Φ + F₁ = 1Φ + 1

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