Another way to find phi

You might want to start by reading my Introduction to Fibonacci Numbers and Fibonacci Rectangles and Spirals post before reading this.

Consider the line above.  We'll start by drawing some line AC.  Then we'll divide that line by placing the B such that the following is true:

   AB is to BC as BC is to AC

We can write an equation to express this relationship as:

   AB/BC = BC/AC.

Let's suppose AB is 1 unit.  Could be 1 inch, 1 millimeter, 1 foot, whatever.  Let's then define BC = x.  Then AC = (1 + x).  Let's rewrite the equation now.

   AB/BC = BC/AC ==> 1 / x = x / (1 + x)

Algebraically rearrange:

   1 / x = x / (1 + x) ==> (1 + x) = x² ==> x² - x - 1

Do you remember basic algebra?  We need the quadratic equation to solve for x:

   ax² + bx + c = 0  ==> x = [-b ± (b² - 4ac)^1/2] / 2a

Solving for x for x² + x - 1 = 0 gives:

   x = [-b ± (b² - 4ac)^1/2] / 2a = [1 ± (-1² + 4)^1/2] / 2 = [1 ± (5)^1/2] / 2

The only positive solution is:

x = [1 + (5)^1/2] / 2 = 1.6180339887498948482045868...

It's Phi (F) - the ratio of successive Fibonacci numbers!

This is why F is called the Golden Ratio.