How exactly does φ become 1.618..?

Question

I'm doing a project on the Golden Ratio thingy. So far i've managed to shown the calculations on how when a and b are two different numbers and a is the larger number the golden ratio states the relationship that (a+b)/a=a/b=φ, and how this equation becomes φsquare-φ-1=0. I know that φ essentially has to be 1.618033989blablabla for this equation to work, but is there anyway to prove this other than telling the audience to try every single number in this world until they find that the only possible number φ can be is 1.619blablabla? Wikipedia states that the only positive solution is for the previous equation to work is for φ to be equal to (1+5root)/2, which is 1.618~, but how is (1+5root)/2 arrived at?
Thanks for the help

Answer 1

The derivation of the value of φ is quite simple, and basically amounts to solving a quadratic equation. First, we remember the fundamental relationship:

(a+b)/a=a/b=φ

Rewrite the equation on the left:

a/a + b/a = a/b

Simplify:

1+1/(a/b) = a/b

φ=a/b, so:

1+1/φ = φ

Multiply by φ:

φ+1=φ²

Add -φ+1/4 to both sides:

5/4 = φ²-φ+1/4

Factor the expression on the right:

5/4 = (φ-1/2)²

Take the square roots of both sides:

±√5/2 = φ-1/2

Add 1/2 to both sides:

φ=(1±√5)/2

Thus we are left with two possibilities for φ: φ=(1+√5)/2 and φ=(1-√5)/2. However, φ is a ratio between lengths, which are both positive, so φ has to be positive. Since (1-√5)/2 is negative, it is not a possible value for φ. We therefore conclude that φ=(1+√5)/2, as claimed.

Answer 2

Check out that wikipedia stuff more carefully. That (1+sqrt(5)/2) comes from geometry of the golden rectangle. It has some special property in how the base and height relate. It's called the 'golden rectangle.'

Here's a good link:

http://en.wikipedia.org/wiki/Golden_rectangle

From that site: constructing a golden rectangle:

# Construct a simple square
# Draw a line from the midpoint of one side of the square to an opposite corner
# Use that line as the radius to draw an arc that defines the height of the rectangle
# Complete the golden rectangle

+ add
What Pascal says is true. The interesting thing here is that the same number is obtained through simple geometric means and also by solving a quadratic equation. It should be emphasized that the golden rectangle and its simple geometric definition far proceeded the analytic formula.

As a further illustration of why people like me enjoy mathematics, consider the fact that the ratio of consequitve Fibonacci numbers tends to phi as a limit! Interesting don't you think? Even as early in the sequence as 89/55=1.16182. It can be shown that as n->infinity F(n+1)/F(n) -> phi.

Cool, huh? In just this one number you have ancient geometry and ideas of beauty, algebraic solutions to a quadratic equation, and limits of real sequences. Quite a range.