i am very interested in Golden Ratio and its value. the Golden Ratio itself is not hard thing to visualize and understand in 5 minutes. But i am trying to reach the historical, logical reasons of origin of this ratio.
my first question is: why is the value of the ratio $\frac{a+b}{a}=\frac{a}{b}=1.618$ ? it is the positive root of $a^2-a-1=0$. can someone pls give me more clues,facts and properties of this phenomenon?
On
Regarding your question about the root...
$\frac{a+b}{a}=\frac{a}{b} \rightarrow \frac{a}{a}+\frac{b}{a}=\frac{a}{b}\rightarrow 1+\frac{1}{\frac{a}{b}}=\frac{a}{b}$. Let ,$\frac{a}{b}=x$ then $1+\frac{1}{x}=x$ and if you multiple both sides with $x$ you get the equation $x+1=x^2\rightarrow x^2-x-1=0$. Now the soloution of this equation: $\Delta =(-1)^2-4\cdot 1 (-1)=5$ and then $x_{positive}=\frac{1+\sqrt{5}}{2}=1,618$ approximately.
and...
if :
you a draw a circle with $radius=\frac{a}{2}$ which has AB as a tangent
draw the line that crosses from K and A
draw the circle with radius AE
you have the desired point in AB
Suppose we have a stick $AB$ of length $1$ and we need to cut that at position $C$ and let be $AC>CB$ by golden cut then we have $${AC\over CB}={AB\over AC}$$If $AB=1,AC=x,CB=1-x$ we get $${x\over 1-x}={1\over x}$$ $$x^2=1-x$$ $$x^2+x-1=0$$ positive solution of this equation $$x=\frac{-1+\sqrt5}{2}=\phi$$ is golden ratio