It is the ratio of a square within a rectangle and it has artistic beauty
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phi = [1+sqrt(5)]/2 sqrt(5) is irrational and so phi is irrational.
(a+b)/a=a/b=phi (the golden ratio, as defined) (a+b)/a=phi (we'll solve this equation) 1+b/a=phi (just changing the form of the left side a little) 1+1/phi=phi (a/b=phi so b/a=1/phi) phi+1=phi2 (multiply both sides by phi) phi2-phi-1=0 (rearrange) From here, we can use the quadratic equation to find the positive solution: phi=(-b+√(b2-4ac))/(2a) phi=(1+√(1+4))/2 phi=(1+√5)/2≈1.618
The main use for the golden ratio is its aesthetic appeal - in art and architecture. Rectangles with the golden ratio as their aspect appeal to the human mind (for some reason). So various aspects of the Parthenon in Athens, for example, have dimensions whose ratio is phi. Phi is closely related to the Fibonacci sequence: the ratio of successive terms of the sequence approaches phi and so, just like the Fibonacci sequence, phi appears in many natural situations. However, there is no particular application based on phi.
The value of phi is NOT 3.14. The value of phi, the Golden ratio, is [sqrt(5)+1]/2 which is approximately 1.62. It is an irrational number and so it has an infinitely long, non-recurring decimal representation.
Fibonacci numbers are important in art and music. The ratio between successive Fibonacci numbers approximates an important constant called "the golden mean" or sometimes phi, which is approximately 1.61803.