In his book, the “Liber Abaci”, he explains the usefulness of Hindu-Arabic numeral system in the tracks of transaction in comparison with the Roman numeral system.The invention of Fibonacci sequence essentially was induced by a commercial interest of rabbit breeding which Fibonacci wrote in his book. The mysteriously aesthetically pleasing ratio is the relation "a b is to a as a is to b" or a b/b=a/b=ÃÂ.
The problem was to determine how the number of rabbits multiplies in a farm.
In the third chapter of the “Liber Abaci”, he describes the problem: A certain man put a pair of rabbits in a place surrounded on all sides by a wall.
For example, 2 and 3 tends to fulfill the rule of golden ratio.
The ratio of 5 and 3 does not equal to the ratio of 3 and 2.
In this sequence, each of the numbers is “the sum of the two preceding numbers,” ().
Fibonacci did not mention the first member, 0, in his book.
The influence of the Fibonacci on modern arts and architectures is huge.
Though the Fibonacci numbers do not properly correspond to the Golden Ratio, they have a lot with it.
As previously stated, the basic equation for phi is a b/a=a/b=ÃÂ. Now returning to our previous equation, a b/a=ÃÂ, we can substitute a for bÃÂ. Substitute these numbers in the quadratic function: x=[-b /-Ã¢ÂÂ(b2-4ac)]/2a and you get ÃÂ=[1 /-Ã¢ÂÂ5]/2.
This allows us to find the roots of the equation; ÃÂ=1.618 033 989 (commonly stated 1.6) and ÃÂ=-0.618 033...