The golden number F  is an irrational number which can be found as follows:

Given the segment

we get the following ratio:

(I)   

if we solve the quadratic equation we get:

the negative root is not a possible solution and as F (PHI) is the proportion ratio (I), we get F =1/x= 1.6180033988. The number 0.618033988 is called the reciprocal golden number and is represented by the Greek letter f.

On the previous page you got two numbers for the ratio: 1.618 and 1.617. This is because we only use up to three decimal places in our calculations.

Both the golden number F and the reciprocal golden number f have been of cultural and aesthetic importance throughout history. They can be

found in buildings such as the Parthenon and the Escorial Monastery in Spain.

In works of art such as Leonardo da Vinci's 'Vitruvian Man' where the radius of the circle is a golden section of the height of the person, i.e. of the height of the square.

In geometric shapes such as the pentagram or five-pointed star, where AB' is the golden section of AC' which is the golden section of AC.

The golden number F also appears in Fibonacci's sequence:1, 1, 2, 3, 5, 8, 13, 21, 34, etc. You can see that a_n/a_n-1 tends to F.