Category Archive 'Golden Ratio'

19 Oct 2014

# The Golden Ratio

### Golden Ratio, Mathematics

Golden Ratio Ï† = (1+sqrt(5))/2 = 1.6180339887498948482â€¦

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0. Two quantities a and b are said to be in the golden ratio Ï† if

(a+b)/a = a/b = Ï†

One method for finding the value of Ï† is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/Ï†:

(a+b)/a = 1+ b/a = 1+1/Ï†

Therefore: 1+1/Ï† = Ï†
Multiplying by Ï† gives: Ï†^2 – Ï† – 1 = 0

Using the quadratic formula, two solutions are obtained::

Ï† = (1- sqrt(5))/2 or Ï† = (1+sqrt(5))/2

Because Ï† is the ratio between positive quantities Ï† is necessarily positive:

Ï† = (1+sqrt(5))/2 = 1.6180339887498948482â€¦

See more at Golden Ratio.

Image: Phi (golden number) by Steve Lewis.

From Ratak Monodosico.

the Archives of Never Yet Melted in the 'Golden Ratio' Category.

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