Question

Explain about the golden ratio?

Answer 1

Hint: In this question we need to define the golden ratio and write its algebraic equation. It has some other names which we can also write. Its properties can also be explained. Calculation can also be shown and explained.

Complete answer:
Two quantities are said to be in golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Greek letter phi ($\Phi $) represents the golden ratio.
Its algebraic equation with quantities a and b where$a > b > 0$,
$\Phi = \dfrac{{a + b}}{a} = \dfrac{a}{b}$
It is an irrational number which is solution to the quadratic equation ${x^2} - x - 1 = 0$, with value
\[\Phi = \dfrac{{1 + \sqrt 5 }}{2}\] $ = 1.618033$
Its other names are:
$\bullet $ Golden mean
$\bullet $ Golden section
$\bullet $ Extreme and mean ratio
$\bullet $ Medial section
$\bullet $ Divine proportion
$\bullet $ Golden proportion
$\bullet $ Golden cut
$\bullet $ Golden number
Calculation:
Two quantities a and b are said to be in the golden ratio if
$\dfrac{{a + b}}{a} = \dfrac{a}{b} = \Phi $
By taking right fraction,
$\dfrac{a}{b} = \Phi $
$\dfrac{b}{a} = \dfrac{1}{\Phi }$-------(1)
One method for finding the value of $\Phi $ is to start with the left fraction.
$ = \dfrac{{a + b}}{a}$
$ = \dfrac{a}{a} + \dfrac{b}{a}$
$ = 1 + \dfrac{b}{a}$
By putting the value of $\dfrac{b}{a}$ using equation (1), we get
$ = 1 + \dfrac{1}{\Phi }$
Therefore,
$1 + \dfrac{1}{\Phi } = \Phi $
Multiply by $\Phi $ on both sides,
$\Phi + 1 = {\Phi ^2}$
After rearranging, we get
${\Phi ^2} - \Phi - 1 = 0$
Using quadratic formula two solutions are obtained:
$ = \dfrac{{1 + \sqrt 5 }}{2}$ $ = 1.618033$ OR $ = \dfrac{{1 - \sqrt 5 }}{2}$$ = - 1.618033$
Hence, the golden ratio is the ratio between positive quantities so its value should also be positive.

Note: The properties of the golden ratio includes its appearance in the dimensions of a regular pentagon and in a golden rectangle. Golden rectangles can be cut into a square and a smaller rectangle with the same aspect ratio. Golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts.