Mathematics-Golden Ratio

The value of Golden Ratio in Mathematics is approximately equal to 1.618. It is expressed using the Greek symbol φ ( phi ).

In mathematics, especially in geometry and architecture, the golden ratio is defined as  " The ratio of larger quantity to the smaller quantity is equal to the ratio of the sum of the two quantities to the larger quantity " .

It can be expressed in the algebratation ( x + y ) / x = x / y

Proof ::-

If the quantities like x & y are in Golden Ratio, then we can write it as →

( x + y ) / x = ( x / y ) = φ --------------------( 1 )

( x + y ) / x = 1 + ( y / x ) = ( 1 + 1 / φ ) ----------------------( 2 )

So we can equate the above 2 equations ( 1 )  & ( 2 ).

( 1 + 1 / φ ) = φ ----------------------( 3 )

Multiplying the equation ( 3 ) with φ, so we get the equation below

φ + 1 = φ²  ------------------------------ ( 4 )

φ² - φ - 1 = 0 ----------------------------- ( 5 )

By applying quadratic equation formula, the roots of the equation ( 5 ) is given below :

φ =  ( 1 ± √5 ) / 2

So we get the answer is 1.6180339887498......

So we can use the approximate value for the φ ( Golden Ratio ) is 1.618