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Concept

Golden Ratio

Consider the positive numbers and where Those numbers are said to be in the golden ratio if the following equation holds true.

Given the equation, the golden ratio is a ratio of two positive numbers such that the ratio of the larger number to the smaller number is equal to the ratio of their sum to the larger number. The golden ratio is represented by the Greek letter and its value is written as a quotient.

The golden ratio is an irrational number. Its approximate numerical value is represented as a decimal.
Additionally, a rectangle whose side lengths are in the golden ratio is called a golden rectangle.

Why

Numerical Value of the Golden Ratio
Start by noting that the ratio of to equals the reciprocal of
The definition of the golden ratio can be manipulated to be written in terms of
A quadratic equation for can be obtained by multiplying the above equation by
The value of can be found by solving this equation. One way of solving it is by using the Quadratic Formula. To do so, the coefficients in the standard form are determined.
Therefore, and The Quadratic Formula can now be applied.
The values of and are then substituted into the formula.
Evaluate right-hand side
Since the golden ratio is defined as the ratio of positive numbers, it must also be positive. Therefore, only the positive solution will be taken.