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Concept

Golden Ratio

Consider the positive numbers

m

and

n,

where

m > n .

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

$\frac{m}{n} = \frac{m + n}{m}$

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 $φ$ and its value is written as a quotient.

$φ = \frac{1 + 5}{2}$

The golden ratio

φ

is an irrational number. Its approximate numerical value is represented as a decimal.

φ = 1.618033 \dots

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

n

to

m

equals the reciprocal of

φ .

φ = \frac{m}{n} ⇓ \frac{1}{φ} = \frac{n}{m}

The definition of the golden ratio can be manipulated to be written in terms of

φ .

\frac{m}{n} = \frac{m + n}{m}

Write as a sum of fractions

\frac{m}{n} = \frac{m}{m} + \frac{n}{m}

$\frac{m}{n} = φ$ , $\frac{n}{m} = \frac{1}{φ}$

φ = \frac{m}{m} + \frac{1}{φ}

$\frac{a}{a} = 1$

φ = 1 + \frac{1}{φ}

A quadratic equation for

φ

can be obtained by multiplying the above equation by

φ .

φ = 1 + \frac{1}{φ}

$LHS \cdot φ = RHS \cdot φ$

φ^{2} = (1 + \frac{1}{φ}) φ

Distribute $φ$

φ^{2} = φ + \frac{φ}{φ}

$\frac{a}{a} = 1$

φ^{2} = φ + 1

$LHS - φ = RHS - φ$

φ^{2} - φ = 1

$LHS - 1 = RHS - 1$

φ^{2} - φ - 1 = 0

The value of

φ

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

a x^{2} + b x + c = 0

are determined.

φ^{2} - φ - 1 = 0 ⇕ 1 φ^{2} + (- 1) φ + (- 1) = 0

Therefore,

a = 1,

b = - 1,

and

c = - 1 .

The Quadratic Formula can now be applied.

φ = \frac{- b \pm b ^{2} - 4 a c}{2 a}

The values of

a,

b,

and

c

are then substituted into the formula.

φ = \frac{- b \pm b ^{2} - 4 a c}{2 a}

SubstituteValues

Substitute values

φ = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 ( 1 ) ( - 1 )}{2 ( 1 )}

Evaluate right-hand side

Calculate power

φ = \frac{- ( - 1 ) \pm 1 - 4 ( 1 ) ( - 1 )}{2 ( 1 )}

Multiply

φ = \frac{- ( - 1 ) \pm 1 - 4 ( - 1 )}{2}

MultNegNegOnePar

$- a (- b) = a \cdot b$

φ = \frac{1 \pm 1 + 4}{2}

Add terms

φ = \frac{1 \pm 5}{2}

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.

$φ = \frac{1 + 5}{2}$

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