In Mathematics, golden ratio – also known as golden mean, golden section, divine proportion – is a special number, which is often represented using the symbol “ϕ” (phi). The golden ratio finds its application in various fields such as arts, architecture, geometry, and so on. In this article, we are going to learn what the golden ratio is, the golden ratio formula, derivation and how the golden ratio is related to the Fibonacci sequence, in detail.
Table of Contents:
- What is the Golden Ratio?
- Formula
- Derivation
- Relation between Golden Ratio and Fibonacci Sequence
- Practice Question
- FAQs
Golden Ratio Definition
Two quantities are said to be in golden ratio, if their ratio is equal to the ratio of their sum to the larger of the two quantities. The golden ratio is approximately equal to 1.618. For example, if “a” and “b” are two quantities with a>b>0, the golden ratio is algebraically expressed as follow:
\(\begin{array}{l}\frac{a}{b} = \frac{a+b}{a}=\phi\end{array} \)
The golden ratio is an irrational number, which is the solution to the quadratic equation x2-x-1=0.
i.e. Golden ratio,
\(\begin{array}{l}\phi = \frac{1+\sqrt{5}}{2} = 1.618033988…\end{array} \)
Some other terms that represent golden ratio include extreme and mean ratio, divine section, medial section, golden cut, and so on.
For example, divide the line into two sections. The two sections are in golden ratio if the ratio of the length of the larger section (say, “a”) to the length of the smaller section, (say, “b”) is equal to the ratio of their sum “a + b” to the larger section “a”.
Golden Ratio Formula
The golden ratio formula is used to calculate the value of the golden ratio.
From the definition of golden ratio,
\(\begin{array}{l}\frac{a}{b} = \frac{a+b}{a}=\phi\end{array} \)
, we get two equations.i.e. a/b = ϕ …(1)
(a+b)/a = ϕ …(2)
Equation (2) can be written as:
(a/a) + (b/a) = ϕ
1 + (1/ϕ) = ϕ …(3) [From equation (1), we can get b/a = 1/ϕ ]
Therefore, the golden ratio formula is given by:
ϕ = 1 + (1/ϕ) |
Golden Ratio Value Derivation
To derive the golden ratio value, multiply ϕ on both sides of equation (3), we get
ϕ +1 = ϕ2
Rearrange the above equation, we get
ϕ2 -ϕ – 1 = 0, which is the form of quadratic equation.
Use the quadratic formula,
\(\begin{array}{l}x = \frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\end{array} \)
.Here, x = ϕ, a = 1, b=-1, c = -1
So, we get,
\(\begin{array}{l}\phi = \frac{-(-1))\pm \sqrt{(-1))^{2}-4(1)(-1)}}{2(1)}\end{array} \)
\(\begin{array}{l}\phi = \frac{1\pm \sqrt{1+4}}{2}\end{array} \)
\(\begin{array}{l}\phi = \frac{1\pm \sqrt{5}}{2}\end{array} \)
Hence, the two solutions obtained are:
\(\begin{array}{l}\phi = \frac{1+\sqrt{5}}{2}\end{array} \)
and\(\begin{array}{l}\phi = \frac{1-\sqrt{5}}{2}\end{array} \)
ϕ = 1.618033.. and ϕ = -0.618033…
As ϕ is the ratio between two positive quantities, the value of ϕ should be the positive one.
Hence, the value of golden ratio ϕ is approximately equal 1.618.
Interesting Facts:
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Relation between Golden Ratio and Fibonacci Sequence
We know that the Fibonacci sequence is a special type of sequence in which each term in the sequence is obtained by adding the sum of two previous terms. Let us take the first two terms 0 and 1, then the third term is obtained by adding 0 and 1, which is equal to 1. The fourth term is found by adding the second term and third term (i.e. 1+1 = 2), and so on.
Hence, the Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21,..
There exists a special relation between the Fibonacci sequence and the golden ratio.
If we take two successive terms in the Fibonacci sequence, their ratio is very close to the golden ratio. If we take the bigger pair of Fibonacci numbers, the approximation is very close to the golden ratio.
Now, let us start with the term 2 in the Fibonacci sequence.
Term 1 | Term 2 | Ratio = Term 2 / Term 1 |
2 | 3 | 1.5 |
3 | 5 | 1.666666… |
5 | 8 | 1.6 |
… … | … … | … … |
144 | 233 | 1.6180555 |
… … | … … | … … |
Practice Question
1. Which of the following represent the golden ratio formula?
- a/b = (a+b)/b
- a/b = (a+b)/a
- a/b = (a-b)/b
- a/b = (a-b)/a
2. The golden ratio ϕ is equal to
- ϕ – 1
- ϕ + 1
- 1 + (1/ϕ)
- 1 – (1/ϕ)
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Frequently Asked Questions about Golden Ratio
Q1
What is the golden ratio?
In Mathematics, two quantities are said to be in golden ratio, if their ratio is equal to the ratio of their sum to the larger of the two quantities.
Q2
Which symbol is used to represent the golden ratio?
The symbol used to represent golden ratio is ϕ (phi).
Q3
What is the value of the golden ratio?
The value of the golden ratio is approximately equal to 1.618.
Q4
How is the golden ratio related to the fibonacci sequence?
There exists a relation between the golden ratio and Fibonacci sequence, such that the ratio of two successive terms in the Fibonacci sequence is very close to the golden ratio.
Q5
Is the divine proportion the same as the golden ratio?
Yes, the divine proportion is the same as the golden ratio. The golden ratio is often represented using the terms, such as divine proportion, golden mean, golden proportion, golden section and so on.
