Golden number

The golden number , or F number as it is also known, was the first rare number or in other words an irrational number that was discovered many years ago by Greek mathematicians. The golden number, which is also known as the “golden number” or “divine proportion” , is a constant that we can observe and perceive on a daily basis, although most of the time we are not aware of it. We can see it in the proportions of buildings, paintings, sculptures, and even in the human body.

What is the golden number?

The golden number is an irrational algebraic number that has great properties of relation and proportion between two segments on a line , in other words, within a geometric construction and we also find it in nature .

  • Definition
  • Characteristics of the golden number
  • Source
  • History
  • Who discovered it
  • Properties
  • Applications of the golden number
  • How much does it cost
  • How it is represented
  • Importance
  • Examples
  • Curiosities


The golden number is an algebraic number that have different properties and that was discovered in ancient times , as a ratio or proportion. This proportion can be found in some geometric figures and also in nature in elements such as shells, ribs of the leaves of some trees, the thickness of the branches, etc.

Throughout history , the golden number has been attributed great importance in various works of architecture and in other arts , although some of these cases have been objectionable to mathematics and archeology . In other words, it is the point at which the arts and mathematics meet .

Characteristics of the golden number

Some of the main characteristics of the golden number are the following:

  • The golden number is an irrational algebraic number .
  • The decimal representation of the number has no period .
  • The ratio of the golden number can be found in some geometric figures and also in nature.
  • The number is represented by the Greek letter φ (phi).
  • It is also known as an irrational or decimal algebraic number with no period.
  • Its mathematical value is approximately 1.618.
  • It is also known as the golden number, extreme and average ratio, golden ratio, golden ratio, golden mean, golden ratio, divine section, golden section, divine proportion.


Many scholars think that it originated in Babylon and Assyria from around 2000 BC. C. although there is no historical documentation that shows us that it was used in the elaboration of the stelae . It is believed that the origin of the number was born from a formal study done by Euclid who thought that it was a line that had been cut in extreme and average ratio when the line was whole.


The first to make a formal study of the golden number was Euclid who showed that the number could not be described as the ratio of two whole numbers ; in other words, it was an irrational number. In 1509 the Italian mathematician and theologian Luca Pacioli published De Divina Proportione raises some reasons why he considers it appropriate to consider the golden number divine:

  • The uniqueness ; for he compared it to the oneness of God.
  • It is defined by three line segments , and that is why it is associated with the Trinity.
  • The incommensurability that was equivalent to the incommensurability of God.
  • The self-similarity associated with the golden number compared to the omnipresence and invariability of God.

The first use of the adjective aureo, gold, or gold, to refer to a number is made by the German mathematician Martin Ohm , in the second edition of 1835 of his book Die Reine Elementar Matematik .

Who discovered it

Leonardo Fibonacci , was the one who discovered the series that leads us to phi . In the 12th century, it was he who discovered a simple number series that would be the basis of the incredible relationship that we find behind phi. Starting with 0 and 1, each number in the series is simply the sum of the previous two.


Some of the properties of the golden number are:

  • One of its most important arithmetic properties is that its square (Φ 2 = 2.61803398874988…) and its inverse (1 / Φ = 0.61803398874988…) have the same infinite decimal places.
  • Another of its properties is that an aesthetic character is attributed to objects whose measurements keep the golden ratio. 

Applications of the golden number

This number has been applied to various works of art , for example the Parthenon , we also find it in proportions of the golden rectangle and in the UN building  You can see examples of golden rectangles we can find them on credit cards , in our card identity cards and also on tobacco packs

How much does it cost

The golden number has a value close to 1,618.

How it is represented

The golden number is the numerical value of the proportion between two line segments a and b and that meet the following relationship:

(a + b) / a = a / b


It is important because we even find it in some geometric figures such as shells, ribs of the leaves of some trees, the thickness of the branches, etc. It also has an aesthetic character of the objects that follow the golden ratio , as well as a mystical importance . In history, it has been given great importance in architecture and other arts, although some of these cases have been objectionable to mathematics and archeology .



  • The pyramids of Cheops
  • Eiffel Tower
  • In the parthenon

In nature

  • On the sheets
  • On the planets of the solar system
  • In the morphological proportions of bees

In art

  • Gioconda, painted by Leonardo.

In music

  • Mozart piano sonatas.
  • Beethoven’s Fifth Symphony.

In the human body

  • Between the height of the man and the distance from the navel to the tip of the hand is the golden number.
  • The teeth grow according to the golden ratio.


It has had many different names, but its symbol makes it unequivocal as it is the Greek letter phi, in honor of the Greek sculptor Phidias , whose works were considered the closest to aesthetic perfection , as is the golden ratio.

One of the reasons why the golden number has fascinated those who study it is that it is found naturally in the most unexpected places. For example, the ratio bees female and male in a hive is generally similar to the golden ratio.

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