GoldenRatio

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

General characteristics

The golden ratio ϕ is a constant. It is a positive quadratic irrational real number.

Limit representations

Continued fraction representations

Identities

Functional identities

Udaya Chinthaka Jayatilake

Above Fibonacci recurrence allows any polynomial in ϕ to be reduced to a linear expression.

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Integral transforms

Fourier exp transforms

Inverse Fourier exp transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Inverse Laplace transforms

Summation

Infinite summation

Representations through more general functions

Through Meijer G

Through other functions

Above equation derived in 1994 connects the golden ratio to the Number of the Beast (666):

Above equation derived in 1994 connects the golden ratio to the Number of the Beast (666):

Above equation derived in 1994 connects the golden ratio to the Number of the Beast (666):

Inequalities

Theorems

Approximation of golden ratio theorem

If a number x agrees with ϕ to n decimal places,then (((x^2)+2x)/((x^2)+1))agrees with ϕ to 2n decimal places.

History

– known 2000–3000 years ago: Pythagoras (circa 580 BC – circa 500 BC),;Phidias (490–430 BC); Euclid (c. 325–c. 265 BC)
– Euclid (c. 325–c. 265 BC) names the ratio 1:ϕ the "extreme and mean ratio" in Book VI of the Elements
– Leonardo of Pisa (1170s or 1180s-1250); Johannes Kepler (1571-1630)
– Luca Pacioli (1509) published book "Divina Proportione", which gave new impulse to the theory of the golden ratio, in particular
  he illustrated the golden ratio as applied to the human faces of artists, architects, scientists, and mystics
– Gerolamo Cardano (1545) mentioned golden ratio in the famouse book "Ars Magna", where he solved quadratic and cubic
   equations and was the first who explicitly made calculations with complex numbers
– M. Mästlin (1597) evaluated 1/ϕ approximatly as 0.6180340…
– J. Kepler (1608) independently shows that ratios of Fibonacci numbers approximate ϕ and describes the golden ratio as a "precious jewel"
– R. Simson (1753) gave simple limit representation of golden ratio based on its very simple continued fraction ϕ==1+(1/(1+(1/(1+(1/(1+…))))))
– G.S. Ohm (1835) gives the first known use of the name "golden ratio," believed to have originated earlier in the century from an unknown source
– J. Sulley (1875) first used the term "golden ratio" in English

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