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GoldenRatio

Status: Experimental

present and registered, but lightly documented and not yet covered by dedicated tests.

Description

GoldenRatio
    is the golden ratio phi = (1 + Sqrt[5])/2, with numerical value
    ~= 1.61803.
GoldenRatio is the positive root of x^2 == x + 1. It is a mathematical
constant: it has attributes Constant and Protected, NumericQ[GoldenRatio]
is True, and D[GoldenRatio, x] is 0. N[GoldenRatio, prec] evaluates it to
any precision.

Examples

No verified examples yet for this function.

Implementation notes

  • Attributes Constant, Protected. `Attributes[GoldenRatio] = {Constant,

Attributes: Constant, Protected.

Implementation status

Experimental — present and registered, but lightly documented and not yet covered by dedicated tests.

References

Notes & additional examples

Worked examples

In[1]:= N[GoldenRatio]
Out[1]= 1.61803

In[1]:= N[GoldenRatio, 40]
Out[1]= 1.6180339887498948482045868343656381177203

In[1]:= N[GoldenRatio^2 - GoldenRatio - 1, 40]
Out[1]= 0.0

In[1]:= FromContinuedFraction[{1, {1}}]
Out[1]= 1/2 (1 + Sqrt[5])

In[1]:= Round[N[(GoldenRatio^15 - (1 - GoldenRatio)^15)/Sqrt[5]]]
Out[1]= 610

Notes

GoldenRatio is phi = (1 + Sqrt[5])/2, the positive root of x^2 == x + 1; evaluating that polynomial at phi numerically returns 0.0, confirming the defining relation. It has the simplest possible continued fraction [1; 1, 1, ...], so FromContinuedFraction[{1, {1}}] recovers it exactly. Through Binet's formula Fibonacci[n] == (phi^n - (1 - phi)^n)/Sqrt[5], the closed form at n = 15 rounds to 610 = Fibonacci[15]. It is a protected Constant (so D[GoldenRatio, x] is 0) evaluated to any precision by N.