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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.

Notes 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 (5)

Every input below was run against the current Mathilda build and its output recorded.

Applications (5)

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

In[2]:= N[GoldenRatio, 40]
Out[2]= 1.6180339887498948482045868343656381177203

In[3]:= N[GoldenRatio^2 - GoldenRatio - 1, 40]
Out[3]= 1.9913648889155653462861004972096397569889e-59

In[4]:= FromContinuedFraction[{1, {1}}]
Out[4]= 1/2 (1 + Sqrt[5])

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

Implementation notes

Definition. GoldenRatio is the golden ratio φ = (1 + √5)/2 ≈ 1.6180339887, the positive root of x^2 == x + 1. It is a constant symbol: row { "GoldenRatio", 1.6180339887…, fill_mpfr_goldenratio } of kConstants[] in src/numeric.c supplies its value and the numericalize_init loop stamps it Constant | Protected, so Attributes[GoldenRatio] is {Constant, Protected}. Interned name SYM_GoldenRatio (src/sym_names.c), docstring in src/info.c. GoldenRatio is a genuine symbol — it does not auto-expand to (1+Sqrt[5])/2 (FullForm[GoldenRatio] is GoldenRatio) — but it carries its defining relation numerically, so GoldenRatio^2 == GoldenRatio + 1 returns True. D[GoldenRatio, x] is 0 from Constant.

Representation & numeric value. A bare EXPR_SYMBOL. N[GoldenRatio] returns the tabled machine double; N[GoldenRatio, k] runs fill_mpfr_goldenratio, which forms (1 + sqrt(5))/2 in MPFR at k + 20 guard bits and rounds down to the target precision. Machine value 1.61803; N[GoldenRatio, 50] = 1.61803398874989484820458683436563811772030917980576.

Usage & limits. GoldenRatio is the quadratic surd behind Fibonacci / Lucas closed forms (Binet's formula) and the continued fraction [1; 1, 1, …]; NumericQ[GoldenRatio] is True (whitelisted in is_numeric_quantity, src/core.c). Because it is kept as a symbol rather than expanded to its radical form, algebraic simplifications that need the surd must apply the defining relation (or N) explicitly.

  • Attributes Constant, Protected. Attributes[GoldenRatio] = {Constant, Protected}; the symbol cannot be reassigned.
  • Propagated as an exact, unevaluated symbol; NumericQ[GoldenRatio] is True and D[GoldenRatio, x] = 0.
  • N[GoldenRatio] gives the machine value 1.61803; N[GoldenRatio, prec] gives any precision, e.g. N[GoldenRatio, 50] = 1.61803398874989484820458683436563811772030917980576.

Attributes: Constant, Protected.

References

Notes & additional examples

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.