A003957 The Dottie number: decimal expansion of root of cos(x) = x.

7, 3, 9, 0, 8, 5, 1, 3, 3, 2, 1, 5, 1, 6, 0, 6, 4, 1, 6, 5, 5, 3, 1, 2, 0, 8, 7, 6, 7, 3, 8, 7, 3, 4, 0, 4, 0, 1, 3, 4, 1, 1, 7, 5, 8, 9, 0, 0, 7, 5, 7, 4, 6, 4, 9, 6, 5, 6, 8, 0, 6, 3, 5, 7, 7, 3, 2, 8, 4, 6, 5, 4, 8, 8, 3, 5, 4, 7, 5, 9, 4, 5, 9, 9, 3, 7, 6, 1, 0, 6, 9, 3, 1, 7, 6, 6, 5, 3, 1, 8, 4, 9, 8, 0, 1, 2, 4, 6

Offset: 0

Author

Leonid Broukhis

Keywords

• nonn
• cons

Comments

Let P be the point in quadrant I where the curve y=sin(x) meets the circle x^2+y^2=1. Let d be the Dottie number. Then P=(d,sin(d)), and d is the slope at P of the sine curve. - Clark Kimberling, Oct 07 2011

From Ben Branman, Dec 28 2011: (Start)

The name "Dottie" is of no fundamental mathematical significance since it refers to a particular French professor who--no doubt like many other calculator users before and after her--noticed that whenever she typed a number into her calculator and hit the cosine button repeatedly, the result always converged to this value.

The number is well-known, having appeared in numerous elementary works on algebra already by the late 1880s (e.g., Bertrand 1865, p. 285; Heis 1886, p. 468; Briot 1881, pp. 341-343), and probably much earlier as well. It is also known simply as the cosine constant, cosine superposition constant, iterated cosine constant, or cosine fixed point constant. Arakelian (1981, pp. 135-136; 1995) has used the Armenian small letter ayb (ա, the first letter in the Armenian alphabet) to denote this constant. (End)

Examples

			0.73908513321516064165531208767387340401341175890075746496568063577328...
		

References

• H. Arakelian, The Fundamental Dimensionless Values (Their Role and Importance for the Methodology of Science). [In Russian.] Yerevan, Armenia: Armenian National Academy of Sciences, 1981.
• A. Baker, Theorem 1.4 in Transcendental Number Theory. Cambridge, England: Cambridge University Press, 1975.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000 (terms 0..499 from Ben Branman)
• Hrant Arakelian, New Fundamental Mathematical Constant: History, Present State and Prospects, Nonlinear Science Letters B, Vol. 1, No. 4, pp. 183-193.
• Mohammad K. Azarian, On the Fixed Points of a Function and the Fixed Points of its Composite Functions, International Journal of Pure and Applied Mathematics, Vol. 46, No. 1, 2008, pp. 37-44. Mathematical Reviews, MR2433713 (2009c:65129), March 2009. Zentralblatt MATH, Zbl 1160.65015.
• Trefor Bazett, What is cos( cos( cos( cos( cos( cos( cos( cos( cos( cos( cos( cos(...?? // Banach Fixed Point Theorem, YouTube video, 2022.
• J. Bertrand, Exercise III, in Traité d'algèbre, Vols. 1-2, 4th ed. Paris, France: Librairie Hachette et Cie, p. 285, 1865.
• Steven Finch, Exercises in Iterational Asymptotics II, arXiv preprint (2025). arXiv:2501.06065 [math.NT]
• Samuel R. Kaplan, The Dottie Number, Math. Magazine, 80 (No. 1, 2007), 73-74.
• T. H. Miller, On the imaginary roots of cos x = x, Proc. Edinburgh Math. Soc. 21, 160-162, 1902.
• V. Salov, Inevitable Dottie Number. Iterals of cosine and sine, arXiv preprint arXiv:1212.1027 [math.HO], 2012.
• David R. Stoutemyer, Inverse spherical Bessel functions generalize Lambert W and solve similar equations containing trigonometric or hyperbolic subexpressions or their inverses, arXiv:2207.00707 [math.GM], 2022.
• Eric Weisstein's World of Mathematics, Dottie Number
• Index entries for transcendental numbers.

Crossrefs

Cf. A009442, A177413, A182503, A197002, A200309, A212112, A212113, A217066.

Cf. A330119 (degrees-based analog).

Programs

Maple

evalf(solve(cos(x)=x,x), 140);  # Alois P. Heinz, Feb 20 2024

Mathematica

RealDigits[ FindRoot[ Cos[x] == x, {x, {.7, 1} }, WorkingPrecision -> 120] [[1, 2] ]] [[1]]
FindRoot[Cos[x] == x, {x, {.7, 1}}, WorkingPrecision -> 500][[1, 2]][[1]] (* Ben Branman, Apr 12 2008 *)
N[NestList[Cos, 1, 100], 20] (* Clark Kimberling, Jul 01 2019 *)
RealDigits[Root[{# - Cos[#] &, 0.739085}], 10, 100][[1]] (* Eric W. Weisstein, Jul 15 2022 *)
RealDigits[Sqrt[1 - (2 InverseBetaRegularized[1/2, 1/2, 3/2] - 1)^2], 10, 100][[1]] (* Eric W. Weisstein, Jul 15 2022 *)

PARI

solve(x=0,1,cos(x)-x) \\ Charles R Greathouse IV, Dec 31 2011

Python

from sympy import Symbol, nsolve, cos
x = Symbol("x")
a = list(map(int, str(nsolve(cos(x)-x, 1, prec=110))[2:-2]))
print(a) # Michael S. Branicky, Jul 15 2022

Formula

Equals twice A197002. - Hugo Pfoertner, Feb 20 2024

Extensions

More terms from David W. Wilson

Additional references from Ben Branman, Dec 28 2011