

A003957


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


41



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
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OFFSET

0,1


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 whono doubt like many other calculator users before and after hernoticed 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 wellknown, 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. 341343), 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. 135136; 1995) has used the Armenian small letter ayb (ա, the first letter in the Armenian alphabet) to denote this constant. (End)


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.
Bertrand, J. Exercise III in Traité d'algèbre, Vols. 12, 4th ed. Paris, France: Librairie de L. Hachette et Cie, p. 285, 1865.


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. 183193.
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. 3744. 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.
Samuel R. Kaplan, The Dottie Number, Math. Magazine, 80 (No. 1, 2007), 7374.
T. H. Miller, On the imaginary roots of cos x = x, Proc. Edinburgh Math. Soc. 21, 160162, 1902.
V. Salov, Inevitable Dottie Number. Iterals of cosine and sine, arXiv preprint arXiv:1212.1027 [math.HO], 2012.
Eric Weisstein's World of Mathematics, Dottie Number


EXAMPLE

0.73908513321516064165531208767387340401341175890075746496568063577328...


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 *)


PROG

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


CROSSREFS

Cf. A009442, A177413, A182503, A200309, A212112, A212113, A217066.
Cf. A330119 (degreesbased analog).
Sequence in context: A011330 A093587 A072334 * A021579 A139788 A093525
Adjacent sequences: A003954 A003955 A003956 * A003958 A003959 A003960


KEYWORD

nonn,cons


AUTHOR

Leonid Broukhis


EXTENSIONS

More terms from David W. Wilson
Additional references from Ben Branman, Dec 28 2011


STATUS

approved



