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A003957
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The Dottie number: decimal expansion of root of cos(x) = x.
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41
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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
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0,1
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COMMENTS
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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)
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REFERENCES
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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. 1-2, 4th ed. Paris, France: Librairie de L. Hachette et Cie, p. 285, 1865.
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LINKS
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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.
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.
Eric Weisstein's World of Mathematics, Dottie Number
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EXAMPLE
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0.73908513321516064165531208767387340401341175890075746496568063577328...
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MATHEMATICA
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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 *)
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PROG
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(PARI) solve(x=0, 1, cos(x)-x) \\ Charles R Greathouse IV, Dec 31 2011
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CROSSREFS
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Cf. A009442, A177413, A182503, A200309, A212112, A212113, A217066.
Cf. A330119 (degrees-based analog).
Sequence in context: A011330 A093587 A072334 * A021579 A139788 A093525
Adjacent sequences: A003954 A003955 A003956 * A003958 A003959 A003960
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KEYWORD
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nonn,cons
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AUTHOR
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Leonid Broukhis
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EXTENSIONS
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More terms from David W. Wilson
Additional references from Ben Branman, Dec 28 2011
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STATUS
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approved
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