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A003957 The Dottie number: decimal expansion of root of cos(x) = x. 24


%S 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,

%T 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,

%U 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

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

%C The Kaplan reference gives "Pedagogical examples [about the Dottie number and other universal attracting fixed points] for several courses ranging from Calculus I to Complex Analysis." - _Jonathan Vos Post_, Apr 04 2007

%C 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

%C From Ben Branman, Dec 28 2011: (Start)

%C The name "Dottie" is of no fundamental mathematical significance since it refers to a particular French professor who--no doubt like many other calculator uses 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.

%C 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)

%D 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.

%D A. Baker, Theorem 1.4 in Transcendental Number Theory. Cambridge, England: Cambridge University Press, 1975.

%D Bertrand, J. Exercise III in Traité d'algèbre, Vols. 1-2, 4th ed. Paris, France: Librairie de L. Hachette et Cie, p. 285, 1865.

%H G. C. Greubel, <a href="/A003957/b003957.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..499 from Ben Branman)

%H Hrant Arakelian, <a href="http://www.nonlinearscience.com/paper.php?pid=0000000113">New Fundamental Mathematical Constant: History, Present State and Prospects</a>, Nonlinear Science Letters B, Vol. 1, No. 4, pp. 183-193.

%H Mohammad K. Azarian, <a href="http://www.ijpam.eu/contents/2008-46-1/3/3.pdf">On the Fixed Points of a Function and the Fixed Points of its Composite Functions</a>, 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.

%H Samuel R. Kaplan, <a href="https://www.maa.org/sites/default/files/Kaplan2007-131105.pdf">The Dottie Number</a>, Math. Magazine, 80 (No. 1, 2007), 73-74.

%H T. H. Miller, <a href="http://dx.doi.org/10.1017/S0013091500030868">On the imaginary roots of cos x = x</a>, Proc. Edinburgh Math. Soc. 21, 160-162, 1902.

%H V. Salov, <a href="http://arxiv.org/abs/1212.1027">Inevitable Dottie Number. Iterals of cosine and sine</a>, arXiv preprint arXiv:1212.1027 [math.HO], 2012.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DottieNumber.html">Dottie Number</a>

%e 0.7390851332151606...

%t RealDigits[ FindRoot[ Cos[x] == x, {x, {.7, 1} }, WorkingPrecision -> 120] [[1, 2] ]] [[1]]

%t FindRoot[Cos[x] == x, {x, {.7, 1}}, WorkingPrecision -> 500][[1, 2]]][[1]] - _Ben Branman_, Apr 12 2008

%o (PARI) solve(x=0,1,cos(x)-x) \\ _Charles R Greathouse IV_, Dec 31 2011

%Y Cf. A009442, A177413, A182503, A200309, A212112, A212113, A217066.

%K cons,nonn

%O 0,1

%A _Leonid Broukhis_

%E More terms from _David W. Wilson_

%E Additional references from _Ben Branman_, Dec 28 2011

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Last modified January 20 17:05 EST 2019. Contains 319335 sequences. (Running on oeis4.)