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A003957
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Decimal expansion of root of cos x = x.
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16
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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
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| The unique root of cos(x)=x has been called the Dottie number. This root is a simple nontrivial example of a universal attracting fixed point. The story of how the Dottie number got its name and mathematical concepts relating to this value can be used as teaching tools. Pedagogical examples are given for several courses ranging from Calculus I to Complex Analysis. [Kaplan] - Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 04 2007
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. [From Clark Kimberling, Oct 7 2011]
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.
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 (first letter in the Armenian alphabet) to denote this constant.
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REFERENCES
| Hrant Arakelian, New Fundamental Mathematical Constant: History, Present State and Prospects, Nonlinear Science Letters B, Vol. 1, No. 4, pp. 183-193; http://www.nonlinearscience.com/paper.php?pid=0000000113.
Samuel R. Kaplan, The Dottie Number, Math. Magazine, 80 (No. 1, 2007), 73-74.
Arakelian, H. The Fundamental Dimensionless Values (Their Role and Importance for the Methodology of Science). [In Russian.] Yerevan, Armenia: Armenian National Academy of Sciences, 1981.
Baker, A. Theorem 1.4 in Transcendental Number Theory. Cambridge, England: Cambridge University Press, 1975.
Miller, T. H. "On the Imaginary Roots of ." Proc. Edinburgh Math. Soc. 21, 160-162, 1902.
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
| Ben Branman (137ben(AT)comcast.net), Apr 12 2008, Table of n, a(n) for n = 0..499
Eric Weisstein's World of Mathematics, Cosine
Eric Weisstein's World of Mathematics, Almost Integer
Eric Weisstein's World of Mathematics, Dottie Number
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EXAMPLE
| 0.7390851332151606...
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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 (137ben(AT)comcast.net), Apr 12 2008
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PROG
| (PARI) solve(x=0, 1, cos(x)-x) \\ Charles R Greathouse IV, Dec 31 2011
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CROSSREFS
| Sequence in context: A011330 A093587 A072334 * A021579 A139788 A093525
Adjacent sequences: A003954 A003955 A003956 * A003958 A003959 A003960
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KEYWORD
| cons,nonn
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AUTHOR
| Leonid Broukhis (leo(AT)mailcom.com)
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EXTENSIONS
| More terms from David W. Wilson (davidwwilson(AT)comcast.net)
Additional references from Ben Branman (137ben(AT)comcast.net), Dec 28 2011
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