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A003957 The Dottie number: decimal expansion of root of cos(x) = x. 22
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; text; internal format)
OFFSET

0,1

COMMENTS

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

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 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 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 (ա, the first letter in the Armenian alphabet) to denote this constant. (End)

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.

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.

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.

Mohammad K. Azarian, Fixed Points of a Quadratic Polynomial, Problem 841, College Mathematics Journal, Vol. 38, No. 1, January 2007, p. 60.  Solution published in Vol. 39, No. 1, January 2008, pp. 66-67.

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

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

Kaplan, Samuel R. The Dottie Number, Math. Magazine, 80 (No. 1, 2007), 73-74.

Miller, T. H. "On the Imaginary Roots of cos x = x." Proc. Edinburgh Math. Soc. 21, 160-162, 1902.

LINKS

_Ben Branman_, Apr 12 2008, Table of n, a(n) for n = 0..499

V. Salov, Inevitable Dottie Number. Iterals of cosine and sine, arXiv preprint arXiv:1212.1027, 2012. - From N. J. A. Sloane, Jan 04 2013

Eric Weisstein's World of Mathematics, Dottie Number

EXAMPLE

0.7390851332151606...

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

PROG

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

CROSSREFS

Sequence in context: A011330 A093587 A072334 * A021579 A139788 A093525

Adjacent sequences:  A003954 A003955 A003956 * A003958 A003959 A003960

KEYWORD

cons,nonn

AUTHOR

Leonid Broukhis

EXTENSIONS

More terms from David W. Wilson

Additional references from Ben Branman, Dec 28 2011

STATUS

approved

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Last modified December 20 08:42 EST 2014. Contains 252241 sequences.