

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
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.


LINKS

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


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 *)
RealDigits[Root[{#  Cos[#] &, 0.739085}], 10, 100][[1]] (* Eric W. Weisstein, Jul 15 2022 *)
RealDigits[Sqrt[1  (2 InverseBetaRegularized[1/2, 1/2, 3/2]  1)^2], 10, 100][[1]] (* Eric W. Weisstein, Jul 15 2022 *)


PROG

(Python)
from sympy import Symbol, nsolve, cos
x = Symbol("x")
a = list(map(int, str(nsolve(cos(x)x, 1, prec=110))[2:2]))


CROSSREFS

Cf. A330119 (degreesbased analog).


KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



