Dottie Number


The Dottie number is the name given by Kaplan (2007) to the unique real root of cosx=x (namely, the unique real fixed point of the cosine function), which is 0.739085... (OEIS A003957). 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 (first letter in the Armenian alphabet) to denote this constant.

This root is a simple nontrivial example of a universal attracting fixed point. It is also transcendental as a consequence of the Lindemann-Weierstrass theorem.

It can be given in closed form as


where I_z^(-1)(a,b) is the inverse of the regularized beta function.

The Dottie number r gives almost integers

 r((160)/pi)^(1/13) approx 0.9999996766

(L. A. Broukhis, pers. comm.). and

 e^(e^(pi/3)-Gamma(r)) approx 5.000000017

(K. Hammond, pers. comm.).

See also

Cosine, Fixed Point

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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.Arakelian, H. "The New Fundamental Constant of Mathematics." Pan-Arm. Sci. Rev., London 3, 18-21, 1995.Baker, A. 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.Briot, C. M. Leons d'algèbre conformes aux programmes officiels de l'enseignement des lycées, 11th ed. Paris, France: Librairie Ch. Delagrave, pp. 341-343, 1881.Gaidash, T. "Why Dottie=...." Feb. 23, 2022., E. Schlüssel zur Sammlung von Beispielen und Aufgaben aus der allgemeinen Arithmetik und Algebra, Volume 2, 3rd ed. Cologne, Germany: Verlag der M. DuMont-Schauberg'schen Buchhandlung, p. 468, 1886.Kaplan, S. R. "The Dottie Number." Math. Mag. 80, 73-74, 2007.Miller, T. H. "On the Numerical Values of the Roots of the Equation cosx=x." Proc. Edinburgh Math. Soc. 9, 80-83, 1890.Miller, T. H. "On the Imaginary Roots of cosx=x." Proc. Edinburgh Math. Soc. 21, 160-162, 1902.Salov, V. "Inevitable Dottie Number. Iterals of Cosine and Sine." 1 Dec 2012., N. J. A. Sequence A003957 in "The On-Line Encyclopedia of Integer Sequences."Stoutemyer, D. R. "Inverse Spherical Bessel Functions Generalize Lambert W and Solve Similar Equations Containing Trigonometric or Hyperbolic Subexpressions or Their Inverses." 2 Jul 2022.

Referenced on Wolfram|Alpha

Dottie Number

Cite this as:

Weisstein, Eric W. "Dottie Number." From MathWorld--A Wolfram Web Resource.

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