The Dottie number is the name given by Kaplan (2007) to the unique real root of (namely, the unique real fixed
point of the cosine function), which is 0.739085...
(OEIS A003957). Starting from any real , the fixed
point iteration
converges to the Dottie number.
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; Briot 1881, pp. 341-343; Heis 1886, p. 468), 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.
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., London3,
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. https://math.stackexchange.com/questions/4389528/why-dottie-sqrt1-left2-text-i-1-frac12-frac-12-frac-32-1-right2.Heis,
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 ." Proc. Edinburgh Math. Soc.9, 80-83,
1890.Miller, T. H. "On the Imaginary Roots of ." Proc. Edinburgh Math. Soc.21, 160-162,
1902.Salov, V. "Inevitable Dottie Number. Iterals of Cosine and
Sine." 1 Dec 2012. https://arxiv.org/abs/1212.1027.Sloane,
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. https://arxiv.org/abs/2207.00707.
(namely, the unique real fixed
point of the cosine function), which is 0.739085...
(OEIS A003957). Starting from any real 
, the fixed
point iteration 
converges to the Dottie number.
is the inverse of the regularized
beta function.
gives almost integers
." Proc. Edinburgh Math. Soc. 9, 80-83,
1890.Miller, T. H. "On the Imaginary Roots of 
." Proc. Edinburgh Math. Soc. 21, 160-162,
1902.Salov, V. "Inevitable Dottie Number. Iterals of Cosine and
Sine." 1 Dec 2012.