Dottie number

The Dottie number is the unique real fixed point of the cosine function.

In mathematics, the Dottie number is a constant that is the unique real root of the equation

$\cos x=x$,

where the argument of $\cos$ is in radians. The decimal expansion of the Dottie number is $0.739085...$.

Since $\cos(x)-x$ is decreasing and its derivative is non-zero at $\cos(x)-x=0$, it only crosses zero at one point. This implies that the equation $\cos(x)=x$ has only one real solution. It is the single real-valued fixed point of the cosine function and is a nontrivial example of a universal attracting fixed point. It is also a transcendental number because of the Lindemann-Weierstrass theorem. The generalised case $\cos z=z$ for a complex variable $z$ has infinitely many roots, but unlike the Dottie number, they are not attracting fixed points.

Using the Taylor series of the inverse of $f(x)=\cos(x)-x$ at ${\textstyle {\frac {\pi }{2}}}$ (or equivalently, the Lagrange inversion theorem), the Dottie number can be expressed as the infinite series ${\textstyle {\frac {\pi }{2}}+\sum _{n\,\mathrm {odd} }a_{n}\pi ^{n}}$ where each $a_{n}$ is a rational number defined for odd n as

${\begin{aligned}a_{n}&={\frac {1}{n!2^{n}}}\lim _{m\to {\frac {\pi }{2}}}{\frac {\partial ^{n-1}}{\partial m^{n-1}}}{\left({\frac {\cos m}{m-\pi /2}}-1\right)^{-n}}\\&=-{\frac {1}{4}},-{\frac {1}{768}},-{\frac {1}{61440}},-{\frac {43}{165150720}},\ldots \end{aligned}}$

The name of the constant originates from a professor of French named Dottie who observed the number by repeatedly pressing the cosine button on her calculator.

If a calculator is set to take angles in degrees, the sequence of numbers will instead converge to $0.999847...$, the root of $\cos \left({\frac {\pi }{180}}x\right)=x$.

Closed form

The Dottie number can be expressed as

${\sqrt {1-\left(2{\text{I}}_{\frac {1}{2}}^{-1}\left({\frac {1}{2}},{\frac {3}{2}}\right)-1\right)^{2}}},$

where $I^{-1}$ is the inverse regularized Beta function. Particularly, in Microsoft Excel and LibreOffice Calc spreadsheets as SQRT(1-(2*BETA.INV(1/2,1/2,3/2)-1)^2), in Mathematica computer algebra system as Sqrt[1 - (2 InverseBetaRegularized[1/2, 1/2, 3/2] - 1)^2].

External links

• Miller, T. H. (Feb 1890). "On the numerical values of the roots of the equation cosx = x". Proceedings of the Edinburgh Mathematical Society. 9: 80–83. doi:10.1017/S0013091500030868.
• Salov, Valerii (2012). "Inevitable Dottie Number. Iterals of cosine and sine". arXiv:1212.1027.
• Azarian, Mohammad K. (2008). "ON THE FIXED POINTS OF A FUNCTION AND THE FIXED POINTS OF ITS COMPOSITE FUNCTIONS" (PDF). International Journal of Pure and Applied Mathematics.