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 $\frac$ (or equivalently, the Lagrange inversion theorem), the Dottie number can be expressed as the infinite series $\frac+\sum_ a_ \pi^$ where each $a_n$ is a rational number defined for odd ''n'' as

:$\begin a_n&=\frac\lim_ \frac \\&=-\frac,-\frac,-\frac,-\frac,\ldots \end$

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 $\cos\left(\fracx\right) = x$.

Closed form

The Dottie number can be expressed as
:$\sqrt,$
where $I^$ 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 _-_(2_InverseBetaRegularized[1/2,_1/2,_3/2-_1)^2.html" ;"title="/2,_1/2,_3/2.html" ;"title=" - (2 InverseBetaRegularized[1/2, 1/2, 3/2"> - (2 InverseBetaRegularized[1/2, 1/2, 3/2- 1)^2">/2,_1/2,_3/2.html" ;"title=" - (2 InverseBetaRegularized[1/2, 1/2, 3/2"> - (2 InverseBetaRegularized[1/2, 1/2, 3/2- 1)^2/code>.


Notes

References

Mathematical constants
Real transcendental numbers
Fixed points (mathematics)


External links


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* {{cite journal , url=https://ijpam.eu/contents/2008-46-1/3/3.pdf , journal=International Journal of Pure and Applied Mathematics , title=ON THE FIXED POINTS OF A FUNCTION AND THE FIXED POINTS OF ITS COMPOSITE FUNCTIONS, first1=Mohammad K. , last1=Azarian, year=2008