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

${\displaystyle \cos x=x}$,

where the argument of ${\displaystyle \cos }$ is in radians.

The decimal expansion of the Dottie number is ${\displaystyle 0.739085133215160641655312087673873404...}$.^[1]

Since ${\displaystyle \cos(x)-x}$ is decreasing and its derivative is non-zero at ${\displaystyle \cos(x)-x=0}$, it only crosses zero at one point. This implies that the equation ${\displaystyle \cos(x)=x}$ 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.^[2] The generalised case ${\displaystyle \cos z=z}$ for a complex variable ${\displaystyle z}$ has infinitely many roots, but unlike the Dottie number, they are not attracting fixed points.

Using the Taylor series of the inverse of ${\displaystyle f(x)=\cos(x)-x}$ 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 ${\displaystyle a_{n}}$ is a rational number defined for odd n as^{[3][4][5][nb 1]}

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

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.^[3]

If a calculator is set to take angles in degrees, the sequence of numbers will instead converge to ${\displaystyle 0.999847...}$,^[6] the root of ${\displaystyle \cos \left({\frac {\pi }{180}}x\right)=x}$.

The Dottie number, for which an exact series expansion can be obtained using the Faà di Bruno formula, has interesting connections with the Kepler and Bertrand's circle problems.^[7]

The Dottie number can be expressed as

${\displaystyle D={\sqrt {1-\left(2I_{\frac {1}{2}}^{-1}\left({\frac {1}{2}},{\frac {3}{2}}\right)-1\right)^{2}}},}$

where ${\displaystyle I^{-1}}$ is the inverse of the regularized beta function. This value can be obtained using Kepler's equation, along with other equivalent closed forms.^[8][7]

Dottie number can be represented as

${\displaystyle D={\sqrt {1-\left(1-\left(\int _{0}^{\infty }{\frac {32(z-\sinh(z))^{2}+24\pi ^{2}}{\left(4(z-\sinh(z))^{2}+3\pi ^{2}\right)^{2}+16\pi ^{2}(z-\sinh(z))^{2}}}\,dz\right)^{-1}\right)^{2}}}}$.^[9]