Dottie number

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

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

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

where the argument of ${\displaystyle \cos }$ is in radians. The decimal expansion of the Dottie number is ${\displaystyle 0.739085...}$.[1]

It can be trivially proven that the equation has only one solution by the intermediate value theorem in the real plane. 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 ${\displaystyle {\frac {\pi }{2}}}$ (or equivalently, the Lagrange inversion theorem), the Dottie number can be expressed as the infinite series ${\displaystyle {\frac {\pi }{2}}+\sum _{n\,\mathrm {odd} }a_{n}\pi ^{n}}$ where each ${\displaystyle a_{n}}$ is a rational number defined for odd n as

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

[3][4][5][nb 1]

The name of the constant originates from Samuel Kaplan (2007) and refers to a French professor who observed the number after by repeatedly pressing the cosine button on her calculator.[3]