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

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

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

Since ${\displaystyle \cos(x)-x}$ is strictly decreasing, 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 French professor 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 ${\displaystyle \cos \left({\frac {\pi }{180}}x\right)=x}$.

Notes[]

References[]

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.