Dottie number

In mathematics, the Dottie number is a constant that is the unique real root of the equation
where the argument of formula_2 is in radians. The decimal expansion of the Dottie number is formula_3.
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. The generalised case formula_4 for a complex variable formula_5 has infinitely many roots, but unlike the Dottie number, they are not attracting fixed points.
Using the Taylor series of the inverse of formula_6 at formula_7 (or equivalently, the Lagrange inversion theorem), the Dottie number can be expressed as the infinite series formula_8 where each formula_9 is a rational number defined for odd n as
The name of the constant originates from Samuel Kaplan (2007) and refers to a French professor who observed the number by repeatedly pressing the cosine button on her calculator.