Fixed points (mathematics) | Mathematical constants | Real transcendental numbers
Dottie number
In mathematics, the Dottie number is a constant that is the unique real root of the equation , where the argument of is in radians. The decimal expansion of the Dottie number is . Since is decreasing and its derivative is non-zero at , it only crosses zero at one point. This implies that the equation 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 for a complex variable has infinitely many roots, but unlike the Dottie number, they are not attracting fixed points. Using the Taylor series of the inverse of at (or equivalently, the Lagrange inversion theorem), the Dottie number can be expressed as the infinite series where each is a rational number defined for odd n as 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 , the root of . (Wikipedia).
