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In mathematics, the Dottie number is a constant that is the unique real root of the equation
where the argument of cos {\displaystyle \cos } is in radians. The decimal expansion of the Dottie number is 0.739085... {\displaystyle 0.739085...}.
Since cos − x {\displaystyle \cos-x} is strictly decreasing, it only crosses zero at one point. This implies that the equation cos = x {\displaystyle \cos=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. The generalised case cos z = z {\displaystyle \cos z=z} for a complex variable z {\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 f = cos − x {\displaystyle f=\cos-x} at π 2 {\textstyle {\frac {\pi }{2}}} , the Dottie number can be expressed as the infinite series π 2 + ∑ n o d d a n π n {\textstyle {\frac {\pi }{2}}+\sum _{n\,\mathrm {odd} }a_{n}\pi ^{n}} where each a n {\displaystyle a_{n}} is a rational number defined for odd n as