Akiva Weinberger

195 $\begingroup$ From Wikipedia A transcendental number is a real or complex number that is not algebraic A transcendental function is an analytic function that does not satisfy a polynomial equation However these definitions are arguably rather cryptic to those who are not familiar with the literature of higher mathematics. So in layman's terms, what exactly does it mean to be transcendental? How would a transcendental number be different from an ordinary number, say 5.

More an intuitive thought than a rigorous proof: According to Crofton's formula the length of rectifiable plane curve $\gamma$ is given as $$\frac{1}{4}\int\int n_\gamma(\varphi, p) ~d\varphi ~dp$$ Where $n_\gamma(\varphi, p)$ is the amount of intersections of a line with orientation (angle) $\varphi$ and position (from the origin) of $p$ with the curve (see article). Let's take a horizontal line and bring it closer to the curve from infinity distance.