Let's say that we proved that the radian measures of angles are real numbers (i.e. if we have an angle $x \text{ rads}$, then $x \in \Bbb R$) since the radian measure of an angle $\theta$ is the length of the arc that subtends an angle of $\theta$ at the centre of a unit circle and the lengths attain real values only. [A good way to visualize this is by drawing a number line tagential to a point on the unit circle and then 'warping' the line along the circle's circumference].
Now, why do we prove this? I would like to know about the implications of this result?
Thanks!