How do we make sense of angles which take irrational measures such as $\sqrt 2 ^\circ$? If you were asked to draw such an angle how would you do so? Would you take it to a limit? Can the degree measure take the value of all real numbers?
 A: You cannot construct this angle with the usual construction tools (straightedge and compasses). The reason behind this is that the Gelfond-Schneider theorem tells us that the complex number $$\cos\sqrt 2^\circ +i\sin\sqrt 2^\circ=\zeta_{360}^{\sqrt 2}$$
(where $\zeta_{360}$ denotes a primitive $360$th root of unity) is transcendental.
A: Another answer addresses the question of why this angle can't be constructed by ruler and compass. However, you also asked what the angle means.
I presume you know what an angle $\pi/n$ means. That is, it's the angle which, if added $n$ times, gives you $\pi$. (Proving such an angle exists is of course another matter.) And it is then easy to see what $\frac{m}{n}\pi$ means.
To say what $\sqrt{2}$ degrees means, it's the only angle that is larger than $q$ degrees, but smaller than $r$ degrees, whenever $q$ and $r$ are rational numbers satisfying $q < \sqrt{2} < r$. Again, I'm not offering a proof that such an angle exists.
To place this in a larger perspective, the real problem is to establish a mapping $\phi$ from the set of real numbers to the set of all rotations about the origin in such a way that $\phi(\alpha)$ represents the counterclockwise rotation of angle $\alpha$. More specifically:


*

*$\phi(\alpha)$ must be a rotation centred at the origin for all $\alpha$. (More explicitly, in case there is any doubt about what a rotation means: it's either the identical transformation or an isometry of the plane whose only fixed point is the origin.)

*We must always have $\phi(\alpha + \beta) = \phi(\alpha) \circ \phi(\beta)$.

*$\phi(360)$ must be the identical transformation.

*Whenever $0 < \alpha < 180$, the rotation $\phi(\alpha)$ must take the point $(1,0)$ to a point in the top half-plane (i.e., the rotation is counterclockwise).
It turns out that there is exactly one way to assign a rotation $\phi(\alpha)$ to each number $\alpha$ in such a way that all four conditions are satisfied. Then $\phi(\sqrt{2})$ is the counterclockwise rotation with angle $(\sqrt{2})^{\circ}$.
