Quaternian rotations

I dont fucking get it

i get that rotating using quats involves x y z and w, where x y and z is a vector that defines an axis about which you rotate by w, but what is w in? its not degrees, and its not radians, and its not cos degress..... arg

Wikipedia says:

It turns out that the angle of rotation α is also easy to read off if we are dealing with a unit quaternion z = a + v: we have a = cos(α/2).