How can I use fractions with hotkeys for translation/rotation/scale? For instance if I want to rotate an object by 360/16 degrees around the z-axis, I would expect to type: R Z 360/16. However the / key acts as a kind of toggle and midway through typing pressing / turns 360 into 1/360, so I end up with a rotation of 1/36016 degrees as the rotation if I were to keep typing.

How do I disable this and allow it to evaluate fractions or other python expressions similar to other transform property boxes?
You simply need to use the = operator to enter Advanced Entry Mode, after that you can enter your fractional value or simple Python expression: