TITLE: Is there a mathematical reason why rotation in the counterclockwise direction positive and clockwise rotation negative?
QUESTION [13 upvotes]: This inquiry has recently come to me in my study of trigonometry and the unit circle. It was said right from the very start that counterclockwise rotation were positive while clockwise rotations are negative, and I was wondering if there was a mathematical reason for this or if it was just picked that way.

REPLY [7 votes]: In a real sense it is an arbitrary choice. But with this choice, we get a nice correspondence between pairs $(x, y)$ and complex numbers $(x + yi)$ in which multiplication of complex numbers rotates in the positive direction.
In other words, $\displaystyle r_1 e^{i\theta_1} \cdot r_2 e^{i\theta_2} = r_1r_2 e^{i(\theta_1 + \theta_2)} $ where we define $e^{i \theta} = \cos \theta + i \sin \theta$.
It seems to me that if our positive direction was clockwise, and we wanted the real axis to be the x-axis, we would be forced to make the imaginary axis point downward.