Is there a mathematical reason why rotation in the counterclockwise direction positive and clockwise rotation negative?

Solution 1:

There is no mathematical reason for this.

The reason for picking it this way (irritating people in other fields, as you can see from the comments!) is that rotating the [positive] $x$-axis onto the [positive] $y$-axis is about the simplest rotation you can think of, so we decide to call it "positive". And clearly rotating the positive $x$-axis onto the positive $y$-axis is, in the normal way we draw Cartesian coordinates, anticlockwise.

There is a bonus. The point $(1,0)$, rotated through an anticlockwise angle $\theta$, ends up at $(\cos\theta,\sin\theta)$, which it wouldn't have if we had defined "positive" rotations as going in the other direction.

Solution 2:

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.