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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.

J. M. ain't a mathematician
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hubble
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    It is definitely more convention than anything else. But I would assume that convention is so that "positive" rotations first take us to points $(x, y)$ where both $x$ and $y$ are positive, if we start on the $x$-axis. Negative rotations take us to points $(x, y)$ where $x$ and $y$ are both negative. – pjs36 Jun 21 '16 at 16:25
  • In graphics, a common convention is that one starts at the top and rotates clockwise. The existence of different conventions increases the probability of error. – André Nicolas Jun 21 '16 at 16:30
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    In physics, we have the "right hand rule of thumb" for combining vectors of certain forces. This rule is counterclockwise in rotation if the resulting vector is "pointing out of the page at the observer" which is very easy for a right hand to imitate. – abiessu Jun 21 '16 at 16:33
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    The fact that counterclockwise rotations are called positive is the combined result of two arbitrary conventions. One is the convention about which rotations count as positive. The other is the convention about how clocks are designed. – Andreas Blass Jun 21 '16 at 17:14
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    @AndreasBlass The direction in which the hands of a clock move is not a convention. It is chosen to coincide with the direction in which the shadow of a sundial moves on the northern hemisphere. – Carlos Esparza Feb 11 '20 at 13:17

5 Answers5

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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.

Martin Kochanski
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    To physicists who complain about this: we will change our convention when you admit that it is absurd that electrons, the main carriers of charge, should be assigned "negative" charge. Make electrons positive, and we will consider our decision. – Martin Kochanski Jun 21 '16 at 17:00
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    +1 for the comment alone :) But I think the "bonus" puts the cart before the horse a bit. I'm quite sure that, had we decided clockwise rotations were positive, the sine and cosine functions would be defined to reflect this. – pjs36 Jun 21 '16 at 20:32
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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.

Eric Wilson
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    I thought that your answer was an order of magnitude better than all the other answers. I hope that more people will read it and upvote it. +1 – user729424 Mar 17 '20 at 14:43
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Besides the right-handed system convention, sometimes we use clockwise sense. Of course the rotation of hour/minute/second hand of a clock. Also, whole-circle bearing is using clockwise direction (though polar coordinates take anti-clockwise as positive).

Interestingly the elliptic integral $\displaystyle \int_{0}^{t} \sqrt{a^2\cos^2 t+b^2\sin^2 t} dt=aE\left( t,\sqrt{1-\frac{b^2}{a^2}} \, \right)$, used in calculating the arc length of an ellipse, are based on $(x,y)=(a\sin t, b\cos t)$ which is clockwise sense.

Ng Chung Tak
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Draw the axes with the $y$ one going downwards, while the $x$ one goes as usual. Now clockwise rotation is positive.

The mathematics behind this behavior is called orientation.

Giuseppe Negro
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Perhaps it merely follows the convention of numbering the quadrants, which increase in a counterclockwise fashion. So trig rotation begins at I and progresses through IV. This is the guidance I give my students and it seems to help them.

MathIsEasyWithMrSmith
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