While studying through the number system, i notice that positive side is from 0 to +ve infinity. The direction is left to right. However, this is opposite in case of angles. The sort of curved number system starts from 0 to 180 ( from right to left). Is their any good reason why unlike a number system, direction of measuring angles is right to left ?

PS: I actually first thought that, it's due to +x +y axis-plane coming on the right side. But then what about -x + y plane coming on the left side. The angles should go negative after 90. ie. -91, -92.... -180.

Logic probably is something different! What is it ?

Thanks V.

  • 1
    $\begingroup$ Well, technically it is counterclockwise, not right to left. But it's still a good question. $\endgroup$
    – 6005
    Jun 11 '13 at 20:05
  • $\begingroup$ I think it has historically become like that. Just like positive rotation in the xy plane is counter clockwise instead of clockwise. I don't know why. $\endgroup$
    – imranfat
    Jun 11 '13 at 20:05
  • $\begingroup$ You might want to wait a day or two before accepting an answer. I didn't really know, and if you wait, someone who does know might come along and tell you. You can un-accept the answer now and re-accept it later if you still like it. $\endgroup$
    – MJD
    Jun 11 '13 at 20:19
  • $\begingroup$ I now question the whole premise here. Of the first page of Google Image results for "protractor" there are 26 protractors that go both ways, two that go clockwise, and none that go counterclockwise. $\endgroup$
    – MJD
    Jun 13 '13 at 14:34
  • $\begingroup$ Btw, to add up, i got an another answer from a source, that when you keep a "protractor" on the "0" of a coordinate system, it's more intuitive to start "angles" from the right side ( being +1 on the right), than anywhere else. One of the reasons probably. $\endgroup$
    – Vishwas
    Jun 14 '13 at 5:50

Mathematicians always measure angles in the counterclockwise direction; a clockwise angle the same size as one of $30^\circ$ is called a $-30^\circ$ angle. The protractor is consistent with this practice.

Why mathematical practice measures counterclockwise, I do not know. I was going to justify it by reference to the common layout of the cartesian plane ($x$ coordinate increasing left to right, as the language is written, and $y$ coordinate increasing bottom to top in accordance with every linguistic metaphor of increase) but on further thought I saw no reason why the zero angle couldn't have been be on the $y$-axis, with angle increasing clockwise, and I wonder why it wasn't done this way, for consistency with existing nautical practice.

As Daniel McLaury points out, this would have made the vertical axis the real axis and the horizontal axis the imaginary axis.

There is a real question of history here, which may or may not be resolvable; some of these things are just mysteries. I believe nobody knows why, when the first automobile traffic lights were constructed, they had red on the top and green on the bottom, opposite to the design of the railroad signal lights they imitated.

railroad vertical signal with green on top and red on the bottom

  • $\begingroup$ It's the right hand rule. They used to do it the other way in the USSR. $\endgroup$
    – Alexander Gruber
    Jun 11 '13 at 20:08
  • $\begingroup$ Which way did they do it in the USSR? $\endgroup$
    – MJD
    Jun 11 '13 at 20:09
  • 4
    $\begingroup$ Ah, so Russians simply disagree as to which root of $x^2 + 1$ is $+i$ and which is $-i$. :) $\endgroup$ Jun 11 '13 at 20:12
  • 2
    $\begingroup$ Yes, $G(\mathbb{C},\mathbb{R})$ swaps America and Russia. :) $\endgroup$
    – Alexander Gruber
    Jun 11 '13 at 20:15
  • 1
    $\begingroup$ Interestingly, if you want to draw (e.g. on a nautical map) a clockwise angle such as NNE $=22.5^\circ$, you place the $22.5$ of the anticlockwise protractor scale on the (vertical) north line. Also, I seem to find onl yimages of protractors with both scales. $\endgroup$ Jun 11 '13 at 20:34

My guess is that it has to do with the original sundials. While sundials that were placed on the floor had shadows moving clockwise, those on the wall had shadows moving counterclockwise. So I would argue that it's because mathematicians like having their sundials on the wall instead of the floor. The reason for this I cannot explain.

  • 3
    $\begingroup$ mathematicians tend to have offices with enough trip hazards $\endgroup$ Jun 11 '13 at 20:12
  • 1
    $\begingroup$ Of course this depends on the hemisphere, but it is safe to assume Greeks or Babylonians at work. $\endgroup$ Jun 11 '13 at 20:29

Well, there is going have to be some arbitrary choice somewhere. That said, the choice of angles is compatible with the way we generally choose to lay out the complex plane: real numbers on the horizontal axis increasing from left to right and imaginary numbers on the vertical axis.

In other words, if +1 is going to be to the right and -1 is going to be to the left, then we have to put the zero to the right, since $1 = e^{0 i}$ and $-1 = e^{\pi i}$. If we want to put $+i = e^{i \pi/2}$ pointing up, then moreover we need to measure counterclockwise. (Of course there's no particularly firm distinction between "$+i$" and "$-i$" -- that's an arbitrary convention too.)

If we want to get touchy-feely about it, I imagine that these choices are probably a consequence of the fact that our conventions were developed by people in 18th century France who read left-to-right rather than right-to-left or top-to-bottom.

  • $\begingroup$ Well, the zero angle has to be on whatever is the real axis, so that real numbers have an argument of zero, but which way is a positive rotation is negotiable. $\endgroup$
    – Kaz
    Jun 11 '13 at 23:40
  • $\begingroup$ And indeed I said as much in my answer. $\endgroup$ Jun 11 '13 at 23:50

I am surely wrong here, but can there be a relation between quaternions and the anticlockwise rotation they produce when one is multiplied by another one? Sounds far-fetched though -_-"

  • $\begingroup$ I strongly assume that protractor scales were in use long before Hamilton finally found the quaternions. And their anticlockwise rotation is also a convention ... $\endgroup$ Jun 11 '13 at 20:36
  • $\begingroup$ Wikipedia reads "(quaternions) were first described by Irish mathematician William Rowan Hamilton in 1843", which leads us to ask; when was the first recorded wide-use of a counterclockwise-scaled protractor? $\endgroup$
    – K. Rmth
    Jun 11 '13 at 20:47
  • $\begingroup$ More importantly, I assume that protractors are mainly used by people not caring for quaternions, shmaternions in their application ... But here's a loose hint towards 1801, but... $\endgroup$ Jun 11 '13 at 20:53

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