I was going through some basic examples of complex numbers (finding the argument and modulus) with my brother yesterday, and he asked

Why is the argument measured anticlockwise rather than clockwise [from the positive real axis]? Surely it's more intuitive to go clockwise.

But I could give no better answer than "by convention". Is there some historical reason that this is the case (is there any advantage of going anticlockwise)?

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    $\begingroup$ But in an Argand diagram, is it not more intuitive to place i above -i? If you do so, then you must go counterclockwise. $\endgroup$ – Asier Calbet Aug 6 '14 at 13:37
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    $\begingroup$ Already in traditional (planar) geometry, anti-clockwise is the "mathematically positive" direction for angles, for labelling of polygons etc. I don't see how clockwise (or anti-clockwise) should be more intuitive. $\endgroup$ – Hagen von Eitzen Aug 6 '14 at 14:05
  • $\begingroup$ We are just used to the way we read clocks. There is no link with intuition. $\endgroup$ – drhab Aug 6 '14 at 14:26

If you want Euler's formula to hold, then it follows from the sine and cosine functions going counterclockwise (otherwise you would have to shove an unnatural negative sign into an otherwise elegant formula). This fact for sine and cosine seems to follow from the convention to label the 'up' $y$ direction and 'right' $x$ directions as being positive, since the sine function needs to take on positive values as you raise the argument from zero.

Mind you I don't know if this is the actual explanation for how it came to be, but it certainly seems like a sufficiently good reason to have it this way.


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