Consider the two imaginary numbers $i$ and $-i$. Is there any fundamental difference between the two of them? If I take the field $\mathbb{C}$ and apply the map $a + bi \mapsto a - bi$ does the image end up meaningfully different from the field I started with? Or when we write out complex numbers are we arbitrarily choosing which of the non-real solutions to $z^4 = 1$ to call $i$ and which to call $-i$?
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3$\begingroup$ In other words, you are asking whether conjugation is an automorphism of $\mathbb{C}$. The answer is yes. $\endgroup$– yohBSFeb 11, 2014 at 20:45
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$\begingroup$ I don't know exactly what you mean by the difference of $i$ and $-i$ but I do know if you use residues to compute some integral and you mix them up, things will go very wrong. So in my view fundamentally spoken, they are each other's conjugates and so they are different. Please correct me if I do not comprehend the question. $\endgroup$– imranfatFeb 11, 2014 at 20:46
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5$\begingroup$ They are the same but different. $\endgroup$– copper.hatFeb 11, 2014 at 20:48
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$\begingroup$ To add to comments made above. It depends what you mean by fundamentally different. If we interpret $\mathbb{C}$ as $\mathbb{R}^{2}$, then this map is just rotation by 180 degrees. So everything is there, just rotated. In particular, this does not affect the field structure. $\endgroup$– acyrlFeb 11, 2014 at 21:04
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2$\begingroup$ @acyrl: Actually, the map is reflection over the $x$ (real) axis, not a rotation. $\endgroup$– Cameron BuieFeb 11, 2014 at 21:32
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2 Answers
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If you have some previous notion of orientation for the plane -- some notion of "clockwise" and "counter-clockwise" -- then you can specify which solution of $z^2+1=0$ is which. And vice-versa: given a choice of $i$ for $\mathbb C$, you get a corresponding orientation for the plane $\mathbb R^2$ using the usual bijection.
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Yes, one of the roots of the polynomial $z^2+1$ is called $i$, the other $-i$; since one is the negative of the other. You can switch the names if you really want to and say that "one root is called $-i$ and the other is called $i$", but this would not make much difference, right? Traditionally, $i$ is drawn in the upper half-plane and $-i$ in the lower, but this is only a tradition. I am not sure what else would you want to know.
answered Feb 11, 2014 at 21:05
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$\begingroup$ This is pretty much what I was looking for. I tried to formalize my question in the description, but what I was really wondering was whether the distinction between $i$ and $-i$ was anything but arbitrary. $\endgroup$ Feb 12, 2014 at 21:32
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$\begingroup$ @kuzzooroo: Their names are as arbitrary as your name at MSE, I suppose, except that one has to be the negative of the other. The latter is not arbitrarily at all! $\endgroup$ Feb 12, 2014 at 21:36