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Is there any intuitive explanation for: $$e^{i\pi} + 1 = 0$$

About whether this question is a duplicate, what is asked for is not a proof but an explanation that helps with the not-so-intuitive aspects of the identity.


marked as duplicate by Zev Chonoles, Andrew D. Hwang, Claude Leibovici, Jonas Meyer, drhab Jun 18 '15 at 16:07

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    $\begingroup$ $-1$ is one unit to the left ($\pi$ radians from the right, i.e. postive direction) of $0$? $\endgroup$ – martini Jun 18 '15 at 12:35
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    $\begingroup$ What do you mean by "intuitive"? $\endgroup$ – user228113 Jun 18 '15 at 12:36
  • $\begingroup$ I am open to any interpretation of the word. $\endgroup$ – user55570 Jun 18 '15 at 12:38
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    $\begingroup$ If you think of $e^{i \theta}$ as a clockwise rotation of angle $\theta$ radians or remember $e^{i \theta} = \cos \theta + i \sin \theta$, then $e^{i \pi}$ is fairly obviously $-1$ $\endgroup$ – Henry Jun 18 '15 at 12:40
  • $\begingroup$ See also Has anyone talked themselves into understanding Euler's identity a bit? $\endgroup$ – Zev Chonoles Jun 18 '15 at 12:46

Unit vectors in the opposite direction along the real line in the complex plane add to zero.

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