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I have just started reading about the Modulus and Argument of Complex Numbers. In the definition, it is said that:

If z is not equal to 0 and $-\pi < \theta \le \pi$ , then $\theta$ is the principal argument of z, written $\theta = \arg(z)$.

My question is about the interval: why do we take $-\pi < \theta \le \pi$, which, if I understand it correctly starts at $180$ degrees, namely $-\pi$ (the most left point on the $x$ axis of the zero circle), and moves counter clockwise until it reaches $180$ degrees again? Why don't we use, say, $[0,2\pi)$?

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  • $\begingroup$ Here's a MathJax tutorial :) $\endgroup$ – Shaun Sep 3 '17 at 12:05
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    $\begingroup$ Not much of an advantage, but in this principal value range, if $\arg(z) = t$ then $\arg(z^*)$ is $-t$. $\endgroup$ – samjoe Sep 3 '17 at 12:08
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    $\begingroup$ Use $\pi$ for $\pi$ and enclose all mathematics in $s $\endgroup$ – Shaun Sep 3 '17 at 12:10
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    $\begingroup$ Note that $-\pi<\Theta\le\pi$ because $\Theta=\pi$ is possible. $\endgroup$ – Shaun Sep 3 '17 at 12:12
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    $\begingroup$ @Vitale $z^*$ is the conjugate of $z$. $\endgroup$ – samjoe Sep 3 '17 at 12:15
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It's just a matter of convention.

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  • $\begingroup$ Thank you. What does this convention refer too, and what meaning lies behind it? $\endgroup$ – Vitale Sep 3 '17 at 12:07
  • $\begingroup$ It's somewhat arbitrary. $\endgroup$ – Shaun Sep 3 '17 at 12:08
  • $\begingroup$ I wouldn't say that it is arbitrary. This convention is related to the principal determination of the complex log function which, rather often, is defined by a cut along the negative $x$ axis. $\endgroup$ – Jean Marie Sep 3 '17 at 16:51

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