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, $[0,2\pi)$?

  • $\begingroup$ Here's a MathJax tutorial :) $\endgroup$ – Shaun Sep 3 '17 at 12:05
  • 4
    $\begingroup$ 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
  • 1
    $\begingroup$ Use $\pi$ for $\pi$ and enclose all mathematics in $s $\endgroup$ – Shaun Sep 3 '17 at 12:10
  • 1
    $\begingroup$ Note that $-\pi<\Theta\le\pi$ because $\Theta=\pi$ is possible. $\endgroup$ – Shaun Sep 3 '17 at 12:12
  • 1
    $\begingroup$ @Vitale $z^*$ is the conjugate of $z$. $\endgroup$ – samjoe Sep 3 '17 at 12:15

It's just a matter of convention.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.