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Difference between revisions of "Argument"

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Given a [[complex number]] <math>z</math>, the '''argument''' <math>\arg z</math> is the measure of the [[signed angle]] the [[ray]] <math>\overrightarrow{0z}</math> makes with the positive [[real axis]].  (Note that this means the argument of the complex number 0 is undefined.)
 
Given a [[complex number]] <math>z</math>, the '''argument''' <math>\arg z</math> is the measure of the [[signed angle]] the [[ray]] <math>\overrightarrow{0z}</math> makes with the positive [[real axis]].  (Note that this means the argument of the complex number 0 is undefined.)
  
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Unfortunately, this means that <math>\arg</math> is not a proper [[function]] but is instead a multi-valued function: for example, any [[positive]] [[real number]] has argument 0, but also has argument <math>2 \pi, -2\pi, 4\pi, \ldots</math>.  This means that the argument may be best considered as an [[equivalence class]] <math>\mathbf r = \{r + 2\pi n, n \in \mathbb{Z}\}</math>.  The advantages of this are several: most importantly, they make <math>\arg</math> into a continuous function.  They also make some properties of the argument "look nicer."  For example, under this interpretation, we can write <math>\arg(w \cdot z) = \arg(w) + \arg(z)</math>.  The other common solution (restricting the [[range]] of <math>\arg</math> to some [[interval]], usually <math>[0, 2\pi)</math> or <math>(-\pi, \pi]</math>) forces us to state this equality [[modulo]] <math>2\pi</math>.
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Unfortunately, this means that <math>\arg</math> is not a proper [[function]] but is instead a "multi-valued function": for example, any [[positive]] [[real number]] has argument 0, but also has argument <math>2 \pi, -2\pi, 4\pi, \ldots</math>.  This means that the argument may be best considered as an [[equivalence class]] <math>\mathbf r = \{r + 2\pi n, n \in \mathbb{Z}\}</math>.  The advantages of this are several: most importantly, they make <math>\arg</math> into a [[continuous function]].  They also make some properties of the argument "look nicer."  For example, under this interpretation, we can write <math>\arg(w \cdot z) = \arg(w) + \arg(z)</math>.  The other common solution is to restrict the [[range]] of <math>\arg</math> to some [[interval]], usually <math>[0, 2\pi)</math> or <math>(-\pi, \pi]</math>.  This forces us to state this equality [[modulo]] <math>2\pi</math>.
  
 
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Revision as of 20:45, 8 November 2006

Given a complex number $z$, the argument $\arg z$ is the measure of the signed angle the ray $\overrightarrow{0z}$ makes with the positive real axis. (Note that this means the argument of the complex number 0 is undefined.)

Unfortunately, this means that $\arg$ is not a proper function but is instead a "multi-valued function": for example, any positive real number has argument 0, but also has argument $2 \pi, -2\pi, 4\pi, \ldots$. This means that the argument may be best considered as an equivalence class $\mathbf r = \{r + 2\pi n, n \in \mathbb{Z}\}$. The advantages of this are several: most importantly, they make $\arg$ into a continuous function. They also make some properties of the argument "look nicer." For example, under this interpretation, we can write $\arg(w \cdot z) = \arg(w) + \arg(z)$. The other common solution is to restrict the range of $\arg$ to some interval, usually $[0, 2\pi)$ or $(-\pi, \pi]$. This forces us to state this equality modulo $2\pi$.

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