Difference between revisions of "Argument"
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− | Given a [[complex number]] <math>z</math>, the '''argument''' <math>\arg z</math> is 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.) |
− | 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 | + | 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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Latest revision as of 20:59, 31 July 2020
Given a complex number , the argument
is the measure of the signed angle the ray
makes with the positive real axis. (Note that this means the argument of the complex number 0 is undefined.)
Unfortunately, this means that 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
. This means that the argument may be best considered as an equivalence class
. The advantages of this are several: most importantly, they make
into a continuous function. They also make some properties of the argument "look nicer." For example, under this interpretation, we can write
. The other common solution is to restrict the range of
to some interval, usually
or
. This forces us to state this equality modulo
.
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