Definition:Argument of Complex Number

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Let $z = x + i y$ be a complex number.

An argument of $z$, or $\arg z$, is formally defined as a solution to the pair of equations:

$(1): \quad \dfrac x {\cmod z} = \map \cos {\arg z}$
$(2): \quad \dfrac y {\cmod z} = \map \sin {\arg z}$

where $\cmod z$ is the modulus of $z$.

From Sine and Cosine are Periodic on Reals, it follows that if $\theta$ is an argument of $z$, then so is $\theta + 2 k \pi$ where $k \in \Z$ is any integer.

Thus, the argument of a complex number $z$ is a continuous multifunction.

Principal Range

It is understood that the argument of a complex number $z$ is unique only up to multiples of $2 k \pi$.

With this understanding, we can limit the choice of what $\theta$ can be for any given $z$ by requiring that $\theta$ lie in some half open interval of length $2 \pi$.

The most usual of these are:

$\hointr 0 {2 \pi}$
$\hointl {-\pi} \pi$

but in theory any such interval may be used.

This interval is known as the principal range.

Principal Argument

Let $R$ be the principal range of the complex numbers $\C$.

The unique value of $\theta$ in $R$ is known as the principal value of the argument, or just principal argument, of $z$.

This is denoted $\Arg z$.

Note the capital $A$.

The standard practice is for $R$ to be $\hointl {-\pi} \pi$.

This ensures that the principal argument is continuous on the real axis for positive numbers.

Thus, if $z$ is represented in the complex plane, the principal argument $\Arg z$ is intuitively defined as the angle which $z$ yields with the real ($y = 0$) axis.

Also known as

The argument of a complex number is also seen as its amplitude.

Some sources use the term phase.


Example: $\arg 3$

$\arg 3 = 0$

Example: $\map \arg {-3}$

$\map \arg {-3} = \pi$

Example: $\map \arg {1 + i}$

$\map \arg {1 + i} = \dfrac \pi 4$

Example: $\map \arg {-1 - i}$

$\map \arg {-1 - i} = -\dfrac {3 \pi} 4$

Example: $\map \arg {2 i}$

$\map \arg {2 i} = \dfrac \pi 2$

Example: $\map \arg {-i}$

$\map \arg {-i} = -\dfrac \pi 2$