Principal Argument of Complex Numbers

Different Angles That Describe the Same Direction

The argument $\theta$ of a nonzero complex number $z = x + iy$ is the angle formed by the vector $z$ with the positive real axis. The zero vector has no direction, so $\text{Arg}(0)$ is undefined.

The same direction can be described by infinitely many coterminal angles.

If $\theta$ is an argument of $z$, then $\theta + 2\pi k$ (where $k$ is an integer: $0, \pm 1, \pm 2, \ldots$) is also an argument of $z$, because adding multiples of $360^\circ$ or $2\pi$ radians results in the same angle on the complex plane.

Example:

The angles $45^\circ$, $405^\circ$ ($45^\circ + 360^\circ$), and $-315^\circ$ ($45^\circ - 360^\circ$) all indicate the same direction.

Because there are infinitely many arguments for a single complex number, we often need a unique standard value. This value is called the Principal Argument.

Definition of Principal Argument

For a nonzero complex number $z = r(\cos \theta + i\sin \theta)$ with $r>0$, the Principal Argument $\operatorname{Arg}(z)$ is the unique angle $\theta$ chosen from $(-\pi, \pi]$. In degrees, the same convention is:

Some references use $[0, 2\pi)$ instead. Both conventions select one representative from each set of coterminal angles. Every example below uses $(-\pi, \pi]$ consistently.

Determining the Principal Argument

Find any argument first, then add or subtract full turns until the result lies in $(-\pi, \pi]$ or $(-180^\circ, 180^\circ]$. An angle of $300^\circ$ lies outside both intervals, and subtracting one full turn brings it to $-60^\circ$, which is the principal value.

Finding the Principal Argument

1. Find the Principal Argument of $z = 1 + i$

   The point $(1, 1)$ is in Quadrant $I$.

   $$\tan \theta = \frac{y}{x} = \frac{1}{1} = 1$$

   $$\theta = \arctan(1) = 45^\circ$$

   Since $45^\circ$ is already within $(-180^\circ, 180^\circ]$, the Principal Argument is:

   $$\text{Arg}(z) = 45^\circ \text{ or } \frac{\pi}{4} \text{ radians}$$

2. Find the Principal Argument of $z = \sqrt{3} + i$

   The point $(\sqrt{3}, 1)$ is in Quadrant $I$.

   $$\tan \theta = \frac{y}{x} = \frac{1}{\sqrt{3}}$$

   $$\theta = \arctan\left(\frac{1}{\sqrt{3}}\right) = 30^\circ$$

   Since $30^\circ$ is already within $(-180^\circ, 180^\circ]$, the Principal Argument is:

   $$\text{Arg}(z) = 30^\circ \text{ or } \frac{\pi}{6} \text{ radians}$$

Equality of Two Complex Numbers in Polar Form

Two nonzero complex numbers $z_1 = r_1(\cos \theta_1 + i\sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i\sin \theta_2)$, with $r_1>0$ and $r_2>0$, are said to be equal if and only if:

1. Their moduli are equal:

   $r_1 = r_2$ (or $|z_1| = |z_2|$)

2. Their arguments are the same or differ by a multiple of $2\pi$ (or $360^\circ$):

   $\theta_1 = \theta_2 + 2k\pi$ or $\theta_1 - \theta_2 = 2k\pi$ for some integer $k$.

If we use the Principal Argument in $(-\pi, \pi]$, the second condition simplifies to $\text{Arg}(z_1) = \text{Arg}(z_2)$.

Checking for Equality

Determine if the following pairs of complex numbers are equal or different?

1. $z_1 = \sqrt{2}(\cos 45^\circ + i\sin 45^\circ)$ and $z_2 = \sqrt{2}(\cos 95^\circ + i\sin 95^\circ)$
2. $z_1 = \cos 30^\circ + i\sin 30^\circ$ and $z_2 = \cos 390^\circ + i\sin 390^\circ$

Solution:

1. Consider:

   • Modulus: $|z_1| = \sqrt{2}$ and $|z_2| = \sqrt{2}$. (Equal)
   • Principal Argument: $\text{Arg}(z_1) = 45^\circ$ and $\text{Arg}(z_2) = 95^\circ$. (Different)

   Since their principal arguments are different ($45^\circ \neq 95^\circ$), then $z_1 \neq z_2$.

2. Consider:

   • Modulus: $|z_1| = 1$ and $|z_2| = 1$. (Equal)
   • Arguments: $\theta_1 = 30^\circ$ and $\theta_2 = 390^\circ$.
   • Difference of arguments: $\theta_1 - \theta_2 = 30^\circ - 390^\circ = -360^\circ$.

   Since the difference of the arguments is a multiple of $360^\circ$ ($-360^\circ = -1 \times 360^\circ$), then $z_1 = z_2$.

   Alternatively, subtract one full turn: $390^\circ - 360^\circ = 30^\circ$. Thus $z_2$ and $z_1$ have the same principal argument.

Exercise

Find the Principal Argument (in degrees) for the following complex numbers. Start from the signs of the real and imaginary part, because those signs decide the quadrant before you apply an inverse trigonometric function.

1. $1 + \sqrt{3}i$
2. $-i$

Worked Solutions

1. For $z = 1 + \sqrt{3}i$:

   The point $(1, \sqrt{3})$ is in Quadrant $I$.

   $$\tan \theta = \frac{\sqrt{3}}{1} = \sqrt{3}$$

   $$\theta = \arctan(\sqrt{3}) = 60^\circ$$

   Since $60^\circ \in (-180^\circ, 180^\circ]$, $\text{Arg}(z) = 60^\circ$.

2. For $z = -i$:

   Write it as $z = 0 - 1i$. The point $(0, -1)$ lies directly on the negative imaginary axis, so no division by $x=0$ is needed.

   $$\text{Arg}(z) = -90^\circ = -\frac{\pi}{2}$$