Principal Argument of Complex Numbers

Understanding the Principal Argument

The argument $\theta$ of a complex number $z = x + iy$ is the angle formed by the vector $z$ with the positive real axis.

However, there's an important point: the argument is not a single value!

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

The Principal Argument of a complex number $z = r(\cos \theta + i\sin \theta)$ is the unique value of the argument $\theta$ that satisfies a specific range.

Principal Argument (denoted $\text{Arg}(z)$) is defined as the argument $\theta$ that satisfies:

Note: Other definitions sometimes use the range $(-\pi, \pi]$ or $(-180^\circ, 180^\circ]$. It's important to always check the definition being used in a specific context.

Determining the Principal Argument

Determining the Principal Argument is the same as finding the regular argument, but we need to ensure the final result is within the range $[0, 2\pi)$ or $[0^\circ, 360^\circ)$.

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 the range $[0^\circ, 360^\circ)$, 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 the range $[0^\circ, 360^\circ)$, 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 complex numbers $z_1 = r_1(\cos \theta_1 + i\sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i\sin \theta_2)$ 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 (with the range $[0, 2\pi)$), 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, we can see that the Principal Argument of $z_2$ is $390^\circ - 360^\circ = 30^\circ$, which is the same as the Principal Argument of $z_1$.

Exercise

Find the Principal Argument (in degrees) for the following complex numbers:

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

Answer Key

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 $60^\circ \in [0^\circ, 360^\circ)$, then $\text{Arg}(z) = 60^\circ$.

2. For $z = -i$:

   Can be written as $z = 0 - 1i$. The point $(0, -1)$ is on the negative imaginary axis.

   $$\tan \theta = \frac{y}{x} = \frac{-1}{0} = \infty$$

   $$\theta = \arctan\left(\infty\right) = 90^\circ$$

   The argument is $270^\circ$ (or $-90^\circ$).

   Since we are looking for the Principal Argument in the range $[0^\circ, 360^\circ)$, then $\text{Arg}(z) = 270^\circ$.