Modulus and Argument of Complex Numbers

What are Modulus and Argument?

A complex number $z = x + iy$ can be represented as a point $(x, y)$ on the complex plane (similar to the Cartesian plane). Besides being a point, we can also view it as a vector starting from the origin $(0, 0)$ to the point $(x, y)$.

This vector has a length and a direction. This length and direction are what we call the Modulus and Argument.

Modulus of a Complex Number

The Modulus of a complex number $z = x + iy$, written as $|z|$, is the distance from the origin $(0,0)$ to the point $(x, y)$ on the complex plane. This is the same as the length of the vector representing $z$.

To calculate the modulus, we can use the Pythagorean Theorem on the right-angled triangle formed by the real part ($x$), the imaginary part ($y$), and the modulus ($|z|$) as the hypotenuse.

Definition of Modulus:

The modulus of the complex number $z = x + iy$ is:

The modulus is always non-negative (never negative) because it represents a distance.

Calculating the Modulus

1. Find the modulus of $z_1 = 3 + 4i$, with $x=3, y=4$

   $$|z_1| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$

2. Find the modulus of $z_2 = -1 - 2i$, with $x=-1, y=-2$

   $$|z_2| = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}$$

3. Find the modulus of $z_3 = 5$, with $x=5, y=0$

   $$|z_3| = \sqrt{5^2 + 0^2} = \sqrt{25} = 5$$

   (The modulus of a real number is its absolute value).

4. Find the modulus of $z_4 = -2i$, with $x=0, y=-2$

   $$|z_4| = \sqrt{0^2 + (-2)^2} = \sqrt{4} = 2$$

Argument of a Complex Number

The Argument of a non-zero complex number $z = x + iy$, written as $\arg(z)$ or $\theta$, is the angle formed by the vector $z$ with the positive real axis on the complex plane. This angle is usually measured in radians or degrees.

From basic trigonometry on the same right-angled triangle as in the modulus visualization, we know the relationships:

To find $\theta$, we can use the arctangent function (or $\tan^{-1}$):

Calculators usually give the $\arctan$ value in the range $(-90^\circ, 90^\circ)$ or $(-\pi/2, \pi/2)$. We need to consider the quadrant where the point $(x, y)$ lies to determine the correct argument.

• Quadrant $I$ ($x>0, y>0$):

  $$\theta = \arctan(y/x)$$

• Quadrant $II$ ($x<0, y>0$):

  $$\theta = 180^\circ + \arctan(y/x) \text{ or } \theta = \pi + \arctan(y/x)$$

• Quadrant $III$ ($x<0, y<0$):

  $$\theta = 180^\circ + \arctan(y/x) \text{ or } \theta = \pi + \arctan(y/x)$$

• Quadrant $IV$ ($x>0, y<0$):

  $$\theta = 360^\circ + \arctan(y/x) \text{ or } \theta = 2\pi + \arctan(y/x)$$

  or simply

  $$\theta = \arctan(y/x)$$

  if a negative angle is desired

Often, we are interested in the Principal Argument (written $\text{Arg}(z)$), which is the argument value in the interval $(-180^\circ, 180^\circ]$ or $(-\pi, \pi]$.

Calculating the Argument

1. Find the argument of $z_1 = 1 + i$

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

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

   $$\theta = \arctan(1) = 45^\circ \text{ or } \frac{\pi}{4} \text{ radians}$$

2. Find the argument of $z_2 = -\sqrt{3} + i$

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

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

   $$\text{Base angle} = \arctan\left(\left|\frac{1}{-\sqrt{3}}\right|\right) = \arctan\left(\frac{1}{\sqrt{3}}\right) = 30^\circ$$

   $$\theta = 180^\circ - 30^\circ = 150^\circ \text{ or } \frac{5\pi}{6} \text{ radians}$$

   (Because it's in Quadrant $II$, we use $180^\circ - \text{base angle}$)

3. Find the argument of $z_3 = -1 - i\sqrt{3}$

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

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

   $$\text{Base angle} = \arctan(\sqrt{3}) = 60^\circ$$

   $$\theta = 180^\circ + 60^\circ = 240^\circ \text{ or } \frac{4\pi}{3} \text{ radians}$$

   (Because it's in Quadrant $III$, we use $180^\circ + \text{base angle}$ . Principal Argument: $-120^\circ$ or $-2\pi/3$ ).

4. Find the argument of $z_4 = 3 - 3i$

   The point $(3, -3)$ is in Quadrant $IV$.

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

   $$\text{Base angle} = \arctan(|-1|) = \arctan(1) = 45^\circ$$

   $$\theta = 360^\circ - 45^\circ = 315^\circ \text{ or } \frac{7\pi}{4} \text{ radians}$$

   (Because it's in Quadrant $IV$, we use $360^\circ - \text{base angle}$ . Principal Argument: $-45^\circ$ or $-\pi/4$ ).

Exercise

Find the modulus and argument (in degrees) of the following complex numbers:

1. $z_a = 2 + 2i$
2. $z_b = -4$
3. $z_c = -i$

Answer Key

1. For $z_a = 2 + 2i$:

   $x=2, y=2$ (Quadrant $I$) Modulus:

   $$|z_a| = \sqrt{2^2 + 2^2} = \sqrt{4+4} = \sqrt{8} = 2\sqrt{2}$$

   Argument:

   $$\tan \theta = \frac{2}{2} = 1 \implies \theta = \arctan(1) = 45^\circ$$

2. For $z_b = -4$:

   $z_b = -4 + 0i$. $x=-4, y=0$ (Negative real axis) Modulus:

   $$|z_b| = \sqrt{(-4)^2 + 0^2} = \sqrt{16} = 4$$

   Argument: The point is on the negative real axis.

   $$\theta = 180^\circ$$

3. For $z_c = -i$:

   $z_c = 0 - 1i$. $x=0, y=-1$ (Negative imaginary axis) Modulus:

   $$|z_c| = \sqrt{0^2 + (-1)^2} = \sqrt{1} = 1$$

   Argument: The point is on the negative imaginary axis.

   $$\theta = 270^\circ$$

   or $-90^\circ$ (Principal Argument).