Modulus and Argument of Complex Numbers

The modulus and argument of a complex number are defined algebraically and interpreted geometrically. Examples with detailed step-by-step solutions hidden in collapsible dropdowns are included.
A modulus and argument calculator may be used for more practice.

Geometric Representation and Definitions

A complex number written in standard form as \( Z = a + ib \) may be plotted on a rectangular system of axes where the horizontal axis represents the real part of \( Z \) and the vertical axis represents the imaginary part of \( Z \). The geometrical representation of complex numbers on a complex plane, also called the Argand plane, is very similar to vector representation in rectangular systems of axes.

modulus and argument in complex plane
Figure 1. Modulus and argument in the complex plane

The modulus of a complex number in standard form \( Z = a + ib \) is defined by:

\[ |Z| = \sqrt{a^2 + b^2} \]

and its argument \( \theta \) is defined by:

\[ \tan(\theta) = \dfrac{b}{a} \]

Note
Since the above trigonometric equation has an infinite number of solutions (because the tangent function is periodic), two major conventions are adopted for the range of \( \theta \):

The four quadrants, as defined in trigonometry, are determined by the signs of \( a \) and \( b \). If the terminal side of \( Z \) is in quadrant I or II, the two conventions give the same value of \( \theta \). If the terminal side of \( Z \) is in quadrant III or IV, convention 1 gives a positive angle and convention 2 gives a negative angle.

In what follows, we primarily use convention (1) where \( \theta \) is in the range \( 0 \le \theta < 2\pi \), while also mentioning convention (2).

The modulus \( |Z| \) is the length of the segment representing the complex number (representing a magnitude if the complex number represents a physical quantity). The argument \( \theta \) is an angle in standard position (starting from the positive real axis) representing the direction of \( Z \).

If we are given the modulus \( |Z| \) and argument \( \theta \) of a complex number \( Z \), then the standard form of \( Z \) is given by:

\[ Z = |Z| (\cos(\theta) + i \sin(\theta)) \]

Example 1

Plot the complex number \( Z = -1 + i \) on the complex plane and calculate its modulus and argument.

View Solution

The complex number \( Z = -1 + i = a + ib \) gives \( a = -1 \) and \( b = 1 \).

modulus and argument of complex number
Figure 2. Plot of \( Z = -1 + i \)

Calculate the modulus \( |Z| \):

\[ |Z| = \sqrt{(-1)^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \]

Find the reference angle \( \theta_r \):

\[ \theta_r = \tan^{-1}\left|\dfrac{b}{a}\right| = \tan^{-1}\left|\dfrac{1}{-1}\right| = \tan^{-1}(1) = \dfrac{\pi}{4} \]

Since the real part is negative and the imaginary part is positive, the terminal side of \( \theta \) lies in quadrant II. Thus:

\[ \theta = \pi - \theta_r = \pi - \dfrac{\pi}{4} = \dfrac{3\pi}{4} \]

Conclusion: Modulus: \( |Z| = \sqrt{2} \), Argument: \( \theta = \dfrac{3\pi}{4} \).

Note Both conventions (1) and (2) give the same value for the argument \( \theta \).

Example 2

Calculate the modulus and argument of the complex numbers:
a) \( i \)
b) \( -2 \)
c) \( -i \)
d) \( -1 - 2i \)
e) \( 1 - i \)

View Solution
  • a) \( Z = i \):
    \( a = 0, b = 1 \implies |Z| = \sqrt{0^2 + 1^2} = 1 \).
    Since the point lies on the positive imaginary axis, argument \( \theta = \dfrac{\pi}{2} \).
    modulus and argument complex number i
    Plot of \( Z = i \)
  • b) \( Z = -2 \):
    \( a = -2, b = 0 \implies |Z| = \sqrt{(-2)^2 + 0^2} = 2 \).
    Since the point lies on the negative real axis, argument \( \theta = \pi \).
    modulus and argument complex number -2
    Plot of \( Z = -2 \)
  • c) \( Z = -i \):
    \( a = 0, b = -1 \implies |Z| = \sqrt{0^2 + (-1)^2} = 1 \).
    Since the point lies on the negative imaginary axis, argument \( \theta = \dfrac{3\pi}{2} \) (Convention 2 gives \( -\dfrac{\pi}{2} \)).
    modulus and argument complex number -i
    Plot of \( Z = -i \)
  • d) \( Z = -1 - 2i \):
    \( a = -1, b = -2 \implies |Z| = \sqrt{(-1)^2 + (-2)^2} = \sqrt{5} \).
    Reference angle \( \theta_r = \tan^{-1}\left|\dfrac{-2}{-1}\right| = \arctan(2) \).
    In quadrant III, \( \theta = \pi + \arctan(2) \approx 4.25 \) (Convention 2 gives \( -\pi + \arctan(2) \approx -2.03444 \)).
    modulus and argument complex number -1-2i
    Plot of \( Z = -1 - 2i \)
  • e) \( Z = 1 - i \):
    \( a = 1, b = -1 \implies |Z| = \sqrt{1^2 + (-1)^2} = \sqrt{2} \).
    Reference angle \( \theta_r = \tan^{-1}\left|\dfrac{-1}{1}\right| = \dfrac{\pi}{4} \).
    In quadrant IV, \( \theta = 2\pi - \dfrac{\pi}{4} = \dfrac{7\pi}{4} \) (Convention 2 gives \( -\dfrac{\pi}{4} \)).
    modulus and argument complex number 1 - i
    Plot of \( Z = 1 - i \)

Example 3

Plot each of the complex numbers given by its modulus and argument and write it in standard form:

View Solution
plot complex numbers given modulus and argument
Figure 3. Plot of complex numbers given by modulus and argument

Writing in standard form (\( Z = |Z|(\cos\theta + i\sin\theta) \)):

  • a) \( Z_1 = 3(\cos(0) + i\sin(0)) = 3 \)
  • b) \( Z_2 = 4(\cos(135^{\circ}) + i\sin(135^{\circ})) = 4\left(-\dfrac{\sqrt{2}}{2} + i\dfrac{\sqrt{2}}{2}\right) \approx -2.83 + 2.83i \)
  • c) \( Z_3 = 2\left(\cos\left(\dfrac{7\pi}{6}\right) + i\sin\left(\dfrac{7\pi}{6}\right)\right) = 2\left(-\dfrac{\sqrt{3}}{2} - \dfrac{1}{2}i\right) = -\sqrt{3} - i \approx -1.73 - i \)
  • d) \( Z_4 = 2.5(\cos(300^{\circ}) + i\sin(300^{\circ})) = 2.5\left(\dfrac{1}{2} - i\dfrac{\sqrt{3}}{2}\right) = 1.25 - 1.25\sqrt{3}i \approx 1.25 - 2.17i \)

Practice Questions

Question 1

Calculate the modulus and argument (in degrees and radians) of the complex numbers:

  1. \( z_1 = -1 \)
  2. \( z_2 = -2i \)
  3. \( z_3 = -\sqrt{3} - i \)
  4. \( z_4 = -3 + 3\sqrt{3}i \)
  5. \( z_5 = 7 - 7i \)

Question 2

Write in standard form the complex numbers given by their modulus and argument:

  1. \( |Z_1| = 0.5 \), \( \theta_1 = 2.1 \)
  2. \( |Z_2| = 3.4 \), \( \theta_2 = \pi/2 \)
  3. \( |Z_3| = 4 \), \( \theta_3 = 0 \)
  4. \( |Z_4| = 12 \), \( \theta_4 = 122^{\circ} \)
  5. \( |Z_5| = 200 \), \( \theta_5 = 5\pi/3 \)
  6. \( |Z_6| = 3/7 \), \( \theta_6 = 330^{\circ} \)

Solutions to Practice Questions

Solutions to Question 1

View Solutions
  1. \( |z_1| = 1 \), \( \theta_1 = \pi \) or \( 180^{\circ} \) (Convention 2 gives the same values)
  2. \( |z_2| = 2 \), \( \theta_2 = \dfrac{3\pi}{2} \) or \( 270^{\circ} \) (Convention 2 gives \( -\dfrac{\pi}{2} \) or \( -90^{\circ} \))
  3. \( |z_3| = 2 \), \( \theta_3 = \dfrac{7\pi}{6} \) or \( 210^{\circ} \) (Convention 2 gives \( -\dfrac{5\pi}{6} \) or \( -150^{\circ} \))
  4. \( |z_4| = \sqrt{(-3)^2 + (3\sqrt{3})^2} = \sqrt{9 + 27} = \sqrt{36} = 6 \), \( \theta_4 = \dfrac{2\pi}{3} \) or \( 120^{\circ} \) (Convention 2 gives the same values)
  5. \( |z_5| = \sqrt{7^2 + (-7)^2} = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2} \), \( \theta_5 = \dfrac{7\pi}{4} \) or \( 315^{\circ} \) (Convention 2 gives \( -\dfrac{\pi}{4} \) or \( -45^{\circ} \))

Solutions to Question 2

View Solutions
  1. \( Z_1 = 0.5(\cos(2.1) + i\sin(2.1)) \approx -0.25 + 0.43i \)
  2. \( Z_2 = 3.4\left(\cos\left(\dfrac{\pi}{2}\right) + i\sin\left(\dfrac{\pi}{2}\right)\right) = 3.4i \)
  3. \( Z_3 = 4(\cos(0) + i\sin(0)) = 4 \)
  4. \( Z_4 = 12(\cos(122^{\circ}) + i\sin(122^{\circ})) \approx -6.36 + 10.18i \)
  5. \( Z_5 = 200\left(\cos\left(\dfrac{5\pi}{3}\right) + i\sin\left(\dfrac{5\pi}{3}\right)\right) = 200\left(\dfrac{1}{2} - i\dfrac{\sqrt{3}}{2}\right) = 100 - 100\sqrt{3}i \)
  6. \( Z_6 = \dfrac{3}{7}(\cos(330^{\circ}) + i\sin(330^{\circ})) = \dfrac{3}{7}\left(\dfrac{\sqrt{3}}{2} - \dfrac{1}{2}i\right) = \dfrac{3\sqrt{3}}{14} - \dfrac{3}{14}i \)

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