Let \( Z \) be a complex number given in standard form by
\( Z = a + i \)
The modulus \( |Z| \) of the complex number \( Z \) is given by
\( |Z| = \sqrt {a^2 + b^2} \)
and the argument of the complex number \( Z \) is angle \( \theta \) in standard position.
Note
Since the above trigonometric equation has an infinite number of solutions (since \( \tan \) function is periodic), there are two major conventions adopted for the rannge of \( \theta \) and let us call them conventions 1 and 2 for simplicity.
Convention (1) define the argumnet \( \theta \) in the range: \( 0 \le \theta \lt 2\pi \)
Convention (2) defines the argument \( \theta \) in the range : \( (-\pi, +\pi ] \)
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 one gives a positive angle and covention (2) gives a negative angle related by
\( \theta_{\text{convention 2}} = \theta_{\text{convention 1}} - 2\pi\)
This calculator calculates \( \theta \) for both conventions.

Use of the calculator to Calculate the Modulus and Argument of a Complex Number

1 - Enter the real and imaginary parts of complex number \( Z \) and press "Calculate Modulus and Argument". The outputs are the modulus \( |Z| \) and the argument, in both conventions, \( \theta \) in degrees and radians.

Use the calculator of Modulus and Argument to Answer the Questions

Use the calculator to find the arguments of the complex numbers \( Z_1 = -4 + 5 i \) and \( Z_2 = -8 + 10 i \) . Why are they equal?

Find the arguments of the complex numbers \( Z_1 = 3 - 9 i \) and \( Z_2 = - 3 + 9i \). Why is the difference between the two arguments equal to \( 180^{\circ} \)?

Find the ratio of the modulii of the complex numbers \( Z_1 = 8 + 16 i \) and \( Z_2 = 2 + 4 i \). Why is the ratio equal to \( 4 \)?

Find the ratio of the modulii of the complex numbers \( Z_1 = - 8 - 16 i \) and \( Z_2 = 2 + 4 i \). Why is the ratio equal to \( 4 \)?

Use the above results and other ideas to compare the modulus and argument of the complex numbers \( Z \) and \( k Z \) where \( k \) is a real number not equal to zero.