# Magnitude and Argument

Consider the complex number \(z = 3 + 4i\). Let us find the *distance* of *z* from the origin:

Clearly, using the Pythagoras Theorem, the distance of *z* from the origin is \(\sqrt {{3^2} + {4^2}} = 5\) units. Also, the angle which the line joining *z* to the origin makes with the positive Real direction is \({\tan ^{ - 1}}\left( {\frac{4}{3}} \right)\).

Similarly, for an arbitrary complex number \(z = x + yi\), we can define these two parameters:

**Modulus of**. This is the distance of*z**z*from the origin, and is denoted by \(\left| z \right|\).**Argument of**. This is the angle between the line joining*z**z*to the origin and the positive Real direction. It is denoted by \(\arg \left( z \right)\).

Let us discuss another example. Consider the complex number \(z = - 2 + 2\sqrt 3 i\), and determine its magnitude and argument. We note that *z* lies in the second quadrant, as shown below:

Using the Pythagoras Theorem, the distance of *z* from the origin, or the magnitude of *z*, is

\[\left| z \right| = \sqrt {{{\left( { - 2} \right)}^2} + {{\left( {2\sqrt 3 } \right)}^2}} = \sqrt {16} = 4\]

Now, let us calculate the angle between the line segment joining the origin to *z* (*OP*) and the positive real direction (ray *OX*). Note that the angle *POX'* is

\[\begin{array}{l}{\tan ^{ - 1}}\left( {\frac{{PQ}}{{OQ}}} \right) = {\tan ^{ - 1}}\left( {\frac{{2\sqrt 3 }}{2}} \right) = {\tan ^{ - 1}}\left( {\sqrt 3 } \right)\\ \qquad\qquad\qquad\qquad\qquad\;\;\,\,\,\,\,\,\,\,\,\, = {60^0}\end{array}\]

Thus, the argument of *z* (which is the angle *POX*) is

\[\arg \left( z \right) = {180^0} - {60^0} = {120^0}\]

It is easy to see that for an arbitrary complex number \(z = x + yi\), its modulus will be

\[\left| z \right| = \sqrt {{x^2} + {y^2}} \]

To determine the argument of *z*, we should plot it and observe its quadrant, and then accordingly calculate the angle which the line joining the origin to *z* makes with the positive Real direction.

**Example 1:** Determine the modulus and argument of \(z = 1 + 6i\).

**Solution:** We have:

\[\left| z \right| = \sqrt {{1^2} + {6^2}} = \sqrt {37} \]

Now, the plot below shows that *z* lies in the first quadrant:

Clearly, the argument of *z* is given by

\[\arg \left( z \right) = \theta = {\tan ^{ - 1}}\left( {\frac{6}{1}} \right) = {\tan ^{ - 1}}6\]

**Example 2:** Find the modulus and argument of \(z = 1 - 3i\).

**Solution:** The modulus is

\[\left| z \right| = \sqrt {{1^2} + {{\left( { - 3} \right)}^2}} = \sqrt {10} \]

Now, we see from the plot below that *z *lies in the fourth quadrant:

The angle \(\theta \) is given by

\[\theta = {\tan ^{ - 1}}\left( {\frac{3}{1}} \right) = {\tan ^{ - 1}}3\]

Can we say that the argument of *z *is \(\theta \)? Well, since the direction of *z *from the Real direction is \(\theta \) measured *clockwise* (and not *anti-clockwise*), we should actually specify the argument of *z *as \( - \theta \):

\[\arg \left( z \right) = - \theta = - {\tan ^{ - 1}}3\]

The significance of the minus sign is in the direction in which the angle needs to be measured. The following example clarifies this further.

**Example 3:** Find the *moduli* (plural of modulus) and arguments of \({z_1} = 2 + 2i\) and \({z_2} = 2 - 2i\).

**Solution:** We have:

\[\begin{align}&\left| {{z_1}} \right| = \sqrt {{{\left( 2 \right)}^2} + {{\left( 2 \right)}^2}} = \sqrt 8 = 2\sqrt 2 \\&\left| {{z_2}} \right| = \sqrt {{{\left( 2 \right)}^2} + {{\left( { - 2} \right)}^2}} = \sqrt 8 = 2\sqrt 2 \end{align}\]

The moduli of the two complex numbers are the same. This is evident from the following figure, which shows that the two complex numbers are *mirror images* of each other in the horizontal axis, and will thus be equidistant from the origin:

Now, we note that

\[{\theta _1} = {\theta _2} = {\tan ^{ - 1}}\left( {\frac{2}{2}} \right) = {\tan ^{ - 1}}1 = \frac{\pi }{4}\]

Thus,

\[\begin{align}&\arg \left( {{z_1}} \right) = {\theta _1} = \frac{\pi }{4}\\&\arg \left( {{z_2}} \right) = - {\theta _2} = - \frac{\pi }{4}\end{align}\]

We have seen examples of argument calculations for complex numbers lying the in the first, second and fourth quadrants. Let us see how we can calculate the argument of a complex number lying in the third quadrant.

**Example 4:** Find the modulus and argument of \(z = - 1 - i\sqrt 3 \).

**Solution:** The modulus of *z *is:

\[\left| z \right| = \sqrt {{{\left( { - 1} \right)}^2} + {{\left( { - \sqrt 3 } \right)}^2}} = \sqrt 4 = 2\]

The plot below shows that *z *lies in the third quadrant:

The angle \(\theta \) is given by

\[\theta = {\tan ^{ - 1}}\left( {\frac{{\sqrt 3 }}{1}} \right) = {\tan ^{ - 1}}\sqrt 3 = \frac{\pi }{3}\]

Thus, the angle between *OP* and the positive Real direction is

\[\phi = \pi - \theta = \pi - \frac{\pi }{3} = \frac{{2\pi }}{3}\]

Now, since the angle \(\phi \) sweeps in the clockwise direction, the actual argument of *z* will be:

\[\arg \left( z \right) = - \phi = - \frac{{2\pi }}{3}\]

We could also have calculated the argument by calculating the magnitude of the angle sweep in the anti-clockwise direction, as shown below:

This way, we can write the argument is

\[\arg \left( z \right) = \pi + \theta = \pi + \frac{\pi }{3} = \frac{{4\pi }}{3}\]

Both ways of writing the arguments are correct, since the two arguments actually correspond to the same direction.

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