Calculating the Argument of a Complex Number Quotient

We are asked to find the argument of the complex number given by the expression:

\( \rm z = \frac {1 - i\sqrt 3}{1 + i \sqrt 3} \)

where \( \rm i = \sqrt {-1} \).

To find the argument of this complex number, we can first simplify the expression by rationalizing the denominator. We multiply both the numerator and the denominator by the conjugate of the denominator, which is \( \rm 1 - i\sqrt 3 \).

\( \rm z = \frac {1 - i\sqrt 3}{1 + i \sqrt 3} \times \frac {1 - i\sqrt 3}{1 - i\sqrt 3} \)

Now, we perform the multiplication in the numerator and the denominator:

Numerator: \( \rm (1 - i\sqrt 3)(1 - i\sqrt 3) = (1)^2 - 2(1)(i\sqrt 3) + (i\sqrt 3)^2 \)

\( \rm = 1 - 2i\sqrt 3 + i^2 (3) \)

Since \( \rm i^2 = -1 \), we have:

\( \rm = 1 - 2i\sqrt 3 + (-1)(3) = 1 - 2i\sqrt 3 - 3 \)

\( \rm = -2 - 2i\sqrt 3 \)

Denominator: \( \rm (1 + i\sqrt 3)(1 - i\sqrt 3) \)

This is in the form \( \rm (a+b)(a-b) = a^2 - b^2 \).

\( \rm = (1)^2 - (i\sqrt 3)^2 \)

\( \rm = 1 - i^2 (3) \)

Since \( \rm i^2 = -1 \), we have:

\( \rm = 1 - (-1)(3) = 1 + 3 \)

\( \rm = 4 \)

So, the complex number \( \rm z \) simplifies to:

\( \rm z = \frac {-2 - 2i\sqrt 3}{4} \)

\( \rm z = \frac {-2}{4} - \frac {2i\sqrt 3}{4} \)

\( \rm z = -\frac {1}{2} - i\frac {\sqrt 3}{2} \)

Now we need to find the argument of the complex number \( \rm z = -\frac {1}{2} - i\frac {\sqrt 3}{2} \). Let the complex number be \( \rm x + iy \), where \( \rm x = -\frac {1}{2} \) and \( \rm y = -\frac {\sqrt 3}{2} \).

To find the argument (\( \theta \)), we first determine the quadrant in which the complex number lies. Since both the real part \( \rm x = -\frac {1}{2} \) and the imaginary part \( \rm y = -\frac {\sqrt 3}{2} \) are negative, the complex number lies in the third quadrant.

The reference angle (\( \alpha \)) is given by \( \rm \alpha = \arctan\left|\frac{y}{x}\right| \).

\( \rm \alpha = \arctan\left|\frac{-\sqrt 3 / 2}{-1 / 2}\right| \)

\( \rm \alpha = \arctan\left|\frac{\sqrt 3}{1}\right| \)

\( \rm \alpha = \arctan(\sqrt 3) \)

The angle whose tangent is \( \rm \sqrt 3 \) is \( \rm 60^\circ \) or \( \rm \frac{\pi}{3} \) radians.

\( \rm \alpha = 60^\circ \)

Since the complex number is in the third quadrant, the argument \( \theta \) can be found using the formula for the third quadrant: \( \rm \theta = 180^\circ + \alpha \) (for an argument between \( \rm 0^\circ \) and \( \rm 360^\circ \)).

\( \rm \theta = 180^\circ + 60^\circ \)

\( \rm \theta = 240^\circ \)

Alternatively, the principal argument (between \( \rm -180^\circ \) and \( \rm 180^\circ \)) would be \( \rm \theta = -180^\circ + \alpha = -180^\circ + 60^\circ = -120^\circ \). However, the options are given as positive angles.

The argument of the complex number \( \rm \frac {1 - i\sqrt 3}{1 + i \sqrt 3} \) is \( \rm 240^\circ \).

Understanding Complex Number Arguments

The argument of a complex number \( \rm z = x + iy \) is the angle \( \theta \) made by the line segment connecting the origin to the point \( \rm (x, y) \) in the complex plane, measured counterclockwise from the positive real axis. It is often denoted as \( \arg(z) \). The principal argument is usually restricted to the interval \( \rm (-180^\circ, 180^\circ] \) or \( \rm (-\pi, \pi] \) radians, but a general argument can be any value \( \rm \theta + 360^\circ n \) for an integer \( \rm n \).

• If \( \rm z \) is in the first quadrant (\( \rm x > 0, y > 0 \)), \( \rm \arg(z) = \arctan\left(\frac{y}{x}\right) \).
• If \( \rm z \) is in the second quadrant (\( \rm x < 0, y > 0 \)), \( \rm \arg(z) = 180^\circ - \arctan\left|\frac{y}{x}\right| \).
• If \( \rm z \) is in the third quadrant (\( \rm x < 0, y < 0 \)), \( \rm \arg(z) = 180^\circ + \arctan\left|\frac{y}{x}\right| \) or \( \rm -180^\circ + \arctan\left|\frac{y}{x}\right| \).
• If \( \rm z \) is in the fourth quadrant (\( \rm x > 0, y < 0 \)), \( \rm \arg(z) = -\arctan\left|\frac{y}{x}\right| \) or \( \rm 360^\circ - \arctan\left|\frac{y}{x}\right| \).
• If \( \rm x > 0, y = 0 \), \( \rm \arg(z) = 0^\circ \).
• If \( \rm x < 0, y = 0 \), \( \rm \arg(z) = 180^\circ \).
• If \( \rm x = 0, y > 0 \), \( \rm \arg(z) = 90^\circ \).
• If \( \rm x = 0, y < 0 \), \( \rm \arg(z) = -90^\circ \) or \( \rm 270^\circ \).

In our case, \( \rm x = -\frac{1}{2} \) and \( \rm y = -\frac{\sqrt{3}}{2} \), which falls into the third quadrant, leading to an argument of \( \rm 240^\circ \).

Revision Table: Key Complex Number Concepts

Additional Information on Complex Number Arguments

When dealing with division of complex numbers \( \rm z = \frac{z_1}{z_2} \), the argument can also be found using the property \( \arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2) \). Let's verify this approach with our problem:

\( \rm z_1 = 1 - i\sqrt 3 \). This is in the 4th quadrant (\( \rm x > 0, y < 0 \)). Reference angle \( \rm \arctan\left|\frac{-\sqrt 3}{1}\right| = 60^\circ \). Argument \( \rm \arg(z_1) = -60^\circ \) or \( \rm 300^\circ \).

\( \rm z_2 = 1 + i\sqrt 3 \). This is in the 1st quadrant (\( \rm x > 0, y > 0 \)). Reference angle \( \rm \arctan\left|\frac{\sqrt 3}{1}\right| = 60^\circ \). Argument \( \rm \arg(z_2) = 60^\circ \).

Using the property:

\( \rm \arg(z) = \arg(z_1) - \arg(z_2) \)

Using \( \rm \arg(z_1) = -60^\circ \):

\( \rm \arg(z) = -60^\circ - 60^\circ = -120^\circ \)

To get a positive argument between \( \rm 0^\circ \) and \( \rm 360^\circ \), we add \( \rm 360^\circ \):

\( \rm -120^\circ + 360^\circ = 240^\circ \)

Both methods yield the same result, \( \rm 240^\circ \), which matches the simplified form calculation. Understanding complex number properties like argument rules for multiplication and division can simplify calculations.