The question asks for the principal argument of the complex number given by \(z = \frac{{1 + 2i}}{{1 - {{\left( {1 - i} \right)}^2}}}\). To find the principal argument, we first need to simplify the complex number \(z\) into the standard form \(a + bi\).

Let's simplify the denominator of the expression for \(z\): \(1 - {{\left( {1 - i} \right)}^2}\).

First, we calculate the term \({\left( {1 - i} \right)}^2\):

\({{\left( {1 - i} \right)}^2} = {1^2} - 2(1)(i) + {i^2}\)

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

\({{\left( {1 - i} \right)}^2} = 1 - 2i - 1 = -2i\)

Now, substitute this back into the denominator:

\text{Denominator} = 1 - (-2i) = 1 + 2i

Now substitute this simplified denominator back into the expression for \(z\):

\(z = \frac{{1 + 2i}}{{1 + 2i}}\)

Since the numerator and the denominator are the same, the fraction simplifies to:

\(z = 1\)

So, the complex number \(z\) is equal to 1. We can write this in the standard form \(a + bi\) as \(z = 1 + 0i\).

Now we need to find the principal argument of \(z = 1 + 0i\). The principal argument of a complex number \(a+bi\) is the angle \(\theta\) such that the complex number lies on the ray from the origin making an angle \(\theta\) with the positive real axis, and \(-\pi < \theta \leq \pi\).

For the complex number \(z = 1 + 0i\), the real part is \(a = 1\) and the imaginary part is \(b = 0\).

A complex number with a positive real part and zero imaginary part lies on the positive real axis in the complex plane.

The angle that the positive real axis makes with itself is 0 radians.

We can also use the formula for the argument. The tangent of the argument \(\theta\) is given by \( \tan \theta = \frac{b}{a} \).

\( \tan \theta = \frac{0}{1} = 0 \)

We need to find an angle \(\theta\) such that \( \tan \theta = 0 \) and \(-\pi < \theta \leq \pi\).

Possible values for \(\theta\) where \( \tan \theta = 0 \) are \(0, \pi, 2\pi, -\pi, -2\pi, \ldots\).

Considering the range for the principal argument \(-\pi < \theta \leq \pi\), the only value from the possibilities that falls within this range is \(0\).

Alternatively, we can consider the location of the complex number \(z = 1 + 0i\) in the complex plane. It is located at the point (1, 0) on the Cartesian plane. This point is on the positive x-axis (real axis). The angle formed with the positive x-axis is 0.

Therefore, the principal argument of \(z = 1 + 0i\) is 0.

Let's summarize the steps:

1. Simplify the denominator of \(z\).
2. Substitute the simplified denominator back into the expression for \(z\).
3. Simplify the expression for \(z\) to get it in the form \(a + bi\).
4. Determine the principal argument of the resulting complex number.

Applying these steps:

• Denominator: \(1 - (1 - i)^2 = 1 - (-2i) = 1 + 2i\)
• \(z = \frac{1 + 2i}{1 + 2i} = 1\)
• \(z = 1 + 0i\)
• Principal argument of \(1 + 0i\) is 0.

The principal argument of \(z\) is 0.

This aligns with one of the given options.

Revision Table: Complex Number Argument Calculation

Understanding the steps to calculate the argument of a complex number is key. Here's a brief revision table:

Additional Information: Principal Argument Explained

The argument of a complex number \(z = a + bi\) (where \(z \neq 0\)) is the angle \(\theta\) that the vector from the origin to the point \((a, b)\) makes with the positive real axis in the complex plane. There are infinitely many possible values for the argument, differing by multiples of \(2\pi\).

The principal argument, denoted as \(\text{arg}(z)\), is the unique value of the argument \(\theta\) that lies in the interval \( (-\pi, \pi] \), i.e., \( -\pi < \theta \leq \pi \).

To find the principal argument \(\theta\) of \(z = a + bi\):

• Calculate \( \alpha = \arctan\left|\frac{b}{a}\right| \), which gives the reference angle in the first quadrant.
• Adjust \(\alpha\) based on the quadrant of \(z\):
  • Quadrant I (\(a > 0, b > 0\)): \(\theta = \alpha\)
  • Quadrant II (\(a < 0, b > 0\)): \(\theta = \pi - \alpha\)
  • Quadrant III (\(a < 0, b < 0\)): \(\theta = -\pi + \alpha\) (or \(\pi + \alpha\), then adjust to \(-\pi < \theta \leq \pi\))
  • Quadrant IV (\(a > 0, b < 0\)): \(\theta = -\alpha\) (or \(2\pi - \alpha\), then adjust to \(-\pi < \theta \leq \pi\))
• For points on the axes:
  • Positive real axis (\(a > 0, b = 0\)): \(\theta = 0\)
  • Positive imaginary axis (\(a = 0, b > 0\)): \(\theta = \pi/2\)
  • Negative real axis (\(a < 0, b = 0\)): \(\theta = \pi\)
  • Negative imaginary axis (\(a = 0, b < 0\)): \(\theta = -\pi/2\)
• The argument of \(z=0\) is undefined.

In this problem, \(z = 1 + 0i\), which is on the positive real axis. According to the rules, the principal argument is 0.