To find the argument of the complex number \(\frac{{1 + i}}{{1 - i}}\), we first need to simplify the complex number into the standard form \(x + iy\).

Complex Number Simplification

The given complex number is \(Z = \frac{{1 + i}}{{1 - i}}\).

To simplify this expression, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(1 - i\) is \(1 + i\).

So, we have:

$$Z = \frac{{1 + i}}{{1 - i}} \times \frac{{1 + i}}{{1 + i}}$$

Now, let's calculate the numerator and the denominator separately.

• Numerator: \((1 + i)(1 + i)\)

Using the algebraic identity \((a+b)^2 = a^2 + 2ab + b^2\):

$$(1 + i)^2 = 1^2 + 2(1)(i) + i^2$$

Since \(i^2 = -1\):

$$1 + 2i + (-1) = 1 + 2i - 1 = 2i$$

• Denominator: \((1 - i)(1 + i)\)

Using the algebraic identity \((a-b)(a+b) = a^2 - b^2\):

$$(1 - i)(1 + i) = 1^2 - i^2$$

Since \(i^2 = -1\):

$$1 - (-1) = 1 + 1 = 2$$

Now, substitute the simplified numerator and denominator back into the expression for \(Z\):

$$Z = \frac{{2i}}{{2}} = i$$

So, the simplified complex number is \(Z = i\).

Argument Calculation

Now that we have \(Z = i\), we can write it in the form \(x + iy\), which is \(Z = 0 + 1i\). Here, the real part \(x = 0\) and the imaginary part \(y = 1\).

The argument of a complex number \(z = x + iy\) is the angle \(\theta\) that the line segment from the origin to the point \((x, y)\) makes with the positive real axis in the complex plane.

We can determine the argument based on the quadrant where the complex number lies:

• If \(x > 0, y > 0\) (Quadrant I), \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\).
• If \(x < 0, y > 0\) (Quadrant II), \(\theta = \pi + \tan^{-1}\left(\frac{y}{x}\right)\).
• If \(x < 0, y < 0\) (Quadrant III), \(\theta = -\pi + \tan^{-1}\left(\frac{y}{x}\right)\) or \(\theta = \pi + \tan^{-1}\left(\frac{y}{x}\right)\).
• If \(x > 0, y < 0\) (Quadrant IV), \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\).
• If \(x > 0, y = 0\) (Positive real axis), \(\theta = 0\).
• If \(x < 0, y = 0\) (Negative real axis), \(\theta = \pi\).
• If \(x = 0, y > 0\) (Positive imaginary axis), \(\theta = \frac{\pi}{2}\).
• If \(x = 0, y < 0\) (Negative imaginary axis), \(\theta = -\frac{\pi}{2}\) or \(\frac{3\pi}{2}\).

In our case, for \(Z = 0 + 1i\), we have \(x = 0\) and \(y = 1\). This means the complex number lies on the positive imaginary axis.

Therefore, the argument of \(Z = i\) is \(\frac{\pi}{2}\).

The final answer is \(\frac{\pi}{2}\).