Begin with the complex number \(z_{1}=x+j y=r e^{j \theta}\). Compute the complex number \(z_{2}=j z_{1}\) in its Cartesian and polar forms. The complex number \(z_{2}\) is sometimes called perp \(\left(z_{1}\right) .\) Explain why by writing perp \(\left(z_{1}\right)\) as \(z_{1} e^{j \theta_{2}} .\) What is \(\theta_{2}\) ? Repeat this problem for \(z_{3}=-j z_{1}\).

The Cartesian form of a complex number is represented as \(z = x + jy\), where \(x\) and \(y\) are real numbers, and \(j\) is the imaginary unit satisfying \(j^2 = -1\). In Cartesian form, we clearly see the real part (\(x\)) and the imaginary part (\(y\)) of the complex number.
When given a complex number in Cartesian form, such as \(z_1 = x + jy\), we can perform various operations straightforwardly. For example, to find the Cartesian form of \(z_2 = jz_1\), we multiply each term by \(j\):
\(z_2 = j(x + jy) = jx + j^2y = jx - y\), simplifying to \(-y + jx\). This new complex number \(z_2\) remains in Cartesian form, making it easy to see its real and imaginary components.

Polar form expresses a complex number in terms of its magnitude (\(r\)) and angle (\(\theta\)). It is written as \(z = re^{j\theta}\). The magnitude is calculated as \(r = \sqrt{x^2 + y^2}\), which is the distance from the origin to the point \((x, y)\) on the complex plane.
The angle or argument (\(\theta\)) represents the direction of the vector, making an angle with the positive real axis. Converting a complex number from Cartesian to polar form involves:

• Computing the magnitude \(r\)
• Finding the argument \(\theta\)

For instance, if \(z_1 = x + jy\) is our original complex number, its magnitude is \(r = |z_1| = \sqrt{x^2 + y^2}\). After converting \(z_1\) to polar form, we can easily represent transformations such as rotation by adding or subtracting angles.
For \(z_2 = jz_1\), the polar form is given by \(z_2 = re^{j(\theta + \frac{\pi}{2})}\), indicating a 90-degree rotation.

Multiplying complex numbers can be elegantly performed using polar form. When multiplying two complex numbers \(z_1 = r_1 e^{j\theta_1}\) and \(z_2 = r_2 e^{j\theta_2}\), their product is:
\(z_1 z_2 = (r_1 r_2)e^{j(\theta_1 + \theta_2)}\).
The magnitudes (\(r_1\) and \(r_2\)) multiply, and the arguments (\(\theta_1\) and \(\theta_2\)) add.
For instance, with \(z_1 = re^{j\theta}\), multiplying by \(j\) (which is \(e^{j \frac{\pi}{2}}\)) results in a 90-degree rotation. Thus, \(jz_1 = z_1 e^{j\frac{\pi}{2}}\), showing how multiplication by \(j\) transforms\(z_1\).
Similarly, for \(z_3 = -j z_1\), the multiplication by \(-j\) (which is \(e^{-j \frac{\pi}{2}}\)) results in a -90-degree rotation: \(-j z_1 = z_1 e^{-j\frac{\pi}{2}}\).
This showcases the power and simplicity of using polar form for complex multiplication.

The argument (or angle) of a complex number is the angle \(\theta\) that the vector representing the number makes with the positive real axis. For a complex number \(z = x + jy\), the argument \(\theta\) can be calculated using:
\(\theta = \tan^{-1}(\frac{y}{x})\).
Rotations of complex numbers correspond to additions or subtractions of these angles.
When multiplying a complex number by \(j\) or \(-j\), we're essentially adding \(\frac{\pi}{2}\) or subtracting \(\frac{\pi}{2}\), respectively, to the original angle. This results in:

• \(z_2 = jz_1\) rotates \(z_1\) by 90 degrees, so \(\theta_2 = \theta + \frac{\pi}{2}\).
• \(z_3 = -jz_1\) rotates \(z_1\) by -90 degrees, so \(\theta_3 = \theta - \frac{\pi}{2}\).

Understanding the argument helps in visualizing and performing operations on complex numbers. Simply put, the argument determines the direction, and manipulating the argument rotates the complex number accordingly.