91Ó°ÊÓ

Understanding the concept of a complex conjugate is essential when working with complex numbers. If you have a complex number in the form of \(z = a + bi \), its complex conjugate is represented as \(\bar{z} = a - bi\). Essentially, the sign of the imaginary part is changed. This transformation has several interesting properties, particularly when you multiply a complex number by its conjugate.

For example, if your complex number is \(z = r(\text{cos} \theta + i \text{sin} \theta) \), the complex conjugate would be: \(\bar{z} = r(\text{cos} \theta - i \text{sin} \theta) \). Here, the imaginary part \(i \text{sin} \theta \) becomes \(-i \text{sin} \theta\).

Multiplying complex numbers can initially seem daunting, but using the polar form simplifies the process. For two complex numbers in polar form like \(z_1 = r_1 (\text{cos} \theta_1 + i \text{sin} \theta_1) \) and \(z_2 = r_2 (\text{cos} \theta_2 + i \text{sin} \theta_2)\), their product is obtained by multiplying their magnitudes and adding their angles:
\[ z_1 z_2 = (r_1 r_2) \big[\text{cos} (\theta_1 + \theta_2) + i \text{sin} (\theta_1 + \theta_2)\big] \]
The result is also in polar form, which makes it straightforward to manage complex numbers geometrically.

In the exercise, when we multiply \(z = r(\text{cos} \theta + i \text{sin} \theta)\) by its conjugate \(\bar{z} = r(\text{cos} \theta - i \text{sin} \theta)\), we apply:
\[ z \bar{z} = r (\text{cos} \theta + i \text{sin} \theta) \times r (\text{cos} \theta - i \text{sin} \theta) = r^2 (\text{cos}^2 \theta + \text{sin}^2 \theta) \]
Since \(\text{cos}^2 \theta + \text{sin}^2 \theta = 1\), this simplifies to \(\text{r}^2 \)

Finding the inverse of a complex number in polar form is a simple step-by-step process. Given the complex number \(z = r(\text{cos} \theta + i \text{sin} \theta)\), the inverse \(1/z\) follows from the relationship:
\[ 1/z = \bar{z}/z \bar{z} \]
Where \(\bar{z}\) is the complex conjugate of \(z\), and \(z \bar{z} = r^2 \). So, you can rewrite the inverse as:
\[ 1/z = \bar{z} / r^2 = \frac{r(\text{cos} \theta - i \text{sin} \theta)}{r^2} = \frac{1}{r} (\text{cos} \theta - i \text{sin} \theta) \]
To express \(1/z\) in polar form, recognize that \(\frac{1}{r} = r^{-1} \), and recalling that the angle \(\theta\) changes sign from positive to negative in the process, you get:
\[ 1/z = r^{-1} (\text{cos} (-\theta) + i \text{sin} (-\theta)) \]
This is the polar form of the inverse of a complex number.