Question

Multiple Choice If \(z_{1}=r_{1} e^{i \theta_{1}}\) and \(z_{2}=r_{2} e^{i \theta_{2}}\) are complex numbers, then \(\frac{z_{1}}{z_{2}}, z_{2} \neq 0,\) equals: (a) \(\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1}-\theta_{2}\right)}\) (b) \(\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1} \cdot \theta_{2}\right)}\) (c) \(\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1}+\theta_{2}\right)}\) (d) \(\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1} / \theta_{2}\right)}\)

Short answer

Option (a) \( \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)} \)

Step-by-step solution

- Understand the problem

We need to divide two complex numbers given in polar form and find the correct expression for the result. The complex numbers are given as: \( z_1 = r_1 e^{i\theta_1} \) \( z_2 = r_2 e^{i\theta_2} \).

- Use the division property of complex numbers in polar form

Complex numbers in polar form can be divided using the property: \[ \frac{z_1}{z_2} = \frac{r_1 e^{i\theta_1}}{r_2 e^{i\theta_2}} \].

- Simplify the expression

To divide, you divide the magnitudes and subtract the angles: \[ \frac{r_1 e^{i\theta_1}}{r_2 e^{i\theta_2}} = \frac{r_1}{r_2} \times e^{i(\theta_1 - \theta_2)} \].

- Compare with the given options

From the options provided: (a) \( \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)} \), (b) \( \frac{r_1}{r_2} e^{i(\theta_1 \theta_2)} \), (c) \( \frac{r_1}{r_2} e^{i(\theta_1 + \theta_2)} \), and (d) \( \frac{r_1}{r_2} e^{i(\theta_1 / \theta_2)} \), option (a) matches our simplified expression: \( \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)} \).

Key concepts

polar form

The polar form of a complex number is a way of expressing complex numbers in terms of their magnitude and angle. Instead of writing a complex number as a sum of a real and an imaginary part, we write it as:
\[ z = re^{i\theta} \]
Here, \( r \) is the magnitude or absolute value of the complex number, and \( \theta \) is the angle formed with the positive real axis, also called the argument. This form is particularly useful for multiplication and division of complex numbers, as it simplifies these operations significantly.

• To multiply two complex numbers in polar form, you multiply their magnitudes and add their angles.
• To divide two complex numbers in polar form, you divide their magnitudes and subtract their angles.

This property makes operations with complex numbers more intuitive in many cases, especially when dealing with periodic functions or rotational transformations.

complex numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are generally written in the form:
\[ z = a + bi \]
where \( a \) is the real part and \( bi \) is the imaginary part. The imaginary unit \( i \) is defined as the square root of -1, i.e., \( i^2 = -1 \). Complex numbers extend the notion of one-dimensional real numbers to a two-dimensional number plane.
This makes them incredibly useful for a variety of applications, including electrical engineering, fluid dynamics, quantum physics, and signal processing.
Some characteristics of complex numbers are:

• Addition and subtraction are performed like combining like terms.
• Multiplication involves the distributive property and the identity \( i^2 = -1 \).
• Division can be done using the conjugate method or by converting to polar form.

Euler's formula

Euler's formula establishes a profound connection between complex numbers and trigonometry. It states:
\[ e^{i\theta} = \text{cos}(\theta) + i \text{sin}(\theta) \]
This formula is exceedingly powerful as it links the exponential function with trigonometric functions. For complex numbers in polar form, Euler's formula simplifies their representation and helps in performing operations like multiplication and division.
For example, a complex number \( z = re^{i\theta} \) can be written using Euler’s formula as:
\[ z = r(\text{cos}(\theta) + i \text{sin}(\theta)) \]
Euler’s formula is a cornerstone in fields such as signal processing, control theory, and quantum mechanics due to its ability to simplify complex periodic functions.

angle subtraction

Angle subtraction is a key operation when dividing complex numbers in polar form. According to the property:
\[ \frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)} \]

This shows that when you divide one complex number by another, you divide their magnitudes and subtract their angles. This angle subtraction is a consequence of the properties of exponents and logarithms applied to complex numbers. For instance:

• If \( z_1 \) has an angle of \( \theta_1 \) and \( z_2 \) has an angle of \( \theta_2 \), the resulting angle after division will be \( \theta_1 - \theta_2 \).
• This subtraction of angles makes visualizing the result straightforward because we are essentially rotating the first complex number by the negative angle of the second one.

Angle subtraction simplifies many operations in signal processing and control systems, where phase differences are crucial.