Question

Suppose \(z_{1}=r_{1} e^{i \theta_{1}}\) and \(z_{2}=r_{2} e^{i \theta_{2}}\) are two complex numbers. Then \(z_{1} z_{2}=\) _______.

Short answer

The product of \( z_1 \) and \( z_2 \) is given by: \( z_1 z_2 = r_1 r_2 e^{i (\theta_1 + \theta_2)} \)

Step-by-step solution

- Understand the Form of Complex Numbers

Recall that a complex number in polar form is written as \( z = re^{i\theta} \), where \(r\) is the magnitude and \(\theta\) is the argument (angle).

- Multiplication of Complex Numbers in Polar Form

When multiplying two complex numbers in polar form, \( z_1 = r_1 e^{i \theta_1} \) and \( z_2 = r_2 e^{i \theta_2} \), the magnitudes are multiplied and the arguments are added. Therefore, \( z_1 z_2 = r_1 r_2 e^{i (\theta_1 + \theta_2)} \)

- Combine the Results

Combine the magnitudes \( r_1 \) and \( r_2 \), and sum the angles \( \theta_1 \) and \( \theta_2 \), resulting in the expression: \( z_1 z_2 = r_1 r_2 e^{i (\theta_1 + \theta_2)} \)

Key concepts

polar form

Let's begin by understanding what polar form is. A complex number can be represented not just in the standard form of \( a + bi \), where \(a\) and \(b\) are real numbers, but also in polar form. The polar form of a complex number is given by \( z = re^{i\theta} \). Here, \(r\) represents the magnitude (or absolute value) of the complex number, and \(\theta\) represents the argument (or angle).

This transformation from cartesian to polar form brings out the geometric properties of complex numbers. It shows the number as a point in the As-per geometry:\

• The magnitude \(r\) is the distance from the origin to the point.
• The argument \(\theta\) is the angle formed with the positive real axis.

magnitude

The magnitude of a complex number \( z \) is a measure of its size or length in the complex plane. If we have a complex number in standard form, \( z = a + bi \), the magnitude can be calculated using the formula: \[ |z| = \sqrt{a^2 + b^2} \]

When the complex number is in polar form, represented as \( z = re^{i\theta} \), the magnitude \(r\) is directly given. This makes computations, especially multiplication and division, more straightforward since we only need to work with the magnitudes and arguments separately.

During multiplication of complex numbers in polar form, you simply multiply their magnitudes. For example, given two complex numbers \( z_1 = r_1 e^{i\theta_1} \) and \( z_2 = r_2 e^{i\theta_2} \), their product \( z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)} \), highlights this step. Here, \(r_1\) and \(r_2\) are multiplied to give the new magnitude.

argument (angle)

The argument of a complex number is the angle that the number makes with the positive real axis in the complex plane. It is usually denoted by \(\theta\). To find the argument of a complex number \(z = a + bi\), you can use the \texttt{atan2(b, a)} function, which efficiently computes the angle even for all four quadrants.

When in polar form, the argument comes in as \(\theta\) in \( z = re^{i\theta} \).

The beautiful part of polar coordinates shows when you multiply complex numbers. The angles (arguments) of the factors add together. For example, if you multiply \( z_1 = r_1 e^{i \theta_1} \) and \( z_2 = r_2 e^{i \theta_2} \), the resulting angle will be the sum of \( \theta_1 \) and \( \theta_2 \): \( \theta_{total} = \theta_1 + \theta_2 \).
This addition of angles makes visualizing and computing complex number multiplications much more intuitive.