Problem 73 - Precalculus

Question

Determine whether the statement is true or false. Let \(z_{1}=r_{1}\left(\cos \theta_{1}+i \sin \theta_{1}\right)=r_{1} e^{i \theta_{i}}\) and \(z_{2}=r_{2}\left(\cos \theta_{2}+i \sin \theta_{2}\right)=r_{2} e^{i \theta_{2}}\) be two complex numbers. Use the properties of exponentials to show that \(z_{1} z_{2}=r_{1} r_{2}\left[\cos \left(\theta_{1}+\theta_{2}\right)+i \sin \left(\theta_{1}+\theta_{2}\right)\right]\)

Step-by-Step Solution

Answer

The statement is true; the multiplication rule holds for complex numbers in polar form.

Step 1: Understanding the Problem

We have two complex numbers given in polar form: \(z_1 = r_1(\cos \theta_1 + i \sin \theta_1)\) and \(z_2 = r_2(\cos \theta_2 + i \sin \theta_2)\). We aim to show that their product \(z_1 z_2\) equals \(r_1 r_2 [\cos (\theta_1 + \theta_2) + i \sin (\theta_1 + \theta_2)]\).

Step 2: Expressing Complex Numbers with Exponentials

Utilizing Euler's formula, the complex numbers can be expressed as \(z_1 = r_1 e^{i \theta_1}\) and \(z_2 = r_2 e^{i \theta_2}\). This simplification will help us leverage the properties of exponents in our proof.

Step 3: Multiplying the Exponential Forms

To find \(z_1 z_2\), multiply the exponential expressions: \(z_1 z_2 = (r_1 e^{i \theta_1})(r_2 e^{i \theta_2}) = r_1 r_2 e^{i (\theta_1 + \theta_2)}\). We use the property \(e^{a} e^{b} = e^{a+b}\).

Step 4: Using Euler's Formula for the Product

Recall that Euler's formula states \(e^{i\theta} = \cos \theta + i \sin \theta\). Therefore, the expression \(e^{i (\theta_1 + \theta_2)}\) becomes \(\cos (\theta_1 + \theta_2) + i \sin (\theta_1 + \theta_2)\).

Step 5: Concluding the Proof

The multiplication of the complex numbers yields: \(z_1 z_2 = r_1 r_2 [\cos (\theta_1 + \theta_2) + i \sin (\theta_1 + \theta_2)]\). This confirms that the product of the two complex numbers follows the expected trigonometric form, thus proving the statement true.

Key Concepts

Polar Form

Complex numbers can be represented in the polar form as \( z = r (\cos \theta + i \sin \theta) \). Here, \( r \) is the modulus of the complex number, indicating its distance from the origin on the complex plane. The angle \( \theta \), known as the argument, is the angle formed between the line representing the complex number and the positive real axis.
By using polar form, multiplication and division of complex numbers become more straightforward. For two complex numbers \( z_1 \) and \( z_2 \) in polar form, their product is obtained by multiplying their moduli and adding their arguments. This feature is extremely helpful in simplifying complex calculations.
Understanding polar form can intuitively connect geometrical concepts to the arithmetic of complex numbers. It provides a powerful tool for visualizing and manipulating complex number operations, especially when dealing with higher-order equations or when plotting them in a plane.

Euler's Formula

Euler's Formula is a fascinating bridge between geometry and algebra, providing that for any real number \( \theta \), the complex exponential \( e^{i\theta} = \cos \theta + i \sin \theta \). This formula gives a concise way to represent complex numbers, allowing for more manageable computations.
Applying Euler's formula transforms complex numbers from the trigonometric form \( \cos \theta + i \sin \theta \) into the exponential form \( e^{i\theta} \). This transformation streamlines multiplicative processes due to the ease of working with exponential properties.
Euler's formula demonstrates the deep interconnections between exponential growth and trigonometric oscillations, revealing the elegant unity between seemingly distinct areas of mathematics. It also serves as a basis for further exploration into complex analysis, signal processing, and other mathematical disciplines.

Exponential Properties

When complex numbers are expressed using Euler's formula, they are transformed into exponential form, \( z = r e^{i \theta} \). This makes it easy to apply the multiplication rule for exponents: \( e^a e^b = e^{a+b} \).
For example, if you have two complex numbers \( z_1 = r_1 e^{i \theta_1} \) and \( z_2 = r_2 e^{i \theta_2} \), their product is \( z_1 z_2 = r_1 r_2 e^{i(\theta_1+\theta_2)} \). This simplifies the multiplication process by eliminating the need for expanding trigonometric expressions.
By translating trigonometric expressions into exponential ones, exponential properties can be exploited for simplifying operations. It showcases the power of exponents in reducing computational complexity in complex number arithmetic.

Trigonometric Form

The trigonometric form of a complex number is an essential building block in the study of complex numbers. It is expressed as \( z = r(\cos \theta + i \sin \theta) \). This form not only highlights the geometric interpretation but also enables easy manipulation through additions and multiplications.
An advantage of the trigonometric form is that it makes it straightforward to visualize operations like rotation and dilations within the complex plane. It emphasizes both the angle and magnitude, providing insight into how complex numbers relate to circles and periodic functions.
Using trigonometric identities within this form can simplify complex relationships. For example, multiplying two complex numbers involves adding their arguments and multiplying their moduli, as reflected in the expression \( z_1 z_2 = r_1 r_2 \left[ \cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2) \right] \), offering an accessible route to understanding complex operations.