Complex Number Conversion Rectangular And Exponential Forms Explained

$\left[6\left(\cos \frac{\pi}{16}+i \sin \frac{\pi}{16}\right)\right]^4$

Introduction

In mathematics, complex numbers are often expressed in different forms, each offering unique advantages for various operations and applications. Two common forms are the rectangular form ($x + yi$) and the exponential form ($re^{i\theta}$), where $x$ and $y$ are real numbers, $i$ is the imaginary unit ($i^2 = -1$), $r$ is the magnitude (or modulus) of the complex number, and $\theta$ is the argument (or angle) in radians. Converting between these forms is a fundamental skill in complex number theory and has practical implications in fields like electrical engineering, physics, and signal processing. In this article, we will explore how to convert a complex number given in trigonometric form into both rectangular and exponential forms. We will use De Moivre's Theorem and Euler's formula as essential tools in this conversion process. We will apply these concepts to the complex number expression $\left[6\left(\cos \frac{\pi}{16}+i \sin \frac{\pi}{16}\right)\right]^4$, systematically transforming it into its rectangular and exponential representations. This process will not only illustrate the practical application of these mathematical principles but also highlight the interconnectedness of different mathematical concepts. Mastering these conversions enhances one's ability to manipulate complex numbers effectively and solve related problems in various mathematical and scientific contexts. The conversion process involves understanding the relationships between trigonometric functions, exponential functions, and complex numbers. By the end of this article, you will have a clear understanding of how to convert complex numbers between these forms, which is a crucial skill in many areas of mathematics and engineering.

Understanding the Forms

Before diving into the conversion, let's briefly define the forms. The rectangular form of a complex number is $x + yi$, where $x$ represents the real part and $y$ represents the imaginary part. This form is particularly useful for addition and subtraction of complex numbers. On the other hand, the exponential form is given by $re^{i\theta}$, where $r$ is the magnitude (or modulus) of the complex number, and $\theta$ is the argument (or angle) in radians. The exponential form is particularly useful for multiplication, division, and raising complex numbers to powers. The magnitude $r$ is calculated as $r = \sqrt{x^2 + y^2}$, and the argument $\theta$ can be found using the arctangent function, $\theta = \arctan(\frac{y}{x})$, taking into account the quadrant in which the complex number lies. Understanding these forms and their respective advantages is crucial for effectively manipulating complex numbers. The exponential form leverages Euler's formula, which connects complex exponentials with trigonometric functions, making it a powerful tool for simplifying complex number operations. By mastering both the rectangular and exponential forms, one gains a comprehensive understanding of complex numbers, enabling efficient problem-solving in various mathematical and scientific contexts. The conversion between these forms often involves trigonometric identities and algebraic manipulations, which further enhances mathematical skills. In the subsequent sections, we will apply these concepts to a specific complex number expression, providing a step-by-step guide to the conversion process.

Applying De Moivre's Theorem

The given expression is $\left[6\left(\cos \frac{\pi}{16}+i \sin \frac{\pi}{16}\right)\right]^4$. To simplify this, we'll use De Moivre's Theorem, which states that for any complex number in the form $r(\cos \theta + i \sin \theta)$ and any integer $n$, we have $(r(\cos \theta + i \sin \theta))^n = r^n(\cos n\theta + i \sin n\theta)$. Applying this theorem to our expression, we get:

$\left[6\left(\cos \frac{\pi}{16}+i \sin \frac{\pi}{16}\right)\right]^4 = 6^4\left(\cos \left(4 \cdot \frac{\pi}{16}\right)+i \sin \left(4 \cdot \frac{\pi}{16}\right)\right)$

This simplifies to:

$6^4\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$

Now, we know that $6^4 = 1296$, $\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, and $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$. Substituting these values, we have:

$1296\left(\frac{\sqrt{2}}{2}+i \frac{\sqrt{2}}{2}\right)$

De Moivre's Theorem is a cornerstone in complex number theory, enabling us to raise complex numbers in polar form to any integer power efficiently. This theorem is not only crucial for simplifying complex number expressions but also for understanding the geometric interpretation of complex number exponentiation. The magnitude of the complex number is raised to the power, and the argument is multiplied by the power, which corresponds to a scaling and rotation in the complex plane. By applying De Moivre's Theorem, we can easily compute complex powers without having to repeatedly multiply the complex number. This significantly reduces computational complexity and provides a clear understanding of the transformation. The theorem is also fundamental in solving polynomial equations and analyzing the roots of complex numbers. In the context of our problem, De Moivre's Theorem allows us to transform the given expression into a simpler trigonometric form, which can then be converted into rectangular and exponential forms.

Converting to Rectangular Form

To convert to rectangular form, we distribute the 1296 across the terms inside the parentheses:

$1296\left(\frac{\sqrt{2}}{2}+i \frac{\sqrt{2}}{2}\right) = 1296 \cdot \frac{\sqrt{2}}{2} + i \cdot 1296 \cdot \frac{\sqrt{2}}{2}$

This simplifies to:

$648\sqrt{2} + 648\sqrt{2}i$

Therefore, the rectangular form of the given expression is $648\sqrt{2} + 648\sqrt{2}i$. Converting to rectangular form involves expressing the complex number in terms of its real and imaginary parts, making it easier to visualize on the complex plane and perform arithmetic operations such as addition and subtraction. The rectangular form provides a direct representation of the complex number's components, which is particularly useful in applications where these components have physical significance, such as in electrical circuit analysis or signal processing. The process of converting from trigonometric form to rectangular form involves evaluating the trigonometric functions and distributing the magnitude, resulting in a complex number in the $x + yi$ format. In our case, we multiplied the magnitude 1296 by the real and imaginary parts obtained from the cosine and sine functions, respectively. This straightforward calculation yields the rectangular form, which clearly shows the real and imaginary components of the complex number.

Converting to Exponential Form

To convert to exponential form, we use the general form $re^{i\theta}$. We already know that $r = 6^4 = 1296$ and $\theta = \frac{\pi}{4}$. Thus, the exponential form is:

$1296e^{i\frac{\pi}{4}}$

The exponential form of a complex number, $re^{i\theta}$, is derived from Euler's formula, which states that $e^{i\theta} = \cos \theta + i \sin \theta$. This form is particularly useful for complex number multiplication and division, as well as exponentiation, due to the properties of exponential functions. The magnitude $r$ represents the distance from the origin in the complex plane, and the argument $\theta$ represents the angle from the positive real axis. Converting to exponential form involves identifying the magnitude and the argument of the complex number. In our case, the magnitude is $6^4 = 1296$, and the argument is $\frac{\pi}{4}$. Substituting these values into the exponential form, we obtain $1296e^{i\frac{\pi}{4}}$. This form concisely represents the complex number and facilitates various operations, such as finding powers and roots. The exponential form also provides a clear connection between complex numbers and trigonometric functions, highlighting the fundamental relationship between exponential and trigonometric behavior.

Final Answer

The rectangular form of the given expression is $648\sqrt{2} + 648\sqrt{2}i$, and the exponential form is $1296e^{i\frac{\pi}{4}}$. Understanding the different forms of complex numbers and how to convert between them is essential for advanced mathematics and engineering applications. The conversion process involves applying De Moivre's Theorem to simplify the expression, and then using trigonometric identities and Euler's formula to move between the rectangular and exponential forms. These skills are fundamental in various fields, including electrical engineering, signal processing, and quantum mechanics, where complex numbers are frequently used to model and analyze systems.

Convert the complex number expression $[6(\cos(\frac{\pi}{16}) + i \sin(\frac{\pi}{16}))]^4$ into rectangular form ($x + yi$) and exponential form ($re^{i\theta}$). What are the resulting expressions?

Complex Number Conversion Rectangular and Exponential Forms Explained