Problem 76 - Precalculus

Question

Determine whether the statement is true or false. Let \(z=r(\cos \theta+i \sin \theta)=r e^{i \theta} .\) Use the propertics of exponents to show that \(w_{k}=r^{1 / n}\left[\cos \left(\frac{\theta}{n}+\frac{2 k \pi}{n}\right)+i \sin \left(\frac{\theta}{n}+\frac{2 k \pi}{n}\right)\right]\).

Step-by-Step Solution

Answer

The statement is true.

Step 1: Understand the Expression

We are given that \( z = r e^{i \theta} \), where \( r \) is the magnitude and \( \theta \) is the argument of the complex number. We need to determine whether this equality can be expressed using the properties of exponentials,

Step 2: Expressing Roots of Complex Numbers

To find the \( n^{th} \) roots of a complex number \( z \), we denote them as \( w_k \), where \( k = 0, 1, 2, \ldots, n-1 \). Using polar form, they are represented as \( w_k = r^{1/n} \left[ \cos \left( \frac{\theta}{n} + \frac{2k\pi}{n} \right) + i \sin \left( \frac{\theta}{n} + \frac{2k\pi}{n} \right) \right] \).

Step 3: De Moivre's Theorem Application

De Moivre's Theorem states that \((re^{i\theta})^n = r^n e^{i n\theta}\). Addressing the \( n^{th} \) root, we have \( re^{i\theta/n} = r^{1/n} e^{i(\theta/n + 2k\pi/n)} \). This implies that the roots can be expressed in terms of cosine and sine functions.

Step 4: Verification of the Expression

The expression simplifies to \( w_k = r^{1/n} e^{i(\theta/n + 2k\pi/n)} \), which matches \( r^{1/n} \left[ \cos \left( \frac{\theta}{n} + \frac{2k\pi}{n} \right) + i \sin \left( \frac{\theta}{n} + \frac{2k\pi}{n} \right) \right] \), confirming the representation as the \( n^{th} \) roots of the original complex number.

Key Concepts

Complex Number Argument

The argument of a complex number is an essential part of its representation in complex analysis. It tells us the direction or angle of the complex number in the complex plane. When you have a complex number expressed in the form \( z = r(\cos \theta + i \sin \theta) \), the \( \theta \) is referred to as the argument. It measures the angle from the positive real axis to the line segment that represents the complex number.

Understanding the argument helps in visualizing complex numbers better. It shows how far the number is rotated from the positive x-axis. This concept plays a major role in converting complex numbers between different forms, especially between rectangular and polar forms. The argument is typically given in radians and can be found using the inverse tangent function based on the real and imaginary components of the complex number.

• For example, if a complex number is \( a + bi \), the argument \( \theta \) is given by \( \theta = \tan^{-1}(\frac{b}{a}) \).
• The standard range for argument values is from \(-\pi\) to \(\pi\).

When dealing with nth roots, the argument becomes crucial because it adjusts with rotations by \( \frac{2k\pi}{n} \) to represent all possible roots.

Polar Form of Complex Numbers

In mathematics, the polar form is particularly helpful for multiplying and dividing complex numbers, and especially for finding powers and roots.

A complex number written in polar form is expressed as \( z = r(\cos \theta + i \sin \theta) \), where:

• \( r \) represents the magnitude, or the distance from the origin to the point in the complex plane.
• \( \theta \), as already mentioned, is the argument, which specifies the angle of rotation from the positive real axis.

The polar form offers a different perspective compared to the traditional rectangular form \( a + bi \) and emphasizes the geometrical properties of complex numbers.

This is particularly useful in understanding complex operations and transformations such as rotation and scaling. By using Euler's formula, the polar form can further be condensed into the exponential form \( re^{i\theta} \). This simplification is elegant and simplifies many mathematical processes, especially those involving exponentiation and nth roots. With the polar form, operations that were previously cumbersome become streamlined, showcasing the power and utility of this representation in complex number theory.

De Moivre's Theorem

De Moivre's Theorem is a fundamental theorem that connects complex numbers and trigonometry, making it easier to compute powers and roots of complex numbers in polar form.

The theorem states that for any real number \( x \) and integer \( n \),

• \((\cos x + i\sin x)^n = \cos(nx) + i\sin(nx)\).

This forms the basis for simplifying calculations involving powers and serves as a powerful tool for deriving expressions for complex roots. The theorem is particularly useful when dealing with complex numbers in their polar or exponential forms.

For finding roots, De Moivre's Theorem is equally advantageous. When looking for the nth roots of \( z = re^{i\theta} \), one approximates as:

• \( w_{k}=r^{1 / n}\left[\cos\left(\frac{\theta}{n} + \frac{2k\pi}{n}\right) + i \sin\left(\frac{\theta}{n} + \frac{2k\pi}{n}\right)\right]\),for \( k = 0, 1, 2, \ldots, n-1 \)

This representation unfolds the geometric distribution of the roots on a circle in the complex plane, spaced evenly due to the symmetry of rotational factors \( \frac{2k\pi}{n} \). Understanding and applying De Moivre's Theorem bridges the gap between algebraic and geometric interpretations of complex numbers.