Problem 67 - Algebra and Trigonometry

Question

Show that each complex \(n\) th root of a nonzero complex number \(w\) has the same magnitude.

Step-by-Step Solution

Answer

Each \( n \)-th root of \( w \) has the magnitude \( r^{1/n} \).

Step 1 - Represent the Complex Number in Polar Form

Express the given complex number \( w \) in polar form: \( w = r e^{i\theta} \), where \( r \) is the magnitude and \( \theta \) is the argument of \( w \).

Step 2 - Compute the n-th Root

The \( n \)-th roots of \( w \) are given by: \( w_k = r^{1/n} e^{i(\theta + 2k\pi)/n} \) for \( k = 0, 1, 2, ... , n-1 \).

Step 3 - Determine the Magnitude

The magnitude of each root \( w_k \) is: \( |w_k| = |r^{1/n} e^{i(\theta + 2k\pi)/n}| \).

Step 4 - Simplify the Magnitude

Since the magnitude of a complex number in polar form \( re^{i\theta} \) is \( r \) regardless of the angle, we have: \( |w_k| = r^{1/n} \).

Step 5 - Confirm Equal Magnitudes

Note that \( r^{1/n} \) is the same for all values of \( k \). Therefore, every \( n \)-th root of \( w \) has the same magnitude: \( r^{1/n} \).

Key Concepts

Polar Form of Complex Numbers

In complex number theory, representing numbers in polar form can simplify many operations. Polar form expresses a complex number using a magnitude and an angle. If you have a complex number, say \( z \), it can be written as \( re^{i\theta} \). Here, \( r \) (magnitude) tells you how far the point is from the origin on the complex plane, and \( \theta \) (argument) tells you the counterclockwise angle from the positive x-axis. This form is particularly useful when multiplying or dividing complex numbers, as it turns multiplication into adding angles and helps with root calculations.

Finding n-th Roots of a Complex Number

Finding the \( n \)-th roots of a complex number means solving for the values that, when raised to the power of \( n \), give the original number. In polar form, if \( w = re^{i\theta} \), its \( n \)-th roots are \( w_k = r^{1/n} e^{i(\theta + 2k\pi)/n} \), where \( k \) ranges from 0 to \( n-1 \). Notice how the magnitude is raised to the \( 1/n \) power, while the angles are split into \( n \) evenly spaced segments. These roots are symmetrically distributed around the origin on the complex plane, forming a regular polygon if plotted.

Magnitude of Complex Numbers

The magnitude (or modulus) of a complex number \( z = a + bi \) is given by \( |z| = \sqrt{a^2 + b^2} \). For a number in polar form \( re^{i\theta} \), the magnitude is simply \( r \). So, when finding the \( n \)-th roots, the magnitude of each root \( |w_k| \) is \( r^{1/n} \). Regardless of the angle \( \theta \), each root's magnitude remains the same, which simplifies problems and shows that all \( n \)-th roots of a nonzero complex number have the same magnitude.

Euler's Formula

Euler's formula is a key concept connecting exponential functions and trigonometry. It states that \( e^{i\theta} = \cos\theta + i\sin\theta \). This formula is very useful when working with complex numbers in polar form. For instance, converting a complex number from Cartesian form \( a + bi \) to polar form \( re^{i\theta} \) uses Euler's formula, allowing it to be represented as \( r(\cos\theta + i\sin\theta) \). This simplifies many operations, especially when dealing with powers and roots of complex numbers.