Problem 49 - Precalculus : Building Concepts and Connections

Question

Let \(z=r(\cos \theta+i \sin \theta)\) be a nonzero complex number, and let \(n\) be a positive integer greater than 1. Verify that each of the following \(n\) numbers is a solution of the equation \(u^{n}=z:\) $$\begin{aligned} &\sqrt[n]{r}\left[\cos \left(\frac{\theta+2 \pi k}{n}\right)+i \sin \left(\frac{\theta+2 \pi k}{n}\right)\right]\\\ &k=0,1,2, \ldots, n-1 \end{aligned}$$ where \(\sqrt[n]{r}\) denotes the positive real number that, when raised to the \(n\) th power, gives \(r .\) (Hint: Use De Moivre's Theorem.)

Step-by-Step Solution

Answer

The short answer is that each of the given \(n\) values is indeed a solution to the equation \(u^n=z\), where \(z\) is a nonzero complex number and \(n\) is a positive integer greater than 1. This is proven using De Moivre's Theorem and by simplifying and matching the said solutions with \(z\).

Step 1: Preparation

Let's start off by setting up our solution in the form given by the exercise: \(\sqrt[n]{r}\left[\cos \left(\frac{\theta+2 \pi k}{n}\right)+i \sin \left(\frac{\theta+2 \pi k}{n}\right)\right]\). This will be our \(u\). Our task is to demonstrate that \(u^n\) equals \(z\). Using De Moivre's theorem, we can rewrite \(u^n\) as \(\sqrt[n]{r^n}\left[\cos \left(n\frac{\theta+2 \pi k}{n}\right)+i \sin \left(n\frac{\theta+2 \pi k}{n}\right)\right]\).

Step 2: Simplifying the expression

The next step is to simplify this expression, in particular, the \(n\)th root of \(r^n\) and the \(n\) in front of the fractions within the parentheses. This will give us: \(r\left[\cos (\theta+2 \pi k)+i \sin (\theta+2 \pi k)\right]\).

Step 3: Matching with given complex number

If we match this with our original \(z\), \(r(\cos \theta+i \sin \theta)\), it's noticeable that the solutions match if we do a full rotation (which is \(2 \pi k\) for \(k = 0,1,2, \ldots, n-1\)), thus proving that each of the \(n\) numbers is indeed a solution to the equation \(u^n=z\).

Key Concepts

Understanding Complex Numbers

Complex numbers are a fundamental concept in mathematics, branching into various fields such as engineering, physics, and computer science. Simply put, a complex number is a number that comprises a real part and an imaginary part. It's generally expressed in the form of a + bi, where a represents the real part, and bi stands for the imaginary part, with i being the imaginary unit equal to the square root of -1.

Complex numbers can be added, subtracted, multiplied, and divided, much like real numbers, but they offer a unique characteristic of having a magnitude and direction when represented in the complex plane. This plane is a two-dimensional space where the horizontal axis is for the real part and the vertical axis is for the imaginary part.

One fascinating aspect of complex numbers is their geometric interpretation. The modulus or absolute value of a complex number corresponds to its distance from the origin in the complex plane, while the argument is the angle that the line from the origin to the number makes with the positive real axis. Understanding these fundamentals is key to grasping the more advanced concepts of complex numbers, such as their roots and how they are expressed in trigonometric form.

Digging into the nth Roots of Complex Numbers

When dealing with the nth roots of complex numbers, we encounter a captivating characteristic: a single complex number has n distinct nth roots. These are essentially the solutions to the equation uⁿ = z, where z is a complex number and u is its nth root.

The process of finding these roots uses the positive real number \(\sqrt[n]{r}\) which, when raised to the power of n, yields r, the modulus of our original complex number z. The respective nth roots lie evenly spaced on the arc of a circle in the complex plane, corresponding to 360 degrees, or 2π radians, divided by n.

Visualizing nth Roots

Imagine a complex number as a point on the complex plane, and picture the circle with radius equal to the modulus of that complex number. The nth roots of this complex number are found at equal intervals along the perimeter of this circle. This visualization helps understand how the roots relate to one another and how the concept of periodicity in the angles arises from the trigonometric form of complex numbers.

Trigonometric Form of Complex Numbers and De Moivre's Theorem

The trigonometric form of complex numbers offers a powerful tool for simplifying computations, particularly when dealing with powers and roots of complex numbers. A complex number z can be written in trigonometric form as \(r(\cos \theta + i \sin \theta)\), where r is the modulus and \theta is the argument of z.

This notation not only succinctly describes the complex number but also sets the stage for De Moivre's Theorem, which states that \(z^n = r^n(\cos n\theta + i \sin n\theta)\) for any positive integer n. This theorem provides a direct way to raise complex numbers to any power by simply multiplying the argument by n and raising the modulus to the power of n.

Application of De Moivre's Theorem

When finding the nth roots of a complex number, De Moivre's Theorem can be employed to simplify the expression and determine the distinct solutions as shown in the exercise. It elegantly demonstrates that the complex roots are not random but follow a pattern around the circle in the complex plane, linking back to the trigonometric representation. This theorem is vital in advanced mathematics and physics, where complex numbers play an integral role in understanding wave functions, quantum mechanics, and signal processing.