Problem 65 - A Graphical Approach to Precalculus with Limits

Question

Find all \(n\) complex solutions of each equation of the form \(x^{n}=k\) $$x^{2}=-18$$

Step-by-Step Solution

Answer

The solutions are \(3\sqrt{2}i\) and \(-3\sqrt{2}i\).

Step 1: Express k in Polar Form

Given the equation \(x^2 = -18\), we recognize \(k = -18\). First, express \(-18\) in polar form. A complex number can be written as \(r(\cos \theta + i\sin \theta)\), where \(r\) is the magnitude and \(\theta\) is the angle. Here, \(-18\) can be expressed as \(18(\cos \pi + i\sin \pi)\). This is because the point \(-18\) lies on the negative real axis.

Step 2: Apply De Moivre's Theorem

To find the square roots (since \(n = 2\)) of a complex number in polar form, use De Moivre’s Theorem. If \(z = r(\cos \theta + i\sin \theta)\), then the \(n\)th roots are \(z_k = r^{1/n}\left(\cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right)\right)\), where \(k = 0, 1, \, ..., n-1\).For \(x^2 = -18 \), \(n = 2\) and \(r = 18\), \(\theta = \pi\).

Step 3: Calculate the Magnitude of the Roots

Calculate \(r^{1/2}\), which is the magnitude of the roots.\[|r|^{1/2} = 18^{1/2} = \sqrt{18} = 3\sqrt{2}\]

Step 4: Determine the Angles for Each Root

Compute the angles for each root. Using \(\theta = \pi\), and \(n = 2\), we get:- For \(k = 0\): \(\theta_0 = \frac{\pi + 2 \cdot 0 \cdot \pi}{2} = \frac{\pi}{2}\)- For \(k = 1\): \(\theta_1 = \frac{\pi + 2 \cdot 1 \cdot \pi}{2} = \frac{3\pi}{2}\)

Step 5: Convert Polar to Rectangular Form

For each root, convert from polar to rectangular form:- Root 1: \(x_0 = 3\sqrt{2}(\cos \frac{\pi}{2} + i\sin \frac{\pi}{2}) = 3\sqrt{2}i\)- Root 2: \(x_1 = 3\sqrt{2}(\cos \frac{3\pi}{2} + i\sin \frac{3\pi}{2}) = -3\sqrt{2}i\) Here, use the fact that \(\cos \frac{\pi}{2} = 0\) and \(\sin \frac{\pi}{2} = 1\), similarly \(\cos \frac{3\pi}{2} = 0\) and \(\sin \frac{3\pi}{2} = -1\).

Key Concepts

De Moivre's Theorem

De Moivre's Theorem is a powerful tool in complex number theory. It connects complex numbers expressed in polar form with their powers and roots. This theorem states that for a complex number in polar form, \(z = r(\cos \theta + i\sin \theta)\), the nth power of this number is given by \[z^n = r^n \left( \cos(n\theta) + i\sin(n\theta) \right)\].

This theorem is incredibly useful when calculating roots of complex numbers. Instead of multiplying or dividing complex numbers repeatedly, De Moivre's provides a direct way to calculate both powers and roots.

• To find nth roots, you express your number in polar form and use \[z_k = r^{1/n} \times \left( \cos \frac{\theta + 2k\pi}{n} + i\sin \frac{\theta + 2k\pi}{n} \right)\], where \(k = 0\) to \(n-1\).
• This allows the calculation of each root by determining their magnitudes and angles.

In our original problem, we use De Moivre's Theorem to find the roots of \(x^2 = -18\) after expressing \(-18\) in polar form. Each step involves straightforward application of finding the appropriate angle for each root using the theorem.

Polar Form

The polar form of a complex number makes it easier to perform various operations, including multiplication, division, and finding roots. A complex number can be expressed in polar form as \(r(\cos \theta + i\sin \theta)\), where \(r\) is the magnitude and \(\theta\) is the angle.

This representation lends itself to visualization since \(r\) shows the distance from the origin in the complex plane, and \(\theta\) shows the direction. Using polar form simplifies finding roots and powers because it aligns closely with trigonometric identities and circle symmetries.

• In the given exercise, \(-18\) was first expressed in polar to facilitate applying De Moivre's Theorem. With \(r = 18\) and \(\theta = \pi\) (as \(-18\) lies on the negative real axis), we can easily determine the roots.
• Transitioning into and out of polar form involves using the identities: \(\cos \theta\) and \(\sin \theta\).

By mastering polar form, complex problems become much easier to solve, especially when dealing with roots as shown in the problem statement.

nth Roots of Unity

The concept of nth roots of unity is foundational in understanding the roots of complex numbers. An nth root of unity is a complex number that satisfies \(z^n = 1\). These numbers are evenly spaced on the unit circle in the complex plane at angles \(\frac{2\pi k}{n}\) where \(k = 0, 1, 2, ..., n-1\).

While the roots of unity focus on unity (1), the principle applies to other numbers by tweaking the radii and angles.

• Using the nth roots of unity in calculations allows breaking complex numbers into parts we can handle individually, projecting solutions as vectors around a circle.
• In the exercise we began discussing, finding the square roots of \(-18\) involved calculating two "roots of unity," adjusted to correspond with the specific magnitude and direction rooted in the problem statement.
• This approach provides a symmetrical perspective on complex numbers, reflecting the harmonic patterns characteristic of complex root plotting.

Understanding this concept lets us understand how complex numbers interact and replicate natural groupings within the complex plane, making computations intuitive and manageable.