Problem 52 - Precalculus

Question

Find all complex solutions to the given equations. $$x^{4}+2 i=0$$

Step-by-Step Solution

Answer

The complex solutions are approximately 1.111 + 0.445i, -1.111 + 0.445i, -1.111 - 0.445i, 1.111 - 0.445i.

Step 1: Equation Simplification

We begin with the equation \(x^4 + 2i = 0\). To simplify, we rewrite it as \(x^4 = -2i\). Our task is now to find the fourth roots of \(-2i\).

Step 2: Express in Polar Form

The complex number \(-2i\) can be expressed in polar form. It has a modulus \(|-2i| = 2\). Since \(-2i\) is purely imaginary and negative, it makes an angle of \(-\frac{\pi}{2}\) or equivalently \(\frac{3\pi}{2}\) radians with the positive x-axis. So, we can express it as \(2\text{cis}(-\frac{\pi}{2})\) or \(2\text{cis}(\frac{3\pi}{2})\).

Step 3: Use De Moivre’s Theorem

To find the fourth roots, we use De Moivre’s Theorem, which tells us that the nth roots of a complex number \(r\text{cis}(\theta)\) are given by \(r^{1/n}\text{cis}(\frac{\theta + 2k\pi}{n})\) for \(k = 0, 1, 2, ..., n-1\). Here, \(n = 4\) and \(r = 2\).

Step 4: Calculate the Modulus

The modulus of our fourth root will be \(2^{1/4}\). Since \(2^{1/4} = \sqrt[4]{2}\), this simplifies to the fourth root of two for all roots.

Step 5: Find the Argument for Each Root

For each root, we calculate the argument as \(\frac{\theta + 2k\pi}{4}\) where \(\theta = -\frac{\pi}{2}\) or \(\theta = \frac{3\pi}{2}\). Consequently, the four roots will be:1. For \(k = 0\): \(\sqrt[4]{2} \text{cis}(-\frac{\pi}{8})\).2. For \(k = 1\): \(\sqrt[4]{2} \text{cis}(\frac{7\pi}{8})\).3. For \(k = 2\): \(\sqrt[4]{2} \text{cis}(\frac{15\pi}{8})\).4. For \(k = 3\): \(\sqrt[4]{2} \text{cis}(\frac{23\pi}{8})\).

Step 6: Convert Polar to Rectangular Form

Convert each root from polar form to rectangular form using \(x = r \cos(\theta) + i r \sin(\theta)\):- \(\sqrt[4]{2}(\cos(-\frac{\pi}{8}) + i\sin(-\frac{\pi}{8}))\)- \(\sqrt[4]{2}(\cos(\frac{7\pi}{8}) + i\sin(\frac{7\pi}{8}))\)- \(\sqrt[4]{2}(\cos(\frac{15\pi}{8}) + i\sin(\frac{15\pi}{8}))\)- \(\sqrt[4]{2}(\cos(\frac{23\pi}{8}) + i\sin(\frac{23\pi}{8}))\).

Step 7: Write Final Solutions

Finally, the complex solutions to the equation \(x^4 + 2i = 0\) are the four roots found:- Approximately \(1.111 + 0.445i\)- Approximately \(-1.111 + 0.445i\)- Approximately \(-1.111 - 0.445i\)- Approximately \(1.111 - 0.445i\).

Key Concepts

De Moivre's Theorem

De Moivre's Theorem is a powerful tool for working with complex numbers, especially when dealing with powers and roots of complex numbers. It provides a straightforward way to calculate the nth root of a complex number.

The theorem states that if a complex number is expressed in its polar form as \( r\text{cis}(\theta) \), then its nth root can be found as \[ r^{1/n}\text{cis}\left(\frac{\theta + 2k\pi}{n}\right) \]where \( k = 0, 1, 2, ..., n-1 \).

This means for each value of \( k \), you will get a different root, providing all possible nth roots of the complex number.

• The angle argument \( \theta \) is the original angle plus whole rotations \( 2k\pi \) divided by \( n \).
• \( r^{1/n} \) is the nth root of the modulus, giving the magnitude of each root.

Using this theorem, finding roots becomes a simple matter of plug-and-chug, provided you have the modulus and angle.

De Moivre's Theorem is central to solving equations involving roots of complex numbers, making it a key concept in complex analysis.

Polar Form

The polar form of a complex number offers an alternative way to represent it, different from the typical rectangular form \( a + bi \). In polar form, a complex number is expressed as \( r\text{cis}(\theta) \), where:

• \( r \) (or modulus) is the distance from the origin to the point in the complex plane.
• \( \theta \) (or argument) is the angle formed with the positive x-axis.

This form is particularly useful when dealing with multiplication, division, and finding powers and roots of complex numbers.

For multiplication and division, the operation is simplified to manipulating polar's modulus and arguments:

• For multiplication: Multiply the moduli and add the arguments.
• For division: Divide the moduli and subtract the arguments.

Finding expressions in polar form is often the first step in solving complex equations, especially those involving roots, as it takes advantage of how easily exponentiation works in this form.

Modulus and Argument

The modulus and argument are key in understanding complex numbers, especially when converting between rectangular and polar forms.

Modulus

The modulus of a complex number, usually denoted as \( |z| \), is its distance from the origin on the complex plane. For a complex number \( z = a + bi \), it is calculated as:\[ |z| = \sqrt{a^2 + b^2} \]The modulus gives you a scalar magnitude of the complex number.

Argument

The argument of a complex number is the angle between its line representation and the positive x-axis. It is denoted as \( \text{arg}(z) \), and calculated using trigonometry:\[ \theta = \arctan\left(\frac{b}{a}\right) \]Adjustments might be necessary depending on the quadrant where the complex number lies.
In the given problem, understanding the modulus and argument helps re-interpret the complex number \(-2i\) in polar form, which simplifies the task of finding its roots. These concepts bridge the gap between rectangular coordinates and polar expressions, making analysis and computation of complex numbers more intuitive.