Problem 91 - Precalculus Mathematics for Calculus

Question

Solve the equation. $$z^{4}+1=0$$

Step-by-Step Solution

Answer

The roots are \( \frac{\sqrt{2}}{2} \pm i\frac{\sqrt{2}}{2} \) and \( -\frac{\sqrt{2}}{2} \pm i\frac{\sqrt{2}}{2} \).

Step 1: Rearrange the Equation

The original equation is given as \( z^{4} + 1 = 0 \). First, we subtract 1 from both sides to isolate the term with \( z \): \( z^{4} = -1 \).

Step 2: Express -1 in Exponential Form

To solve \( z^{4} = -1 \), express \(-1\) using Euler's formula: \(-1 = e^{i\pi} \). This allows us to work in exponential form, where complex solutions are more accessible.

Step 3: Equate Powers

Since \( z^{4} = e^{i\pi} \), we want to find the fourth roots of \( e^{i\pi} \). In exponential form, the roots are \( z = e^{i(\pi + 2k\pi)/4} \) for \( k = 0, 1, 2, 3 \).

Step 4: Calculate Roots for Each \( k \)

Substitute \( k = 0, 1, 2, 3 \) into the formula: - For \( k = 0 \): \( z = e^{i\pi/4} = \cos(\pi/4) + i\sin(\pi/4) = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \)- For \( k = 1 \): \( z = e^{i3\pi/4} = \cos(3\pi/4) + i\sin(3\pi/4) = -\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \)- For \( k = 2 \): \( z = e^{i5\pi/4} = \cos(5\pi/4) + i\sin(5\pi/4) = -\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} \)- For \( k = 3 \): \( z = e^{i7\pi/4} = \cos(7\pi/4) + i\sin(7\pi/4) = \frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} \)

Key Concepts

Euler's Formula

Euler's Formula is a powerful tool in mathematics that links complex numbers and trigonometry beautifully. This formula is expressed as: \[ e^{i\theta} = \cos(\theta) + i\sin(\theta) \] The expression shows how rotation on the complex plane is represented. Here, \( e \) is the base of the natural logarithm, \( i \) is the imaginary unit (where \( i^2 = -1 \)), and \( \theta \) is the angle in radians. Through Euler's formula, we can express complex numbers in exponential form. This makes calculations with complex numbers much easier, especially for multiplication and finding roots. For example, the complex number \(-1\) can be expressed as \( e^{i\pi} \) since it represents a half-turn rotation (180 degrees) on the unit circle in the negative direction. Understanding this transformation is crucial as it simplifies the process of finding roots and solving complex equations.

Fourth Roots

The concept of fourth roots involves finding values that, when raised to the power of four, give the original number. In the context of complex numbers, this involves identifying all possible complex number solutions. To find the fourth roots of a complex number expressed in exponential form, such as \( e^{i\pi} \), we use the general formula for roots: \[ z = e^{i(\pi + 2k\pi)/4} \] Here, \( k \) is an integer that ranges from 0 to 3, as we are finding fourth roots (four solutions). Each \( k \) value corresponds to a different fourth root.

• For \( k = 0 \), the root expression simplifies to \( e^{i\pi/4} \).
• For \( k = 1 \), it becomes \( e^{i3\pi/4} \).
• For \( k = 2 \), it changes to \( e^{i5\pi/4} \).
• For \( k = 3 \), we get \( e^{i7\pi/4} \).

By plugging each \( k \) into the formula, we obtain the four different roots, all rotating equally around the unit circle.

Exponential Form

Exponential Form is a representation of complex numbers using Euler's formula. Expressing complex numbers in this way makes many mathematical operations more straightforward. A complex number \( z \) in exponential form is written as \( z = re^{i\theta} \), where:

• \( r \) is the modulus or magnitude of the complex number, given by \( |z| \).
• \( \theta \) is the argument or angle of rotation on the complex plane, measured in radians.

Choosing the exponential form is particularly advantageous when multiplying or finding roots of complex numbers. The multiplication becomes a simple addition of exponents, and roots can be calculated using uniform angle divisions. In the context of finding roots, it utilizes the properties of rotation on the unit circle and the periodic nature of trigonometric functions. This form reduces the complexity of solving equations like \( z^4 = -1 \), simplifying them to a problem of finding equal divisions of rotations on the complex plane.