Solving the Complex Equation x^4 1 - i: A Comprehensive Guide

In this article, we will explore the process of solving the complex equation x4 1 - i. This involves understanding the equation's representation, using De Moivre's Theorem for finding the roots, and expressing them in both trigonometric and algebraic forms. By the end of this article, you will have a clear understanding of how to tackle such complex equations.

Representation of 1 - i

The given equation is x4 1 - i. First, we can represent the complex number 1 - i in polar form. The polar form of a complex number z a bi is given by z r(cos θ i sin θ), where r is the modulus and θ is the argument.

To find the modulus r, we use the formula r √(a2 b2). In this case, a 1 and b -1, so:

r √(12 (-1)2) √2

Next, we find the argument θ. The argument is the angle made with the positive real axis. Here, tan θ b/a -1/1 -1. Since sin θ -1/√2 and cos θ 1/√2, we have θ -π/4.

Therefore, we can represent 1 - i as:

1 - i √2 (cos(-π/4) i sin(-π/4)) √2 e-πi/4

Using De Moivre's Theorem

De Moivre's Theorem states that for a complex number in polar form z r(cos θ i sin θ), the nth root of z is given by:

z1/n r1/n (cos((θ 2kπ)/n) i sin((θ 2kπ)/n))

In our case, n 4, r √2, and θ -π/4. The equation x4 1 - i can be written as x4 (√2)4 e-πi 2 e-πi/4. Taking the fourth root, we get:

x (2)1/4 e -πi/4(1 2k)

where k 0, ±1, ±2.

Solving for x

Let's solve for each value of k.

Case 1: k 0

x (2)1/4 e -πi/16 (2)1/8 (cos(-π/16) i sin(-π/16))

This simplifies to:

x (2)1/8 cos(-π/16) - i (2)1/8 sin(-π/16)

or more approximately:

x ≈ 1.0695539 - 0.2127475i

Case 2: k 1

x (2)1/4 e 23πi/16 (2)1/8 (cos(23π/16) i sin(23π/16))

This simplifies to:

x (2)1/8 cos(23π/16) - i (2)1/8 sin(23π/16)

or more approximately:

x ≈ -0.2127475 - 1.0695539i

Case 3: k -1

x (2)1/4 e 15πi/16 (2)1/8 (cos(15π/16) i sin(15π/16))

This simplifies to:

x (2)1/8 cos(15π/16) - i (2)1/8 sin(15π/16)

or more approximately:

x ≈ -1.0695539 - 0.2127475i

Case 4: k -2

x (2)1/4 e 7πi/16 (2)1/8 (cos(7π/16) i sin(7π/16))

This simplifies to:

x (2)1/8 cos(7π/16) i (2)1/8 sin(7π/16)

or more approximately:

x ≈ 0.2127475 1.0695539i

In summary, the four solutions to the equation x4 1 - i are:

x ≈ 1.0695539 - 0.2127475i x ≈ -0.2127475 - 1.0695539i x ≈ -1.0695539 - 0.2127475i x ≈ 0.2127475 1.0695539i

These solutions are derived by using the polar form of the complex numbers and applying De Moivre's Theorem. Understanding these concepts provides a comprehensive approach to solving complex algebraic equations.