Solving the Equation x4 1 0: Insights into Complex Roots

The equation x4 1 0 is a classic example that delves into the realm of complex numbers. Let's explore the roots of this equation and understand the mathematical concepts involved.

Real vs. Complex Roots

It has been established that every equation of the form xn a (where a 0) has at least one real root. However, the equation x4 1 0 does not have any real roots. This is because x4 0 for all real values of x. Therefore, adding 1 to a non-negative number results in a positive number, which cannot be zero.

Complex Roots of the Equation

Despite having no real roots, the equation x4 1 0 has four distinct complex roots. To find these roots, we can rewrite the equation as x4 -1. The complex number -1 can be expressed in its trigonometric form as -1 cos(π) i sin(π).

Finding the Roots Using De Moivre's Theorem

According to De Moivre's theorem, the nth roots of a complex number z_0 cos θ_0 i sin θ_0 are given by:

z_k cos(θ_0 2kπ/n) i sin(θ_0 2kπ/n), where k 0, 1, ..., n-1

For our equation, z_0 -1, which translates to cos(π) i sin(π). Thus, the four roots can be derived as follows:

z_0 cos(π/4) i sin(π/4) ≈ 0.7071 0.7071i

z_1 cos(3π/4) i sin(3π/4) ≈ -0.7071 0.7071i

z_2 cos(5π/4) i sin(5π/4) ≈ -0.7071 - 0.7071i

z_3 cos(7π/4) i sin(7π/4) ≈ 0.7071 - 0.7071i

These roots can also be expressed in degrees as:

z_0 cos(45°) i sin(45°)

z_1 cos(135°) i sin(135°)

z_2 cos(225°) i sin(225°)

z_3 cos(315°) i sin(315°)

Verification of the Roots

To verify that these roots satisfy the original equation, we can substitute each root back into the equation:

(0.7071 0.7071i)4 ≈ -1

(-0.7071 0.7071i)4 ≈ -1

(-0.7071 - 0.7071i)4 ≈ -1

(0.7071 - 0.7071i)4 ≈ -1

Geometric Interpretation

Geometrically, these roots can be visualized as four points on the unit circle in the complex plane Oxy, where the angles between them are π/2 radians (90°) apart. This alignment can be described using compass directions:

z_0 is in the first quadrant, at NE (45°) z_1 is in the second quadrant, at NW (135°) z_2 is in the third quadrant, at SW (225°) z_3 is in the fourth quadrant, at SE (315°)

Applications of Complex Numbers

Complex numbers play a pivotal role in higher algebra, appearing in the solution of algebraic equations, especially when the discriminant is negative. In advanced mathematics, particularly in higher-level courses at colleges and universities, complex numbers are essential in the discussion of exponentials, logarithms, series, and complex functions. The study of functions of a complex variable (T.F.C.V.) has been a rich field, with notable contributions by mathematicians like Simion Stoilow and Petru Caraman.

The exploration of the roots of the equation x4 1 0 showcases the beauty and depth of complex number theory, a fundamental aspect of both theoretical and applied mathematics.