1. What Are Complex Roots?

Complex roots are solutions to polynomial equations that involve complex numbers. For the equation \( z^n - a = 0 \), there are exactly n distinct complex roots (when a ≠ 0), known as the nth roots of the complex number a.

2. How Does the Calculator Work?

The calculator finds all roots using De Moivre's Theorem:

Where:

• \( r \) — Magnitude of the complex number \( a \)
• \( \theta \) — Argument (angle) of \( a \)
• \( n \) — The root degree (exponent)
• \( k \) — Integer from 0 to \( n-1 \)

Explanation: The roots are equally spaced around a circle in the complex plane with radius \( r^{1/n} \).

3. Importance of Complex Roots

Details: Complex roots are fundamental in many areas of mathematics, physics, and engineering, including signal processing, control theory, and quantum mechanics.

4. Using the Calculator

Tips: Enter the exponent n (must be positive integer), and the real and imaginary parts of the complex number a. The calculator will display all n roots in rectangular form.

5. Frequently Asked Questions (FAQ)

Q1: What is the principal root?
A: The principal root is the one with the smallest positive argument (k=0).

Q2: How are roots distributed in the complex plane?
A: The roots lie on a circle centered at the origin, equally spaced at angles of \( 2\pi/n \) radians.

Q3: What happens when a is real?
A: For positive real numbers, one root is real and positive; for negative real numbers with odd n, one root is real and negative.

Q4: What about multiple roots?
A: The equation \( z^n - a = 0 \) always has n distinct roots when a ≠ 0.

Q5: Can this be used for matrices?
A: No, this calculator is for scalar complex numbers only. Matrix roots require different methods.