Roots Of Complex Numbers Calculator

Complex Roots Formula:

\[ z^{1/n} = r^{1/n} e^{i (\theta + 2\pi k)/n}, \quad k = 0, 1, \ldots, n-1 \]

Roots of Complex Number:

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1. What Are Roots of Complex Numbers?

The roots of a complex number are all the solutions to the equation z^n = w, where z and w are complex numbers. Every non-zero complex number has exactly n distinct nth roots.

2. How Does the Calculator Work?

The calculator uses De Moivre's Theorem to find all roots:

\[ z^{1/n} = r^{1/n} \left[ \cos\left(\frac{\theta + 2\pi k}{n}\right) + i \sin\left(\frac{\theta + 2\pi k}{n}\right) \right], \quad k = 0, 1, \ldots, n-1 \]

Where:

\( r \) — Modulus (magnitude) of the complex number
\( \theta \) — Argument (angle) of the complex number
\( n \) — The root being calculated (square root, cube root, etc.)
\( k \) — Index for each distinct root (from 0 to n-1)

3. Importance of Complex Roots

Applications: Complex roots are essential in electrical engineering, quantum mechanics, signal processing, and solving polynomial equations.

4. Using the Calculator

Instructions: Enter the real and imaginary parts of your complex number, and the desired root (n). The calculator will display all n roots in rectangular form (a + bi).

5. Frequently Asked Questions (FAQ)

Q1: Why are there multiple roots for complex numbers?
A: Due to the periodic nature of trigonometric functions, adding multiples of 2π gives distinct solutions when divided by n.

Q2: What's the principal root?
A: The root with k=0 is called the principal root, having the smallest non-negative argument.

Q3: How are the roots geometrically arranged?
A: The roots lie equally spaced on a circle in the complex plane with radius r^(1/n).

Q4: What happens when n=1?
A: You simply get back the original complex number as the only root.

Q5: Can this calculator handle real numbers?
A: Yes, real numbers are complex numbers with imaginary part zero.