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Finding the Roots of Complex Numbers

Just as a positive real number has two square roots (one positive, one negative), a complex number has 'n' distinct n-th roots. Our Complex Root Calculator is a powerful tool for finding all of these roots using De Moivre's formula, a cornerstone of complex analysis.

🧮 How to Use the Calculator

1. Choose Input Format: Select whether you want to enter your complex number in Cartesian form (a + bi) or Polar form (r, φ).
2. Enter the Complex Number:
   • Cartesian: Input the real part 'a' and the imaginary part 'b'.
   • Polar: Input the modulus 'r' (the distance from the origin) and the argument 'φ' in degrees.
3. Enter the Root 'n': Specify which root you want to find (e.g., 2 for square root, 3 for cube root).
4. Calculate: Click the "Calculate Roots" button.

The calculator will display a table with all 'n' roots in both polar and Cartesian forms, along with a step-by-step explanation of the calculation.

De Moivre's Formula for Roots

To find the n-th roots of a complex number, we first express it in polar form: z = r(cos(φ) + i⋅sin(φ)). De Moivre's formula states that the 'n' distinct n-th roots are given by:

zₖ = ⁿ√r ⋅ [cos((φ + 2πk)/n) + i⋅sin((φ + 2πk)/n)]

for k = 0, 1, 2, ..., n-1.

This means:

• Modulus of Roots: All 'n' roots have the same modulus (length), which is the n-th real root of the original modulus 'r' (ⁿ√r).
• Argument of Roots: The roots are evenly spaced around a circle. The first root (k=0) has an angle of φ/n. Each subsequent root is found by adding 360°/n (or 2π/n radians) to the angle of the previous root.

Our calculator performs these steps, starting by converting your input to polar form if necessary, then applying the formula for each value of 'k' from 0 to n-1.

💡 Real-World Applications

Solving Polynomial Equations

The Fundamental Theorem of Algebra states that a polynomial of degree 'n' has 'n' complex roots. Finding the roots of an equation like zⁿ = c is a direct application of this calculator.

Engineering and Physics

In fields like electrical engineering and control systems theory, analyzing the roots of characteristic polynomials is essential for determining system stability and response.

Computer Graphics and Fractals

Algorithms for generating fractals, such as the Mandelbrot set or Julia sets, often involve iterating complex functions. The roots of these functions can define the shape and behavior of the fractal.