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Roots Calculator Complex

Reviewed by Calculator Editorial Team

This complex roots calculator finds all nth roots of any complex number in rectangular or polar form. It handles both real and complex roots, visualizes them on a complex plane, and explains the mathematical principles behind complex roots.

How to Use This Calculator

To find the roots of a complex number:

  1. Enter the real part of your complex number in the first field
  2. Enter the imaginary part in the second field
  3. Select the root degree (n) you want to calculate
  4. Choose whether to display roots in rectangular or polar form
  5. Click "Calculate Roots"

The calculator will display all n distinct roots of your complex number, formatted according to your selection. The results include both real and complex roots when they exist.

Complex Roots Theory

Finding roots of complex numbers extends the concept of square roots to higher dimensions. For any complex number z = a + bi and positive integer n, there are exactly n distinct nth roots of z.

Key properties of complex roots:

  • Roots are equally spaced around a circle in the complex plane
  • For real numbers, roots are either real or come in complex conjugate pairs
  • The principal root (argument between 0 and 2π) is the first root listed

Complex roots are calculated using polar form representation where z = r(cosθ + i sinθ). The nth roots are then given by:

z_k = r^(1/n) [cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)] for k = 0, 1, ..., n-1

This formula ensures roots are equally spaced around the circle of radius r^(1/n) in the complex plane.

Formula Used

The calculator uses the following steps to find complex roots:

  1. Convert the input complex number to polar form: z = a + bi → r = √(a² + b²), θ = atan2(b, a)
  2. Calculate the magnitude of each root: r_k = r^(1/n)
  3. Calculate the angle for each root: θ_k = (θ + 2πk)/n for k = 0 to n-1
  4. Convert back to rectangular form if requested: z_k = r_k(cosθ_k + i sinθ_k)

This method ensures all roots are found and properly spaced in the complex plane.

Worked Example

Let's find the cube roots of the complex number -1 + i√3.

Step 1: Convert to Polar Form

First, calculate the magnitude and angle:

r = √((-1)² + (√3)²) = √(1 + 3) = 2 θ = atan2(√3, -1) ≈ 2.6779 radians (153.4349°)

Step 2: Calculate Each Root

For k = 0, 1, 2:

r_k = 2^(1/3) ≈ 1.2599 θ_k = (2.6779 + 2πk)/3 Root 1 (k=0): θ_0 ≈ 2.6779/3 ≈ 0.8926 radians z_0 ≈ 1.2599(cos(0.8926) + i sin(0.8926)) ≈ 0.6299 + 1.0911i Root 2 (k=1): θ_1 ≈ (2.6779 + 6.2832)/3 ≈ 2.9605 radians z_1 ≈ 1.2599(cos(2.9605) + i sin(2.9605)) ≈ -1.2599 + 0i Root 3 (k=2): θ_2 ≈ (2.6779 + 12.5664)/3 ≈ 5.2431 radians z_2 ≈ 1.2599(cos(5.2431) + i sin(5.2431)) ≈ 0.6299 - 1.0911i

Final Roots

The three cube roots of -1 + i√3 are approximately:

  • 0.6299 + 1.0911i
  • -1.2599 + 0i
  • 0.6299 - 1.0911i

Frequently Asked Questions

What is the difference between real and complex roots?
Real roots are purely real numbers, while complex roots have both real and imaginary components. For real numbers, roots are either real or come in complex conjugate pairs.
How are complex roots visualized?
The calculator displays roots on a complex plane chart, showing their positions relative to each other and the origin.
Can I find roots of negative numbers?
Yes, the calculator handles negative numbers correctly by properly calculating their polar angles.
What happens when I enter zero as the complex number?
The calculator will return zero as the only root for any degree n, since 0^n = 0 for all n > 0.
How accurate are the results?
The calculator uses JavaScript's Math functions which provide approximately 15 decimal digits of precision.