Complex Root Calculator

Find the n-th roots of z = a + bi

+i
Other Titles
Finding the Roots of a Complex Number
An exploration of De Moivre's Theorem and the process of finding the n-th roots of a complex number.

Understanding the Complex Root Calculator: A Comprehensive Guide

  • What are the n-th roots of a complex number?
  • The role of De Moivre's Theorem
  • From rectangular to polar form
Just as a positive real number has two square roots (one positive, one negative), a complex number has 'n' distinct n-th roots. For example, a number has 3 cube roots, 4 fourth roots, and so on. These roots are themselves complex numbers and are spread out evenly on a circle in the complex plane.
De Moivre's Theorem for Roots
To find these roots, we use a formula derived from De Moivre's Theorem. The first step is to convert the complex number from its standard (rectangular) form z = a + bi to its polar form z = r(cos(θ) + isin(θ)).

Conceptual Example

  • To find the cube roots of 8, we first write 8 as a complex number: 8 + 0i.
  • Its polar form is r=8, θ=0°. De Moivre's theorem is then used to find the three cube roots.
  • One root will be the real number 2. The other two will be complex numbers, all lying on a circle of radius ∛8 = 2.

Step-by-Step Guide to Using the Complex Root Calculator

  • Entering the complex number and the root degree
  • Interpreting the polar form and the list of roots
Our calculator handles the conversion and calculation for you. Here's the process:
Inputs
Outputs
The calculator first displays the polar coordinates (r and θ) of your number. It then lists all 'n' unique roots in the standard 'a + bi' format.

Using the Calculator

  • To find the 4th roots of 16i (or 0 + 16i):
  • Enter a=0, b=16, and n=4.
  • The calculator will find r=16, θ=90°. It will then compute the four 4th roots, which all lie on a circle of radius ⁴√16 = 2.

Real-World Applications of Complex Roots

  • Polynomial factorization
  • Electrical engineering and signal analysis
  • Fractal geometry
Solving Polynomials
The Fundamental Theorem of Algebra states that a polynomial of degree 'n' has 'n' roots (counting multiplicity). Finding the roots of an equation like zⁿ - c = 0 is equivalent to finding the n-th roots of 'c'. This is crucial in many areas of engineering and physics.
Signal Processing
In digital signal processing, algorithms like the Fast Fourier Transform (FFT) heavily rely on the properties of the 'roots of unity' (the n-th roots of 1). These roots are fundamental to how the algorithm efficiently analyzes signal frequencies.
Art and Geometry
The calculation of complex roots is fundamental to generating some types of fractals, such as Newton fractals. The intricate boundaries in these images are determined by which root of a polynomial a starting point converges to under iteration.

Geometric Interpretation

  • The n-th roots of a complex number z are equally spaced on the circumference of a circle with radius r^(1/n).
  • The angle between successive roots is always 360°/n (or 2π/n radians).
  • For the cube roots of 8 (r=8, n=3), the roots lie on a circle of radius 2. They are separated by 360/3 = 120°. The roots are 2, -1 + √3 i, and -1 - √3 i.

Common Misconceptions and Correct Methods

  • Thinking there is only one root
  • Errors in converting to polar form
  • Forgetting to add 2πk
Misconception 1: Only One Root
A common mistake, especially when coming from real numbers, is to assume there's only one principal root. In the complex plane, there are always 'n' distinct n-th roots.
Misconception 2: Argument (Angle) Calculation
Simply using atan(b/a) to find the angle θ can be incorrect, as it doesn't distinguish between quadrants. For example, atan(1/-1) and atan(-1/1) both give -45°, but the numbers (-1 + i) and (1 - i) are in different quadrants. Using the atan2(b, a) function is essential as it correctly identifies the angle in all four quadrants.
Misconception 3: Forgetting the 2πk Term
The formula for the angle of the roots is (θ + 2πk) / n. The + 2πk part is what generates the different roots. Forgetting it will only give you the principal root (for k=0).

Correct Angle Calculation

  • For z = -1 + i: a=-1, b=1. The point is in Quadrant II.
  • Incorrect: atan(1/-1) = -45°.
  • Correct: atan2(1, -1) = 135°. Using this correct angle is crucial for finding the correct location of the roots.

Mathematical Derivation of the Root Formula

  • The formula for n-th roots
  • Detailed worked example
The N-th Root Formula
Given a complex number in polar form z = r(cos(θ) + isin(θ)), its n-th roots z_k are given by the formula:
zₖ = r¹/ⁿ [cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)]
Where k = 0, 1, 2, ... , n-1.

Detailed Worked Example: Find the Cube Roots of z = i

  • 1. Standard Form: z = 0 + 1i.
  • 2. Polar Form: r = √(0²+1²) = 1. θ = atan2(1,0) = π/2 radians (or 90°). So z = 1(cos(π/2) + isin(π/2)).
  • 3. Apply Formula (n=3): The roots are zₖ = 1¹/³ [cos((π/2 + 2πk)/3) + i sin((π/2 + 2πk)/3)].
  • 4. k=0: z₀ = cos(π/6) + i sin(π/6) = √3/2 + 0.5i.
  • 5. k=1: z₁ = cos((π/2 + 2π)/3) + i sin((π/2 + 2π)/3) = cos(5π/6) + i sin(5π/6) = -√3/2 + 0.5i.
  • 6. k=2: z₂ = cos((π/2 + 4π)/3) + i sin((π/2 + 4π)/3) = cos(9π/6) + i sin(9π/6) = cos(3π/2) + i sin(3π/2) = 0 - 1i = -i.