Find Roots Of Complex Numbers Calculator

Table of the n-th roots of the complex number.

k	Root (Polar)	Root (Rectangular: x + yi)
Roots will be listed here.

Plot of the roots on the complex plane.

What is a Roots of Complex Numbers Calculator?

A Roots of Complex Numbers Calculator is a tool used to find the 'n' distinct nth roots of a complex number given in the form a + bi. When we talk about the nth root of a complex number Z, we are looking for all complex numbers 'w' such that wⁿ = Z. Unlike real numbers, a complex number has exactly 'n' distinct nth roots.

This calculator is useful for students of mathematics, engineering, and physics, as well as anyone working with complex number theory. It helps visualize the roots, which lie on a circle in the complex plane and are equally spaced.

Common misconceptions include thinking there's only one root (like with positive real numbers and square roots) or that the roots are always real numbers. In fact, most roots of a complex number are themselves complex.

Roots of Complex Numbers Formula and Mathematical Explanation

To find the nth roots of a complex number Z = a + bi, we first convert it to its polar form, Z = r(cos θ + i sin θ), where:

• r = |Z| = √(a² + b²) is the modulus (magnitude) of Z.
• θ = atan2(b, a) is the argument (angle) of Z, usually in radians (-π < θ ≤ π) or degrees (-180° < θ ≤ 180°).

According to De Moivre's Theorem for roots, the n distinct nth roots of Z are given by:

w_k = r^1/n [ cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n) ]

or in degrees:

w_k = r^1/n [ cos((θ + 360°k)/n) + i sin((θ + 360°k)/n) ]

for k = 0, 1, 2, …, n-1.

Here, r^1/n is the real nth root of the positive real number r.

Variables Table

Variable	Meaning	Unit	Typical Range
a	Real part of the complex number Z	–	Any real number
b	Imaginary part of the complex number Z	–	Any real number
n	The degree of the root being calculated	–	Integer ≥ 2
r	Modulus (magnitude) of Z	–	Non-negative real number
θ	Argument (angle) of Z	Radians or Degrees	-π < θ ≤ π or -180° < θ ≤ 180°
k	Index for the roots	–	0, 1, 2, …, n-1
wk	The k-th root of Z	–	Complex number
r^1/n	Magnitude of each root	–	Non-negative real number

Practical Examples (Real-World Use Cases)

Example 1: Cube Roots of 8i

Let's find the cube roots (n=3) of Z = 0 + 8i. So, a=0, b=8.

• r = √(0² + 8²) = 8
• θ = atan2(8, 0) = π/2 radians (90°)
• Magnitude of roots: 8^1/3 = 2
• Angles of roots (k=0, 1, 2):
  • k=0: (90° + 360°*0)/3 = 30°
  • k=1: (90° + 360°*1)/3 = 150°
  • k=2: (90° + 360°*2)/3 = 270° (-90°)
• The roots are:
  • w₀ = 2(cos 30° + i sin 30°) = 2(√3/2 + i 1/2) = √3 + i
  • w₁ = 2(cos 150° + i sin 150°) = 2(-√3/2 + i 1/2) = -√3 + i
  • w₂ = 2(cos 270° + i sin 270°) = 2(0 – i 1) = -2i

Example 2: Fourth Roots of -16

Let's find the fourth roots (n=4) of Z = -16 + 0i. So, a=-16, b=0.

• r = √((-16)² + 0²) = 16
• θ = atan2(0, -16) = π radians (180°)
• Magnitude of roots: 16^1/4 = 2
• Angles of roots (k=0, 1, 2, 3):
  • k=0: (180° + 360°*0)/4 = 45°
  • k=1: (180° + 360°*1)/4 = 135°
  • k=2: (180° + 360°*2)/4 = 225°
  • k=3: (180° + 360°*3)/4 = 315°
• The roots are:
  • w₀ = 2(cos 45° + i sin 45°) = 2(√2/2 + i √2/2) = √2 + i√2
  • w₁ = 2(cos 135° + i sin 135°) = 2(-√2/2 + i √2/2) = -√2 + i√2
  • w₂ = 2(cos 225° + i sin 225°) = 2(-√2/2 – i √2/2) = -√2 – i√2
  • w₃ = 2(cos 315° + i sin 315°) = 2(√2/2 – i √2/2) = √2 – i√2

Our Roots of Complex Numbers Calculator can quickly find these for you.

How to Use This Roots of Complex Numbers Calculator

1. Enter the Real Part (a): Input the real component 'a' of your complex number a + bi.
2. Enter the Imaginary Part (b): Input the imaginary component 'b'.
3. Enter the Root (n): Specify the degree of the root you want to find (e.g., 3 for cube root). 'n' must be an integer greater than or equal to 2.
4. Calculate: The calculator automatically updates as you type or you can click "Calculate Roots".
5. View Results:
   • The "Primary Result" shows the principal root (for k=0).
   • "Intermediate Results" display the modulus (r), argument (θ), and the magnitude of the roots (r^1/n).
   • The table lists all 'n' roots in both polar and rectangular form.
   • The chart visually plots the roots on the complex plane. They form vertices of a regular n-gon centered at the origin.
6. Reset: Click "Reset" to clear the fields to default values.
7. Copy Results: Click "Copy Results" to copy the main results and intermediate values to your clipboard.

Using the Roots of Complex Numbers Calculator provides a quick and visual way to understand the nature and distribution of complex roots.

Key Factors That Affect Roots of Complex Numbers Results

• Real Part (a) and Imaginary Part (b): These values determine the position of the original complex number in the complex plane, which in turn defines its modulus (r) and argument (θ). Changes in 'a' or 'b' shift and rotate the set of roots.
• Root (n): The value of 'n' determines the number of distinct roots and the angle between them (360°/n or 2π/n). A larger 'n' means more roots, spaced closer together on the circle of radius r^1/n.
• Modulus (r): The modulus r = √(a² + b²) affects the magnitude of all roots (r^1/n). A larger 'r' means the roots are further from the origin.
• Argument (θ): The argument θ = atan2(b, a) determines the starting angle for the first root (k=0), which is θ/n. The other roots are then equally spaced from this first root.
• Choice of k: The index 'k' (from 0 to n-1) distinguishes between the n different roots. Each value of 'k' gives a unique root.
• Coordinate System (Polar vs. Rectangular): While the roots are the same, their representation differs. The calculator provides both for convenience.

Frequently Asked Questions (FAQ)

How many nth roots does a complex number have?

A non-zero complex number has exactly 'n' distinct nth roots.

What is the principal root?

The principal root is usually the root obtained when k=0, which has the smallest non-negative argument (θ/n).

What do the roots look like when plotted on the complex plane?

The n nth roots of a complex number lie on a circle centered at the origin with radius r^1/n. They are equally spaced, forming the vertices of a regular n-sided polygon.

Can I find the roots of a real number using this calculator?

Yes, real numbers are just complex numbers with an imaginary part of zero (b=0). For example, to find the cube roots of -8, enter a=-8, b=0, n=3.

What happens if I enter n=1?

The 1st root is the number itself, but the calculator is designed for n ≥ 2, as the concept of multiple distinct roots applies for n ≥ 2.

Why is the argument θ calculated using atan2(b, a)?

atan2(b, a) correctly determines the angle θ in the correct quadrant, covering the full range from -π to π radians (-180° to 180°).

Are the roots always complex?

Not necessarily. For example, the cube roots of 8 (8+0i) are 2, -1 + i√3, and -1 – i√3. One is real, two are complex. The Roots of Complex Numbers Calculator shows all forms.

What are some applications of finding roots of complex numbers?

They are used in solving polynomial equations, electrical engineering (AC circuit analysis), signal processing, and various areas of physics and mathematics, like understanding the De Moivre's Theorem more deeply.