## Introduction to Complex Roots Calculator

Complex roots calculator is an online tool that is used to find the complex root of any degree with the help of complex analytic De Moivre's formula in a couple of minutes.

Our complex root calculator **evaluates both real and imaginary number** square roots of any number (till nth number roots) of polar coordinate or geometric expressions or cartesian coordinates.

## Definition of Complex Root:

A complex root is the **sum of real or imaginary roots** that is used to find the square root of a function. The complex roots represent a+bi as a is the real root Re(a) and bi is the imaginary root Imz(bi).

You can find the complex root numbers using the Demovire formula where you can find any number of roots, especially in square and cubic roots.

## Formula Behind Complex Root Calculator:

For finding complex roots from the complex roots calculator, **De Moivre formula** is used. It helps to find the square root of complex number roots for the nth degree which is given as

$$ (e^{i \theta})^n \;=\; e^{i n \theta} $$

$$ \biggr(r (cos \theta + i sin \theta) \biggr)^n \;=\; r^n (cos n \theta + i sin n\theta ) $$

$$ r^{\frac{1}{n}} \biggr( cos \biggr( \frac{\theta + 2 \pi k}{n} \biggr) + i\;sin \biggr(\frac{\theta + 2 \pi k}{n} \biggr) \biggr), k \;=\; \bar{0..n - 1} $$

Here,

R shows the real number

(cos+isin ) shows the imaginary root

n and k value varies from …-1,0,1,2,3…

n represents the number of roots

For positive integer n for polar coordinates

For real numbers, r>0, and n is a positive integer for cartesian coordinates

## Evaluation Process in Imaginary Roots Calculator:

The Imaginary root calculator evaluates different numbers of real or imaginary roots from one to the nth number quickly because it has up-to-date software on its server. This feature enables you to evaluate the complex number roots in both cartesian or polar coordinates form.

When you give the input value as a complex function of **polynomial or algebraic function** in a complex root finder, it analyzes the nature of the function. After identification, the given function is in polar expression or cartesian coordinates.

Then roots of unity calculator apply the Demovire formula to solve both (cartesian or polar function coordinates) and adjusts this formula as per the number of roots. If you give root number 3 as given in the below example then n=3 and k=0,1,2.

After adding values in the formula complex square root calculator calculates the value of 𝜃 when you give **the value of k=0,1,2** one by one. At last, it will give the three cubic roots (it may be imaginary or real or mixed of both real or imaginary roots).

You can get the output in the form of both polar or cartesian coordinates from this complex roots calculator. Let's observe an example to check the mechanism of calculation in it.

## Solved Example of Complex Root

An example of a complex root number is given to let you know about the analysis of the answer from the imaginary roots calculator. It will also help in manual calculations.

### Example:

Determine the three cube roots of i. Determine all z such that:

$$ z^3 \;=\; i $$

**Solution:**

Convert each number to polar form:

$$ (r e^{i \theta})^3 \;=\; r^3 e^{3i \theta} \;=\; 1e^{\frac{i \pi}{2}} $$

For the second equation, divide by i and then write the equation as follows,

$$ 3 \theta \;=\; \frac{\pi}{2} + 2 \pi l \;for\; l \;=\; 0,1,2 $$

$$ \theta \;=\; \frac{\pi}{6} + \frac{2}{3} \pi l for l \;=\; 0,1,2 $$

For l = 0:

$$ \theta \;=\; \frac{\pi}{6} + \frac{2}{3} \pi(0) \;=\; \frac{\pi}{6} $$

For l = 1:

$$ \theta \;=\; \frac{\pi}{6} + \frac{2}{3} \pi(1) \;=\; \frac{5}{6}\pi $$

For l = 2:

$$ \theta \;=\; \frac{\pi}{6} + \frac{2}{3} \pi(2) \;=\; \frac{3}{2} \pi $$

Therefore the three roots are:

$$ 1e^{\frac{i\pi}{6}}, 1e^{i \frac{5}{6}}, 1e^{i \frac{3}{2}} \pi $$

Writing in standard form,

$$ \frac{\sqrt{3}}{2} + i \frac{1}{2}, - \frac{\sqrt{3}}{2}, -i $$

## How to Use a Complex Roots Calculator?

Complex root calculator has a user-friendly interface that enables everyone to easily **use this calculator** to solve complex root number problems.

You should abide by our guidelines before using this complex fourth roots calculator so that you do not get into any trouble in the calculation process. These guidelines are:

- Enter the complex number function in its input field of the complex cube root calculator.
- Select the number of roots from the given list.
- Choose the types of output you want to evaluate from the given problem.
- Review your input function (complex number root) before hitting the calculate button.
- Click on the “Calculate” button of the imaginary roots calculator to get the solution of complex roots both in imaginary or real numbers.
- Press the “Recalculate” button that brings you back to a new page for the calculation of more complex number questions.

## Output from Imaginary Root Calculator:

You will get the **result of the complex roots** from the complex roots calculator according to your given complex analytic function (even the nth root) in a few seconds. It may include as

**Result box**

Result box of the complex root finder provides the solution of your given complex number roots in both real and imaginary parts.

**Possible steps option**

Steps give you a solution of the complex roots in step by step method.

**Plot option**

This option provides you solution in the form of a graph so that you can easily understand this concept.

## Benefits of Complex Root Finder:

You just need to enter your complex function from first to nth number of square root and you will get the result in no time. Complex root calculator provides you with a ton of benefits while you are using it to calculate the complex root numbers. These advantages are:

- The imaginary roots calculator is a
**trustworthy tool**as it always provides precise results with less or no human error in calculation. - Our roots of unity calculator is a speedy tool that provides real or imaginary numbers in no time.
- You do not need to pay any fee because it is a free online tool to find the complex root value.
- You can use the imaginary root calculator for more practice questions so that you get a strong hold on the complex number root concept.
- Complex Roots calculator has a user-friendly interface even a beginner can use it easily.