## Introduction to Complex Number Calculator with Steps

Complex number calculator is an online complex variable tool that is used to find real and imaginary roots in complex analysis in less than a minute.

Our imaginary number calculator helps you to **evaluate complex numbers** in addition, subtraction, multiplication, and division with the help of properties that give you accurate solutions to given problems.

## Complex Number Definition:

Complex numbers are the **sum of ordered pairs** of real numbers (a, bi), where a is called the real part and bi is called the imaginary part. It denotes an ordered pair of a complex number as a+bi, where i shows to imaginary number unit.

This imaginary root value is i^{2} = -1 or i = √-1. These Imaginary numbers are complex numbers whose real parts are zero, and then the complex number is written as bi.

## Notation Used by Imaginary Number Calculator:

Complex number is represented with a z letter. While solving such problems the Complex number calculator uses the following **notation**,

$$ z \;=\; a + bi $$

where

a: real number

b: imaginary root of a complex number

i: given number is the imaginary number

## Rules Followed by Complex Numbers Calculator

Finding the Complex number problem’s solution in arithmetic operations (addition, subtraction, multiplication, division) uses **different rules and properties**. When calculating complex number problems the imaginary numbers calculator follows some rules that are given,

### Commutative Law:

Commutative law used by the a+bi calculator holds in both addition and multiplication if we suppose a,b,c complex number

$$ a + b \;=\; b + a $$

$$ a . b \;=\; b.a $$

### Associative Law:

Our complex number calculator with steps **uses the associative law** which satisfies its properties. Suppose a,b, and c are complex numbers where they satisfy associative property in both sum and multiplication problems.

$$ a+(b+c) \;=\; (a+b)+c $$

$$ a.(b.c) \;=\; (a.b).c $$

### Distributive Law:

The complex numbers calculator uses the distributive law to solve problems. Such as, Let a,b, and c be the three complex numbers that satisfy the distributive law of complex numbers.

$$ a.(b+c) \;=\; a.b + a.c $$

### Additive Inverse:

Let a complex number z = a + ib, there exists a complex number -z = -a -ib such that z + (-z) = (-z) + z = 0. Here -z is the additive inverse

$$ a + (-a) \;=\; 0 $$

**Note:** Remember all these properties are only used in multiplication and addition methods only not for multiplication and division problems in complex numbers.

## Evaluation Method of Imaginary Numbers Calculator:

Complex calculator is the simplest method for the evaluation of complex number problems in addition, subtraction, multiplication, or division. Our tool is equipped with all the properties of complex numbers.

You just need to enter the **input function of the complex number** in the Complex number calculator and the rest of the work will be done automatically. It uses different properties for solving various types of complex numbers.

Now let's discuss these different types of complex numbers of arithmetic operations.

### For the Addition of a Complex Number:

If the complex number is given in the form of addition then the a+bi form calculator adds one complex number to another complex number as z1+z2. There is a condition real part is added to the real part or the imaginary in the imaginary part.

### For Subtraction of a Complex Number:

In the **subtraction of complex numbers**, the imaginary number calculator uses the same method to add complex numbers where z^{1}-z^{2} is shown in the example below.

### For the Multiplication of Complex Numbers:

For complex number multiplication, Our imaginary calculator uses two complex numbers but no condition is applied for real and imaginary part multiplication separately.

For the Division of the Complex Number

In the division first function, the calculator with imaginary numbers divides z^{1} by the second complex function z^{2} as z^{1}/z^{2}.

### For the Magnitude of a Complex Number:

Complex number calculator with steps takes the **modulus of z ^{1} and z^{2}** separately to give you the distance. These methods are used to solve complex number problems in arithmetic operations.

Let's see the example where all these arithmetic operations of complex numbers are involved and observe the working process of calculation that occurs in our imaginary numbers calculator.

## Example of Complex Number with solution:

An example of a complex number with a step-by-step manual solution is given as,

### Example:

If z1 = -2 + 3i & z2 = 3 + 4i, determine the following,

$$ z_1 + z_2, z_1 - z_2, z_1 z_2, \frac{z_1}{z_2}, |z_1| \;and\; |z_2| $$

**Solution:**

Using the rules and definitions,

$$ z_1 + z_2 \;=\; (-2 + 3i) + (3 + 4i) $$

$$ \;=\; 1 + 7i $$

$$ z_1 - z_2 \;=\; (-2 + 3i) - (3 + 4i) $$

$$ \;=\; -5 - i $$

$$ z_1 z_2 \;=\; (-2 + 3i)(3 + 4i) $$

$$ \;=\; ((-2)(3) - (3)(4)) + ((-2)(4) + (3)(3)i $$

$$ \;=\; - 18 + i $$

$$ \;=\; \frac{z_1}{z_2} \;=\; \frac{-2 + 3i}{3+ 4i} $$

$$ \;=\; \frac{(-2)(3)+(3)(4) + (3)(3)-(-2)(4))i}{3^2 + 4^2} $$

$$ \;=\; \frac{6}{25} \frac{17}{25}i $$

$$ |z_1| \;=\; \sqrt{(-2)^2 + 3^2} $$

$$ \sqrt{13} $$

$$ |z_2| \;=\; \sqrt{3^2 + 4^2} $$

$$ 5 $$

## How to Use the Complex Number Calculator?

The imaginary number calculator is simply designed to solve various types of complex number problems instantly. For this you need to follow our **instructions** then you won't find any trouble during calculation. These steps are:

- Enter the complex function in the input field
- Check your function before pressing the calculate button to the actual solution as per your given input.
- Click the “Calculate” button for the solution of a complex number
- Click the “Recalculate” button for more evaluation of a complex number
- You can use the load example to see the evaluation process of complex numbers in our complex numbers calculator.

## Outcomes Obtained from a+bi Calculator

Complex number calculator with steps gives you a **solution to your input function** in less than a minute when you click on the calculate button because the calculation process has been started. It may include as:

**In result box**

You get the solution of a complex number problem in the result section of the imaginary numbers calculator.

**Steps box**

If you click on the step option then you get given a solution in a step-by-step method

**Plot box**

It gives you a solution of complex numbers in the form of a graph

## Benefits of Our Complex Calculator

When you write in terms of i calculator then you would be able to solve complex functions as this tool gives multiple **benefits whenever you use** it for calculating various types of complex numbers using arithmetic operations. These advantages are:

- The a+bi form calculator is a trustworthy tool as it always provides you with accurate results of the complex number function.
- It has a simple design anyone even a beginner can easily use it.
- The imaginary number calculator is a speedy tool that evaluates the analytic function a couple of times without making any manual effort for its solution.
- Our complex numbers calculator has advanced features that allow you to solve various types of complex functions.
- It is a free tool so that you can use it to calculate complex number problems without paying anything for evaluation.
- Complex number calculator is used for
**practice to solve different kinds of examples**of analytic functions without any limit.