## Introduction to Complex Conjugate Calculator

Complex conjugate calculator is an online tool that is used to **find the reciprocal function** of a complex number.

Complex number conjugate calculator evaluates the complex number whose real part is the same but the imaginary part has the opposite sign to get conjugate values in a fraction of a second.

## What is Conjugacy in Complex Numbers?

The Complex conjugate is a complex number that has both **real and imaginary roots** and magnitudes but in opposite directions.

You can get the original complex number if you multiply the two conjugate complex numbers. You can find all the arithmetic operations for complex conjugate numbers as you do in complex numbers.

## Notation Used by Complex Number Conjugate Calculator:

Complex conjugate represents with **the bar on z**. The Complex conjugate calculator uses the following formula:

$$ \bar{z} \;=\; a - bi $$

Here,

- a: the real number
- -bi: imaginary root but in the opposite sign

## Calculating through Complex Conjugate Theorem Calculator:

The conjugate complex numbers calculator has a built-in formula of complex analysis in its server that allows it to solve various types of complex conjugate problems in no time. It uses the simplest method for solving complex conjugate problems so that you can easily grasp the concept of conjugacy.

When you give the **input value of the complex conjugate number** in our conjugate of complex number calculator, it identifies the nature of the complex function. Then according to the nature of the given problem, it applies the arithmetic operations. These mathematical operations are:

### Complex Conjugate in Addition and Subtraction Method:

You can find the solution of two complex conjugate numbers in addition using our Complex conjugate calculator as z̄^{2}+ z̄^{2}. For example if z̄^{1}= 2-3i and z̄^{2} =4-2i then its addition is z̄^{1} + z̄^{2}= (2-3i) + (4-2i) and its solution is 6-5i.

The same rule is applied for the **subtraction of two complex conjugate numbers** represented as z̄^{1} - z̄^{2}. For example z̄^{1} = 3-4i ,z̄^{2}=3-5i then its solution is z̄^{1} - z̄^{2} = 0+1i.

**Note**: Here remember that, in both addition and subtraction you can only add or subtract the real part with the real part and the imaginary in the imaginary part only.

### Complex Conjugacy in Multiplication:

In our complex conjugate theorem calculator, you can find out the **multiplication of two complex conjugate numbers** z̄^{1}z̄^{2} and it gives the original function of a complex number. As z̄^{1} = 2-4i , z̄^{2} =3-2i then the solution of multiply is z̄^{1} z̄^{2} = (2-4i)(3-2i)= 2-16i

**Note**: During multiplication of complex conjugate questions you can multiply real numbers with imaginary and the same imaginary with real numbers.

### Complex Conjugacy in Division:

The complex number conjugate calculator finds the division of complex conjugacy numbers using the rationalization method where then it multiplies and divides the denominator method so that it can easily solved. As

$$ \frac{\bar{z^1}}{\bar{z^2}} $$

Where,

$$ \bar{z^1} \;=\; 1 + i, \bar{z^2} \;=\; 1-i $$

Then,

$$ \frac{\bar{z^1}}{\bar{z^2}} \;=\; \frac{1+i}{1-i} \times \frac{1-i}{1-i} \;=\; \frac{(1-i)(1+i)}{2} $$

Here

$$ i \;=\; \sqrt{-1} $$

value after the square root of i^{2} = -1 in complex number and complex conjugate as the negative number has no square root.

## Example of Complex Conjugate with Solution:

An example of the conjugate number along with solution obtained from the complex conjugate calculator that gives the solution in a step-wise process.

### Example 1:

Evaluate the real and imaginary parts of the following equation:

$$ z \;=\; \frac{i-4}{2i-3} $$

**Solution:**

$$ z \;=\; \frac{i-4}{2i-3} $$

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

$$ \frac{-2 + 3i - 8i - 12}{-4-9} $$

$$ \frac{14}{13} + i\frac{5}{13} $$

Hence,

$$ Re (z) \;=\; \frac{14}{13} \;and\; Im(z) \;=\; \frac{5}{13} $$

Multiplying (4+7i) by (4-7i):

$$ (4+7i)(4-7i) \;=\; 16 - 28i + 28i - 49i^2 $$

$$ 16 + 49 $$

$$ 65 $$

### Example 2:

Add and subtract the numbers:

$$ 3 + 4i \;and \; 2 - 7i $$

**Solution:**

Addition:

$$ (3 + 4i) + (2 - 7i) $$

$$ (3+2) + i(4-7) \;=\; 5 - 3i $$

Subtraction:

$$ (3 + 4i) - (2 - 7i) $$

$$ (3 - 2) + i(4 + 7) \;=\; 1 + 11i $$

## Steps to Use in the Complex Conjugate Calculator:

The conjugate of complex number calculator has a user-friendly design that allows you or even a beginner to evaluate the conjugate of different complex numbers. Follow some simple steps that help you to solve complex functions easily. These steps are:

**Enter your complex conjugate number**in the respective box.- Review your complex number before hitting the calculate button so that you get the accurate solution as per your input function.
- Click the “Calculate” button to get the solution of the complex conjugacy problem.
- Press the “Recalculate” button to bring you back to the new page for more evaluation of a complex conjugate number.
- If you want to check out our calculator then you can use the load example to see the working procedure of complex numbers.

## Outcome from Conjugate Complex Numbers Calculator:

You will get the outcome from the complex conjugate calculator after adding the input in it and get the solution within a few seconds. It may contain the following:

- Result box gives you the
**solution to the complex conjugate**problem - Possible steps option provides you with solutions in a step-by-step method of conjugating complex numbers from the calculator.
- Plot option draws a graph as per the solution given values of conjugacy in the number

## Benefits of Conjugate of Complex Number Calculator

Conjugate complex numbers calculator provides multiple **benefits** whenever you use it to find the solution of complex conjugate numbers. These benefits are:

- The complex conjugate theorem calculator has a simple design so anyone can easily accessible to solve the reciprocal of complex number functions.
- You do not need to charge any fee while you using this calculator to find the complex conjugates use it for finding its solution
- The complex number conjugate calculator saves the time you consume to solve complex conjugate problems manually and gives you the result of complex numbers in a few seconds.
- This calculator is a reliable tool as it provides an accurate solution of complex conjugate numbers.
- Complex conjugate calculator enables you to practice more examples of a complex number so that you get familiar with this concept for better understanding.