A mathematical operation of adding a complex number to another complex number is called the addition of complex numbers.

## Introduction

The complex numbers are often appeared in an expression by connecting them with a plus sign. It expresses that we have to calculate the sum of them mathematically. So, it is essential to know how to add two or more complex numbers.

### Steps

There are three fundamental steps that we use to find the sum of the complex numbers in mathematics.

1. Write the addition of complex numbers as an expression by displaying a plus sign between every two complex numbers.
2. Write the real part of all complex numbers closer, also imaginary part of the all complex numbers closely and then take out the imaginary unit common from them.
3. Find the sum of the real part of the all complex numbers and also evaluate the sum of the imaginary numbers in the expression.

### Algebraic form

$a+ib$ and $c+id$ are two complex numbers in algebraic form. Let us learn how to add them mathematically

##### Step – 1

Write the complex numbers in a line but display a plus sign between every two complex numbers.

$(a+bi)+(c+di)$

$\implies$ $(a+bi)+(c+di)$ $\,=\,$ $a+bi+c+di$

##### Step – 2

Write the real and imaginary parts of the complex numbers closer to perform the addition of them.

$\implies$ $(a+bi)+(c+di)$ $\,=\,$ $a+c+bi+di$

##### Step – 3

Take out the imaginary unit common from the imaginary numbers in the expression.

$\implies$ $(a+bi)+(c+di)$ $\,=\,$ $a+c+i(b+d)$

Finally, add the real numbers in the real and imaginary parts of the expression to get the sum of them.

$\implies$ $(a+bi)+(c+di)$ $\,=\,$ $(a+c)+i(b+d)$

$\implies$ $(a+bi)+(c+di)$ $\,=\,$ $(a+c)+(b+d)i$

Thus, we find the sum of the complex numbers in mathematics.

Let’s take $a+c = x$ and $b+d = y$, then express it in terms of $x$ and $y$.

$\implies$ $(a+bi)+(c+di)$ $\,=\,$ $x+yi$

It proves that the sum of two or more complex numbers is also a complex number.

#### Example

Find the sum of $2+3i$ and $4+5i$

$(2+3i)+(4+5i)$

$= \,\,\,$ $2+3i+4+5i$

$= \,\,\,$ $2+4+3i+5i$

$= \,\,\,$ $2+4+i(3+5)$

$= \,\,\,$ $6+i(8)$

$= \,\,\,$ $6+i8$

$= \,\,\,$ $6+8i$

It is evaluated that the sum of the complex numbers $2+3i$ and $4+5i$ is also a complex number $6+8i$.

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