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Complex Number Calculator
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Complex Number Calculator is an online tool used to find real part and imaginary part separately by using basic operations(Addition, subtraction, multiplication and Division).

Complex numbers are numbers made up of two parts, that is real part and imaginary part.

Real numbers include all integers, rational numbers, and irrational numbers.

Imaginary numbers are those numbers that can only be realized theoretically, and not practically, because they do not exist.

A complex number is written as [z = a + ib] , where ' [a] ' and ' [b] ' are any real numbers and ' [i] ' is the imaginary number $\sqrt{-1}$. The real part of the above complex number is [Re (z) = a], and the imaginary part is [Im (z) = b].

You can see a default numbers which are real and imaginary numbers and also with the arithmetic operation given below. Click on "Calculate", as you can see in this case, addition of complex number takes place between first and the second complex number.

## Step by Step Calculation

Step 1 :

Basic Operations on Complex Numbers

Addition: (a + ib) + (c + id) = (a + c) + i (b + d)

Subtraction: (a + ib) - (c + id) = (a - c) + i(b - d)

Multiplication: (a + ib) (c + id) = ac + iad + ibc + i2bd = (ac - bd) + i (ad + bc)

Where i2 = -1

Division:  $\frac{(a+bi)}{(c+di)}$ = $\frac{(ac+bd)}{(c^{2}+d^{2})}$ + $\frac{(bc-ad)}{(c^{2}+d^{2})i}$

Where c and d are not together zero. This is get by multiplying jointly the above digit and the under digit by the conjugate of the denominator c + di, other one is (c - di).

Step 2 :

To find imaginary part and real part separately, perform basic operations for the complex numbers.

## Complex Number Examples

1. ### Find the value for the complex number (12 + 9i) + (10 + 4i).

Step 1 :

Given: Complex number is (12 + 9i) + (10 + 4i).

We have to perform the addition operation for the complex numbers.

Addition: (a + ib) + (c + id) = (a + c) + i (b + d)

Step 2 :

First add the real part separately and add the imaginary part separately.

(12 + 9i) + (10 + 4i) = (12 + 10) + (9i + 4i)

(12 + 9i) + (10 + 4i) = 22 + 13i

The value for the complex number (12 + 9i) + (10 + 4i) is 22 + 13i.

2. ### Find the value for the complex number (13 + 9i) - (23 + 91i).

Step 1 :

Given: Complex number is (13 + 9i) - (23 + 91i).

We have to perform the subtraction operation for the complex numbers.

Subtraction: (a + ib) - (c + id) = (a - c) + i(b - d)

Step 2 :

First subtract the real part separately and subtract the imaginary part separately.

(13 + 9i) - (23 + 91i) = (13 - 23) + (9 - 91)i

(13 + 9i)- (23 + 91i) = -10 + (-82) i

(13 + 9i)- (23 + 91i) = -10 - 82 i