**(A) Addition of two Complex Number:**

If and be any two complex numbers, then the sum is defined by following properties:

**(i) The Closure Property:**

The sum of two complex number is a complex numbers, i.e., is a complex number for all numbers and .

**(ii)Commutative Property:**

For any two complex numbers and ,

**(iii) Associative Property:**

For any three complex numbers and ,

**(iv) Existance of Additive Identity:**

There exists the complex number (denoted as 0), called the additive identity or the zero complex number, such that for every complex number ,

**(v) Existance of Additive Inverse:**

To, every complex number , we have the complex (denoted as ) called the additive inverse or negative of i.e.,

**(B) Difference of Two Complex Numbers:**

Any two complex numbers and the difference is defined as

**(C) Multiplication of Two Complex Numbers:**

Let and be any two complex numbers, then the product

**For Example:**

The multiplication of two complex numbers possesses the following properties:

**(i) Closure Property:**

The product of two complex numbers is a complex number is a complex number, i.e., the product is a complex number for all complex numbers and .

**(ii) Commutative Property:**

For any two complex numbers and ,

**(iii) Associative Property:**

For any three complex numbers and , we have

**(iv) Existance of Multiplicative Identity:**

There exists the complex number ( denoted as ), called the multiplicative identity such that , for every complex number .

**(v) Existance of Multiplicative Inverse:**

For every non-zero complex number, , we have the complex number , called the multiplicative inverse of such that where

**(vi) Distributive Property:**

For any three complex numbers:

(1)

(2)

**(D) Division of Two Complex Numbers:**

Any two complex numbers and , where , the quotient is defined by

**For Example:**

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