### Mathematics Class XI

Unit-I: Sets and Functions
Chapter 1: Sets
Unit-II: Algebra
Chapter 5: Binomial Theorem
Chapter 6: Sequence and Series
Unit-III: Coordinate Geometry
Chapter 1: Straight Lines
Chapter 2: Conic Sections
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability
Chapter 1: Statistics
Chapter 2: Probability

# Algebra of complex numbers

(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 ,

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

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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