Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Algebra of complex numbers

(A) Addition of two Complex Number:

If   z_1=a+ib   and    z_2=c+id   be any two complex numbers, then the sum   z_1+z_2 is defined by following properties:

(i) The Closure Property:

The sum of two complex number is a complex numbers, i.e.,  z_1+z_2    is a complex number for all numbers z_1 and  z_2.

(ii)Commutative Property:

For any two complex numbers  z_1 and  z_2~ z_1+z_2=z_2+z_1

(iii) Associative Property:

For any three complex numbers  z_1, z_2 and z_3 , (z_1+z_2)+z_3=z_1+(z_2+z_3)

(iv) Existance of Additive Identity:

There exists (\exists ) the complex number  0+i0 (denoted as 0), called the additive identity or the zero complex number, such that for every complex number  z ,  z+0=0+z=z

(v) Existance of Additive Inverse:

To, every complex number z=a+ib , we have the complex -a+i (-b) (denoted as  -z) called the additive inverse or negative of  z i.e., z+(-z)=0

(B) Difference of Two Complex Numbers:

Any two complex numbers  z_1 and z_2 the difference  z_1-z_2 is defined as z_1-z_2=z_1+(-z_2)  

(C) Multiplication of Two Complex Numbers:

Let  z_1=a+ib and  z_2=c+id  be any two complex numbers, then the product z_1z_2=(ac-bd)+i(ad+bc)

For Example:

z_1=3+i5,\quad z_2=2+i6

 \begin{aligned} z_1z_2&= (3+i5)(2+i6)\\&=(3 \times 2-5 \times6)+i(3 \times 6+5\times 2)\\&=-24+i28 \end{aligned}

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  z_1z_2 is a complex number for all complex numbers  z_1 and  z_2.

(ii) Commutative Property:

For any two complex numbers  z_1 and  z_2, z_1z_2=z_2z_1

(iii) Associative Property:

For any three complex numbers z_1, z_2 and  z_3, we have  (z_1z_2)z_3=z_1(z_2z_3)

(iv) Existance of Multiplicative Identity:

There exists  (\exists) the complex number 1+i0   ( denoted as   1 ), called the multiplicative identity such that  z \cdot 1=z , for every complex number  z .

(v) Existance of Multiplicative Inverse:

For every non-zero complex number,  z_1=a+ib (a\neq,~ b \neq 0) , we have the complex number z_2=c+id , called the multiplicative inverse of  z_1 such that  z_1 \cdot z_2 =1 \left( \therefore z_2=\dfrac{1}{z_1} \text { or } z_1^{-1} \right) where  c+id=\dfrac{a}{a^2+b^2}+i \dfrac{-b}{a^2+b^2}

(vi) Distributive Property:

For any three complex numbers:

(1) z_1(z_2+z_3)=z_1z_2+z_1z_3

(2)  (z_1+z_2)z_3=z_1z_3+z_2z_3

(D) Division of Two Complex Numbers:

Any two complex numbers  z_1 and  z_2, where  z_2 \neq 0, the quotient \dfrac{z_1}{z_2} is defined by

 \dfrac{z_1}{z_2}=z_1\dfrac{1}{z_2}

For Example:

z_1=6+3i, \quad z_2=2-i

 \begin{aligned} \dfrac{z_1}{z_2}&=z_1 \cdot \dfrac{1}{z_2}=(6+3i)\left( \dfrac{1}{2-i} \right)\\&=(6+3i) \left( \dfrac{2}{2^2+(-1)^2}+i\dfrac{(-1)}{2^2+(-1)^2} \right)\\&=(6+3i)\left( \dfrac{2}{5}+\dfrac{i}{5} \right)=(6+3i)\left( \dfrac{2+i}{5} \right)\\&=\dfrac{1}{5}\left[ (12-3)+i(6+6) \right]\\&=\dfrac{1}{5}(9+12i)=\dfrac{3}{5}(33+4i) \end{aligned}

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