Algebra of complex number

"ALGEBRA OF COMPLEX NUMBERS"

The topic 'Algebra of complex numbers ' contains " Addtion of complex number , Substraction of complex number , Multiplication of complex numbers and Division of complex numbers ".
We are going to discuss about all above operations .

Addition of two complex number

Let z and z' be two complex numbers then ( z+z') represents the addition of two complex numbers z and z'
Explanation:
Let z = a + bi and z' = c + di
Then z + z' = a + bi + c + di
= (a + c) + i (b+ d)
Remark : The real part of the complex number z can be to the real part of the other complex number z' .Similarly , The imaginary part of the complex number z can be added to the imaginary part of the complex number z' (otherwise the result of the addition will be wrong ) .
Example :
Add 3+4i with 5+3i
Solution : 3+4i + 5+3i = (3+5) +(4+3)I
= 8+ 7i
Subtraction of two complex numbers

If z = a+bi and z' = c + di then (z -z' ) represents the result of subtraction .
Explanation: z - z' = (a + bi) - (c + di )
= (a-c) + i ( b -d)
Remark:
In the operation of subtraction the corresponding terms {i,e real part with real part and imaginary part with imaginary part}are operated.
Example:
Subtract (2+3i )from (5+7i )
Solution: (5+7i ) - ( 2+3i)
=( 5-2 ) +(7-3)i
= 3+4i
Multiplication of two complex numbers

Let z = a + bi and z' = c+di be two complex numbers then the product of two complex numbers z and z' is denoted by( z×z' ) and

z×z' = ( a+bi )( c + di )
= a(c + di ) + bi ( c + di )
= ( ac - bd ) + ( ad + bc )i
Which is the result of product of two complex number .
Remark : use the formula given below

i^ (4n +p) = i ^ p

Example:
Multiply ( 1+i ) with ( 2+ 3i )
Solution : ( 1+ i ) × ( 2 + 3i )
= 1 ( 2+ 3i ) + i ( 2+3i )
= 2+ 3i + 2i -3 =( -1 + 5i )
Division of two complex numbers

In the operation division we have to apply rationalise and irrationalise method to solve the division of two complex number
as follows ( by image )