Introduction of Complex Numbers,Properties of Complex Number for Class 11 ,CBSE Board, IITJEE maths and other exams

Complex Number

Complex numbers are the numbers of the form a+ib where $i=\sqrt{(-1)}$ and a and b are real numbers.
Definition:- Complex numbers are defined as an ordered pair of real numbers like (x,y) where
$z=(x,y)=x+iy$
and both x and y are real numbers and x is known as real part of complex number and y is known as imaginary part of the complex number.
Example
z1= 2+4i
z1= 8-4i

Properties of Complex Number

Let z1=x1+iy1 and z2=x2+iy2 then
z1+z2=(x1+x2)+i(y1+y2)
Subtraction
z1-z2=(x1-x2)+i(y1-y2)
Multiplication
(z1.z2)=(x1+iy1).(x2+iy2)
Multiplicative Inverse
for z=x+iy
z-1 is given by
=$(\frac{a}{a^{2}+b^{2}})+i(\frac{-b}{a^{2}+b^{2}})$
Division
To divide complex number by another , first write quotient as a fraction. Then reduce the denominator complex number to multiplicative Inverse and then simple multiplication applies
Example
Find the value of
(1+i)/(1+2i)
Solution:
The multiplicative inverse of (1+2i) is given
as
=$(\frac{a}{a^{2}+b^{2}})+i(\frac{-b}{a^{2}+b^{2}})$
=(1/5-2i/5)
So (1+i)/(1+2i)
=(1+i)(1/5-2i/5)
=1/5-2i/5+ i/5+2/5
=-i/5 +3/5