Complex numbers are the numbers of the form x+iy where $ i=\sqrt{(-1)}$ and x and y are real numbers.

Definition:- Complex numbers are defined as an ordered pair of real numbers like (x,y) where
$z=(x,y)=x+iy$

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.

We also write that as Re(z) =x and Im(z) =y

Two complex Number z_{1} = a + ib and z_{2} = c + id are equal if a = c and b = d.

Example
z_{1}= 2+4i
z_{1}= 8-4i

Algebra of Complex Number

Just like real numbers, Addition ,subtraction,multiplication ,division is also performed on Complex numbers

Addition of complex numbers

Let z_{1}=x_{1}+iy_{1} and z_{2}=x_{2}+iy_{2} then
z_{1}+z_{2}=(x_{1}+x_{2})+i(y_{1}+y_{2})
Therefore , Real part of both the complex number are added and similarly imaginary part of both the complex numbers are added Important Notes

Addition of Complex numbers also follows the closure, commutative,associative Laws.I will leave the proof of them to you.

We also have additive identity (0) for Complex numbers.

Additive inverse also exists for the complex number for z=x+iy, Additive inverse = -x-iy such that z+(-z) =0

Subtraction of complex numbers

z_{1}-z_{2}=(x_{1}-x_{2})+i(y_{1}-y_{2})
Here the real part are subtracted and imaginary part is also subtracted

Mulitiplication of complex numbers

(z_{1}.z_{2})=(x_{1}+iy_{1}).(x_{2}+iy_{2})
=x_{1}x_{2} -y_{1}y_{2} + i(x_{1}y_{2} + x_{2}y_{1})
Multiplication is similar to multiplication of algebraic expression and we just need put out i^{2}=-1 wherever it comes in the equation Important Notes

Multiplication of Complex numbers also follows the closure, commutative,associative Laws.I will leave the proof of them to you.

We also have Multiplicative identity (0) for Complex numbers.

Multiplicative inverse also exists for the complex number .Details are given below

Multiplicative Inverse

for $z=a+ib$
z^{-1} is given by
=$(\frac{a}{a^{2}+b^{2}})+i(\frac{-b}{a^{2}+b^{2}})$

Division of Complex Numbers

To divide complex number by another complex number, first write quotient as a fraction. Then reduce the denominator complex number to multiplicative Inverse and then simple multiplication applies

Example -1
Find the value of
$\frac {1+i}{1+2i}$ Solution:
The multiplicative inverse of (1+2i) is given as
$=(\frac{a}{a^{2}+b^{2}})+i(\frac{-b}{a^{2}+b^{2}})$
$=(\frac{1}{5}-i \frac{2}{5})$
So $\frac {1+i}{1+2i}$
$=(1+i)(\frac{1}{5}-i \frac{2}{5})$
=$\frac {1}{5} -i \frac {2}{5}+ i \frac {1}{5}+\frac {2}{5}$
=$ \frac {=i}{5} + \frac {3}{5}$

Example -2
Find the value of (1+3i)(2-5i) Solution:
(1+3i)(2-5i) = 2-5i+ 6i-15i^{2}
Now i^{2}=-1
Therefore,
=17+i