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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

Addition of complex numbers

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



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