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Conjugate of Complex Numbers and Modulus of Complex numbers for Class 11 ,CBSE Board, IITJEE maths and other exams




Conjugate of Complex Numbers

Let z be the complex number defined as
$z=x+iy$
The conjugate of z is defined as
$\bar{z}=x-iy$
So by defination of Conjugate of any complex number is obtained by replacing i with -i

Properties of Conjugate Number

For $z=x+iy$
1) $\bar(\bar{z})=z$
2) $z+\bar{z}=2x$
3) $z-\bar{z}=2iy$
4) if $\bar{z}=z$ then it means z is real number
5) if $z+\bar{z}=0$ then it means z is pure imaginary number
6) $z\bar{z}=(x^{2}+y^{2})$
7)
8)
9)
10) when $z_{2}$ is not zero
Example:
z=2-3i
Then
$\bar{z}=2+3i$

Modulus Of complex Number

The module of a complex number
$z=x+iy$
is defined as |z|
|z|=$\sqrt{(x^2+y^2)}$
Clearly $|z| \geqslant 0$

Properties of Module of Complex Number

For $z=x+iy$
1) |z| =0 then it mean x=y=0
2) $|z|=|\bar{z}|=|-z|$
3) $z\bar{z}=|z|^{2}$
4)$|z_{1}z_{2}|=|z_{1}||z_{2}|$
5)
6)
7) $|\frac{z_{1}}{z_{2}}|=\frac{|z_{1}|}{|z_{2}|}$ when $z_{2}$ is not zero
The proof of the above properties are quite self explanatory
Example
z=3-4i
Then
|z|=5

Reciprocal or Multiplicative Inverse of Complex Number using Complex Conjugate

z= x+iy
Then
$\frac{1}{z}=\frac{\bar{z}}{|z|^{2}}$

Examples
1)(3 + i)(1 + 7i)
(3 + i)(1 + 7i)= 3×1 + 3×7i + i×1+ i×7i
= 3 + 21i + i + 7i2
= 3 + 21i + i - 7 (because i2 = -1)
= -11 + 22i
We can generalized this multiplication
(a+bi)(x+ui) = (ax-by) + (ay+bx)i

2) Solving the equation
(1+2i)z=(1-i)
Solution:
$z=\frac{1-i}{1+2i}$
Dividing the conjugate for denominator
$z=[\frac{1-i}{1+2i}][\frac{1-2i}{|1-2i}]$
$z==[\frac{1-2i-i+2}{5}]$
$z==[\frac{3-i}{5}]$




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