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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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