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Complex Conjugate

Complex numbers having the same real part and imaginary parts of opposite signs in a complex pair is defined as complex conjugates.

Conjugate of a complex number is denoted by `bar z` or z*. Let z be a complex number, z = a + ib, here, a and b are real numbers and i = `sqrt(-1)` .

The conjugate of z is,

`bar z` = a - ib.

For example, 5 + i and 5 - i are complex conjugates.

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Complex Conjugate Properties

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There are several properties under complex conjugate learning, some of the properties of complex conjugate learning are as follows,

`bar(z + w)` = `barz` + `barw`

`bar(z w)` = `barz` `barw`

`barz` = z, if and only if z is real.

`bar(z^n)` = `(barz)^n` for any integer n.

`bar(|z|)` = |z|

|z|2 = z( `barz` ) = ( `barz` )z

`bar(barz)` = z (i.e., the conjugate of the conjugate of a complex number z is again that number)

z-1 = `barz / |z|^2` , if z is non-zero.

Complex Conjugate Examples

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Below are the examples on complex conjugates -

Solved Examples

Question 1: Solve using complex conjugate, `(8 + 6i) / (2 - 4i)`
Solution:
 
`(8 + 6i) / (2 - 4i)` By taking the conjugate of (2 - 4i), we get (2 + 4i). Now multiple and divide the conjugate (2 + 4i), we get the following expression,

=> `(8 + 6i) / (2 - 4i)` `xx` `(2 + 4i) / (2 + 4i)`

=> `((8 + 6i) xx (2 + 4i)) / ((2 + 4i) xx (2 - 4i))`

=> `(2(4 + 3i) (2 + 4i)) / ((2 + 4i)(2 - 4i))`

=> `(2(4 + 3i)(2 + 4i)) / (2^2 - (4i)^2)`

=> `(2(8 + 16i + 6i + 12i^2)) / (2^2 - 16i^2)`

=> `(2(-4 + 22i)) / (4 - 16(-1))`

=> `(2(-4 + 22i)) / (4 + 16)`

=> `(2(- 4 + 22i)) / 20`

=> `(4(-2 + 11i)) / 20`

The solution is, `(1/5)` (-2 + 11i).
 

Question 2: Solve using complex conjugate, `(4 + 6i) / (2 - 3i)`
Solution:
 
`(4 + 6i) / (2 - 3i)` By taking the conjugate of (2 - 3i), we get (2 + 3i). Now multiple and divide the conjugate (2 + 3i), we get the following expression,

=> `(4 + 6i) / (2 - 3i)` `xx` `(2 + 3i) / (2 + 3i)`

=> `((4 + 6i) xx (2 + 3i)) / ((2 + 3i) xx (2 - 3i))`

=> `(2(2 + 3i) (2 + 3i)) / ((2 + 3i)(2 - 3i))`

=> `(2(2 + 3i)^2) / (2^2 - (3i)^2)`

=> `(2(2 + 3i)^2) / (2^2 - 9i^2)`

=> `(2(2 + 3i)^2) / (4 - 9(-1))`

=> `(2(4 - 9 + 12i)) / (4 + 9)`

=> `(2(- 5 + 12i)) / 13`

=> `(-10 + 12i) / 13`

The solution is, `(1/13)` (-10 + 12i).
 

Question 3: Solve using complex conjugate, `(3 + 4i) / (1 - i)`
Solution:
 
`(3 + 4i) / (1 - i)` By taking the conjugate of (1 - i), we get (1 + i). Now multiple and divide the conjugate (1 + i), we get the following expression,

=> `(3 + 4i) / (1 - i)` `xx` `(1 + i) / (1 + i)`

=> `((3 + 4i) xx (1 + i)) / ((1 + i) xx (1 - i))`

=> `((3 + 4i) (1 + i))/((1 + i)(1 - i))`

=> `(3 + 3i + 4i + 4i^2) / (1 - i^2)`

=> `(-1 +7i) / (1 + 1)`

=> `(-1 + 7i) / 2`

The solution is,`(-1 + 7i) / 2`
 

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