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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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