Properties of complex conjugate is discussed in detail.

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Given `z_1 = a_1+ib_1` and `z_2 = a_2+ib_2`, what is the conjugate of sum `bar(z_1+z_2)`?

- `bar(z_1)+bar(z_2)`
- `bar(z_1)+bar(z_2)`
- `bar(z_1)-bar(z_2)`

The answer is '`bar(z_1)+bar(z_2)`'

• Conjugate of sum is sum of conjugates.

• Conjugate of difference is difference of conjugates.

**Conjugate of Sum or Difference: ** For complex numbers `z_1, z_2 in CC`

`bar(z_1 +- z_2) = bar(z_1) +- bar(z_2)`

Given `z_1 = a_1+ib_1` and `z_2 = a_2+ib_2` what is the conjugate `bar(z_1 xx z_2)`?

- `bar(z_1) xx bar(z_2)`
- `bar(z_1) xx bar(z_2)`
- `bar(z_1) -: bar(z_2)`

The answer is '`bar(z_1) xx bar(z_2)`'

• Conjugate of product is product of conjugates.

• Conjugate of quotient is quotient of conjugates.

**Conjugate of product or quotient: **For complex numbers `z_1, z_2 in CC`

`bar(z_1 xx z_2) = bar(z_1) xx bar(z_2)`

`bar(z_1 -: z_2) = bar(z_1) -: bar(z_2)`

Given `z = a+ib`, what is the conjugate `bar(z^n)`?

- `(bar(z))^n`
- `(bar(z))^n`
- `(bar(z))^(1/n)`

The answer is '`(bar(z))^n`'

• Conjugate of a power is power of conjugate.

• Conjugate of a root is root of conjugate.

**Conjugate of Power or Root: **For a complex number `z in CC`

`bar(z^n) = (bar(z))^n `

`bar(z^(1/n)) = (bar(z_1))^(1/n) `

Given `z = a+ib`, what is the modulus `|bar(z)|`?

- `|z|`
- `|z|`
- `-|z|`

The answer is '`|z|`'

• Modulus of a conjugate equals modulus of the complex number.

**Modulus of a Conjugate: **For a complex number `z in CC`

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

Given `z = a+ib`, what is the argument `text(arg) bar(z)`?

- `text(arg)z`
- `-text(arg)z`
- `-text(arg)z`

The answer is '`-text(arg)z`'

• Argument of a conjugate equals negative of the argument of the complex number.

**Argument of a Conjugate: **For a complex number `z in CC`

`text(arg) bar(z) = - text(arg)z `

Given `z = a+ib`, what is the conjugate of conjugate `bar bar(z)`?

- `z`
- `z`
- `-z`

The answer is '`z`'

• Conjugate of a conjugate is the complex number itself.

** Conjugate or Conjugate: ** For a complex number `z in CC`

`bar (bar(z)) = z `

Given `z = a+ib`, what is the product `z bar(z)`?

- `|z|`
- `|z|^2`
- `|z|^2`

The answer is '`|z|^2`'

• Product of a number and its conjugate is the square of the modulus.

**Product of a number and its conjugate: ** For a complex number `z in CC`

`z bar(z) = |z|^2 `

*Solved Exercise Problem: *

Given `z = a+ib`, what is the argument `text(arg) bar(z)`?

- `text(arg)z`
- `-text(arg)z`
- `-text(arg)z`

The answer is '`-text(arg)z`'

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