# Complex Number Conjugate

## Conjugate of a complex number

z is a complex number whose algebraic format is as follows,

`z = x + i * y`

Then, the conjugate of z, denoted by `bar z`, is defined by,

`bar z = x - i * y`

## Examples

1) `z = 0 , bar z = 0`

2) `z = i , bar z = -i`

3) if z is a real number then `bar z = z`.

4) `z = 1+i , bar z = 1-i`.

## Properties

1) The conjugate of the sum of two complex numbers is the sum of their conjugates.

`bar (y + z) = bar y + bar z`

2) The conjugate of the product of two complex numbers is the product of their conjugates.

`bar (y * z) = bar y * bar z`

3) The conjugate of a quotient of two complex numbers is the quotient of their conjugates.

`bar ((y / z)) = bar y / bar z`

4) `z*bar z = |z|^2`

5) A complex number and its conjugate have the same modulus.

`|bar z| = |z|`

## See also

Modulus of a complex number

Algebraic form of a complex number