Complex Number Conjugate

The conjugate of a complex number \( z = a+ib \) is noted with a bar \( \overline{z} \) (or sometimes with a star \( z^* \)) and is equal to \( \overline{z} = a-ib \) with \( a = \Re{z} \) the real part and \( b = \Im{z} \) the imaginary part.

Example: Consider \( z = 1 + i \) then the conjugate is \( \overline{z} = 1-i \)

On a complex plane, the points \( z \) and \( \overline{z} \) are symmetrical (symmetry with respect to the abscissa x-axis), the 2 points are conjugate pairs.

Using the complex numbers \( z, z_1, z_2 \), the conjugate has the following properties:

$$ \overline{z_1+z_2} = \overline{z_1} + \overline{z_2} $$

$$ \overline{z_1 \cdot z_2} = \overline{z_1} \times \overline{z_2} $$

$$ \overline{\left(\frac{z_1}{z_2}\right)} = \frac{\overline{z_1}}{\overline{z_2}} \iff z_2 \neq 0 $$

A number without an imaginary part is equal to its conjugate:

$$ \Im(z) = 0 \iff \overline{z} = z $$

The modulus of a complex number and its conjugate are equal:

$$ |\overline{z}|=|z| $$

The conjugate \( \overline{a} \) of a real number \( a \) is the number \(a\) itself: \( a=a+0i=a-0i=\overline{a} \)