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.

In other words, the conjugate of a complex is the number with the same real part but with opposite imaginary part.

From the algebraic form of a complex number $ z = a+ib $, the conjugate is calculated $ \overline{z} = a-ib $.

The set of 2 elements: a complex number $ z $ and its conjugate $ \overline{z} $, form a pair of conjugates.

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 multiplication of a complex number $ z = a + ib $ and its conjugate $ \overline{z} = a-ib $ gives: $$ z \times \overline{z} = a^2+b^2 $$

The conjugate of the number $ i $ is the number $ -i $

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

The real part of a complex number always remains unchanged when calculating the conjugate. The real part never changes. $ \Re(z) = \Re(\overline{z}) $