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: Determine the conjugate of $ z = 1+i $ is to calculate $ \overline{z} = 1-i $

In other words, to find the conjugate of a complex number, take that same complex number but with the opposite (minus sign) of its imaginary part (containing $ i $).

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

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

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: $$ z \ times \overline{z} = a ^ 2 + b ^ 2 $$

This number is a real number (no imaginary part $ i $) and strictly positive (addition of 2 squares values necessarily positive)

The conjugate of $ i $ is $ -i $

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

Example: $ \overline{1 + 0 \times i} = 1 $