For complex numbers \(w\) and \(z\) the following hold:
| \(\CCnjgt{w+z}\) | \(=\) | \(\CCnjgt{w} + \CCnjgt{z}\) |
| \(\CCnjgt{w\cdot z}\) | \(=\) | \(\CCnjgt{w} \cdot \CCnjgt{z}\) |
Moreover, the real and imaginary parts of \(z\) are given by
\[\RePrt{z} = \tfrac{1}{2}(z + \CCnjgt{z})\quad \text{and}\quad \ImPrt{z} = -\tfrac{i}{2}(z-\CCnjgt{z})\]We prove these properties by computation. If \(w=u+vi\) and \(z=x+yi\), we find
| \(\CCnjgt{w+z}\) | \(=\) | \(\CCnjgt{(u+vi) + (x+yi)}\) |
| \(\) | \(=\) | \(\CCnjgt{u+x + (v+y)i}\) |
| \(\) | \(=\) | \(u+x - (v+y)i\) |
| \(\) | \(=\) | \(u-vi\ +\ x-yi\) |
| \(\) | \(=\) | \(\CCnjgt{w} + \CCnjgt{z}\) |
This proves the conjugation commutes with addition of complex numbers. As for multiplication,
| \(\CCnjgt{w\cdot z}\) | \(=\) | \(\CCnjgt{(u+vi) \cdot (x+yi)}\) |
| \(\) | \(=\) | \(\CCnjgt{ux-vy + (uy+vx)i}\) |
| \(\) | \(=\) | \(ux-vy\ -\ (uy+vx)i\) |
| \(\) | \(=\) | \(ux - (-v)(-y)\ +\ (u(-y) + v(-x))i\) |
| \(\) | \(=\) | \((u-vi)(x-yi)\) |
| \(\) | \(=\) | \(\CCnjgt{w} \cdot \CCnjgt{z}\) |
So conjugation commutes with multiplication of complex numbers as well.
The compute the real part of \(z = x + yi\):
\[\dfrac{1}{2}(z + \CCnjgt{z}) = \dfrac{1}{2}(x+yi + (x-yi)) = \dfrac{1}{2}\cdot 2x = x = \RePrt{z}\]Similarly, to compute the imaginary part of \(z\):
\[-\dfrac{i}{2}(z - \CCnjgt{z}) = -\dfrac{i}{2}(x+yi - (x-yi)) = -\dfrac{i}{2}\cdot (2yi) = y = \ImPrt{z}\]