Proposition: Extracting the Real and the Imaginary Part of a Complex Number

Let \(z\in\mathbb C\) be a complex number. Because by definition \[z:=\Re(z) + \Im (z) i\] and because from the definition of complex conjugate we have that \[ z^*:=\Re(z) - \Im (z) i,\] it follows (by adding or subtracting both equations) that

\[\Re(z)=\frac 12(z+ z^*)\] and that \[\Im(z)=\frac 1{2i}(z- z^*).\]

Proofs: 1