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
\[\Im(z)=\frac 1{2i}(z- z^*).\]

| | | | | created: 2015-04-26 18:30:01 | modified: 2020-06-14 13:59:21 | by: bookofproofs | references: [581]

1.Proof: (related to "Extracting the Real and the Imaginary Part of a Complex Number")