Proving the real and imaginary part formulas

[CrazyNeutrino]

Can someone please prove the formulas:

The real part of z= 1/2(z+z*)
And the imaginary part of z= 1/2i(z-z*)

I can't understand why it is like this. Could someone please give me the proof?

 

[Muphrid]

What are $z$ and $z^*$ in terms of their real and imaginary parts?

 

[DonAntonio]

Can someone please prove the formulas:

The real part of z= 1/2(z+z*)
And the imaginary part of z= 1/2i(z-z*)

I can't understand why it is like this. Could someone please give me the proof?

I write $\,\overline z\,$ instead of your z*. Put

$$z=x+iy\,\,,\,\,x,y\in \Bbb R\Longrightarrow \,\,Re(z)=x\,\,,\,\,Im(z)=y\Longrightarrow$$

$$z+\overline z=x+iy+x-iy=2x\;\;,\;\;z-\overline z=x+iy-(z-iy)=2yi$$

Now end the exercise.

DonAntonio

 

[CrazyNeutrino]

So 1/2 2x = re part = 1/2(z+z*)
And 1/2i 2yi= I am part= 1/2i(z-z*)

Thanks!