# Complex numbers

1. Oct 22, 2012

### 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?

2. Oct 22, 2012

### Muphrid

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

3. Oct 22, 2012

### DonAntonio

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

4. Oct 22, 2012

### CrazyNeutrino

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

Thanks!