Complex numbers

  • #1
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?
 

Answers and Replies

  • #2
834
2
What are ##z## and ##z^*## in terms of their real and imaginary parts?
 
  • #3
606
1
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 [itex]\,\overline z\,[/itex] 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
So 1/2 2x = re part = 1/2(z+z*)
And 1/2i 2yi= Im part= 1/2i(z-z*)

Thanks!
 

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