Proof of Complex Number Formulas for Real & Imaginary Parts

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Discussion Overview

The discussion centers around the proof of formulas for the real and imaginary parts of complex numbers, specifically the expressions for the real part of \( z \) as \( \frac{1}{2}(z + z^*) \) and the imaginary part as \( \frac{1}{2i}(z - z^*) \). Participants seek clarification and proof of these formulas.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • One participant requests a proof for the formulas regarding the real and imaginary parts of complex numbers.
  • Another participant asks for clarification on the definitions of \( z \) and \( z^* \) in terms of their real and imaginary components.
  • A third participant provides a derivation using \( z = x + iy \) and expresses the real part as \( \frac{1}{2}(z + \overline{z}) \) and the imaginary part as \( \frac{1}{2i}(z - \overline{z}) \), concluding the exercise.
  • A subsequent reply confirms the derived expressions for the real and imaginary parts, reiterating the formulas presented.

Areas of Agreement / Disagreement

Participants appear to agree on the derivation of the formulas, but there is no explicit consensus on the need for further proof or clarification, as some participants are still seeking understanding.

Contextual Notes

The discussion does not address potential limitations or assumptions in the derivation process, nor does it clarify the notation differences between \( z^* \) and \( \overline{z} \).

Who May Find This Useful

Individuals studying complex numbers, particularly those looking for clarification on the properties of real and imaginary parts, may find this discussion beneficial.

CrazyNeutrino
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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?
 
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What are ##z## and ##z^*## in terms of their real and imaginary parts?
 
CrazyNeutrino said:
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
 
So 1/2 2x = re part = 1/2(z+z*)
And 1/2i 2yi= I am part= 1/2i(z-z*)

Thanks!
 

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