Demonstration for Re(z), Im(z), Abs(z) and Arg(z)

In summary: I will not tolerate anyone's continued disregard for our rules.In summary, the thread is closed due to the OP's lack of effort in showing their work and following the rules of the forum. They have also shown a disregard for the help and hints given by others.
  • #1
Jhenrique
685
4
Someone can demonstrate me why

##Re(z) = \frac{1}{2} \left ( z+\bar{z} \right )##
##Im(z) = \frac{1}{2i} \left ( z-\bar{z} \right )##
##Abs(z)=\sqrt{z\bar{z}}##
##Arg(z)=-i ln\left ( \frac{z}{\sqrt{z\bar{z}}} \right )##

?

2# Is correct to affirm that

##Arg(z)=-i ln\left (\sqrt{\frac{z}{\bar{z}}} \right)##

?
 
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  • #2
Try writing ##z## as either ##x + iy## or ##r e^{i\theta}##, whichever is most suitable in each case. If you get stuck somewhere, please show what you tried.
 
  • #3
Jhenrique said:
Someone can demonstrate me why

##Re(z) = \frac{1}{2} \left ( z+\bar{z} \right )##
##Im(z) = \frac{1}{2i} \left ( z-\bar{z} \right )##
##Abs(z)=\sqrt{z\bar{z}}##
##Arg(z)=-i ln\left ( \frac{z}{\sqrt{z\bar{z}}} \right )##

?

2# Is correct to affirm that

##Arg(z)=-i ln\left (\sqrt{\frac{z}{\bar{z}}} \right)##

?

PF Rules require you to show your work. We do not do homework here; we just give hints and suggestions.
 
  • #4
Ray Vickson said:
PF Rules require you to show your work. We do not do homework here; we just give hints and suggestions.

homework? no comments...

I'm asking for a demonstration. Demonstration in math is a serious thing.
 
  • #5
Jhenrique said:
homework? no comments...

I'm asking for a demonstration. Demonstration in math is a serious thing.
That's not relevant. If the question is about homework or textbook problems, the rules here require that you show what you have tried.
 
  • #6
I find it odd that the OP is asking for proof for the representation of Arg(z) as well as Re(z). The difference in difficulty between the two of these is quite profound.Start with answering these questions:
What is the standard representation for z, a complex number? Hint: It's already been said in this thread.
What is Re(z) equal to? What about Im(z)?
What is [itex]\bar{z}[/itex]?
 
  • #7
I already show my hypotheses a lot of times in others topics but, in general, the answer that I have received are, nearly always, a specie of subterfuge. If someone ask how much is 2+2 the answers are (in general) "the sum was the first discovery of man...", "the equality is reflexive, replacement, transitive, symmetric..." etc,etc,etc. But the answer 2+2 is equal to 2 not is given.
 
  • #8
Jhenrique said:
I already show my hypotheses a lot of times in others topics but, in general, the answer that I have received are, nearly always, a specie of subterfuge. If someone ask how much is 2+2 the answers are (in general) "the sum was the first discovery of man...", "the equality is reflexive, replacement, transitive, symmetric..." etc,etc,etc. But the answer 2+2 is equal to 2 not is given.
No wonder, because 2 + 2 ≠ 2.

Since you have refused to show any sort of effort on this problem, I am closing this thread.
 

1. What do Re(z), Im(z), Abs(z) and Arg(z) stand for?

Re(z) stands for the real part of a complex number z, while Im(z) stands for the imaginary part. Abs(z) represents the absolute value or magnitude of the complex number, and Arg(z) represents the argument or angle of the complex number in the polar coordinate system.

2. How are Re(z) and Im(z) related to each other?

Re(z) and Im(z) are related through the complex conjugate property, where the real part of a complex number is equal to the average of the number and its conjugate, and the imaginary part is equal to the difference between the number and its conjugate divided by two.

3. Can you give an example of a complex number and its corresponding Re(z), Im(z), Abs(z), and Arg(z) values?

Yes, for the complex number z = 3 + 4i, Re(z) = 3, Im(z) = 4, Abs(z) = 5, and Arg(z) = arctan(4/3) = 53.13 degrees.

4. How is the absolute value of a complex number calculated?

The absolute value or magnitude of a complex number is calculated using the Pythagorean theorem, where the absolute value is equal to the square root of the sum of the squares of the real and imaginary parts.

5. What is the significance of the argument of a complex number?

The argument or angle of a complex number is important in representing the number in the polar coordinate system, and it is also used in various mathematical operations involving complex numbers, such as multiplication and division.

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