Proving various properties of complex numbers

  • Thread starter Yosty22
  • Start date
  • #1
185
4

Homework Statement



This problem is very easy, but I'm not sure how best to "prove" it. This part of the question just states:
Prove that (1/z)* = 1/(z*) where z* is the complex conjugate of z.

Homework Equations





The Attempt at a Solution



So the complex conjugate of z = x + iy is defined is z* = x - iy. That is, the complex conjugate of something is just making the complex part the opposite sign. So clearly, if you have 1/(x+iy) the complex conjugate would be to make the imaginary part negative, so 1/(x-iy), which clearly is equal to 1/z*.

The answer seems far too straight forward to warrant some formal "proof" but as the question is asking for a proof, I was wondering how you would go about doing it? For previous parts of this problem, such as proving that z* = re-i\theta, I could prove by graphing z and z* in the complex plane, then using Euler's identity of sines and cosines to rewrite those terms in terms of exponential functions and "prove" that way. However, I'm not sure how to "prove" this part of the problem as it seems way too straight forward.

Thanks for any advice.
 

Answers and Replies

  • #2
SteamKing
Staff Emeritus
Science Advisor
Homework Helper
12,798
1,670

Homework Statement



This problem is very easy, but I'm not sure how best to "prove" it. This part of the question just states:
Prove that (1/z)* = 1/(z*) where z* is the complex conjugate of z.

Homework Equations





The Attempt at a Solution



So the complex conjugate of z = x + iy is defined is z* = x - iy. That is, the complex conjugate of something is just making the complex part the opposite sign. So clearly, if you have 1/(x+iy) the complex conjugate would be to make the imaginary part negative, so 1/(x-iy), which clearly is equal to 1/z*.

The answer seems far too straight forward to warrant some formal "proof" but as the question is asking for a proof, I was wondering how you would go about doing it? For previous parts of this problem, such as proving that z* = re-i\theta, I could prove by graphing z and z* in the complex plane, then using Euler's identity of sines and cosines to rewrite those terms in terms of exponential functions and "prove" that way. However, I'm not sure how to "prove" this part of the problem as it seems way too straight forward.

Thanks for any advice.

I don't think you have to graph anything. By using z and z* in rectangular form, you should be able to establish the stated identity algebraically.
 
  • #3
RUber
Homework Helper
1,687
344
Prove that (1/z)* = 1/(z*) ...So clearly, if you have 1/(x+iy) the complex conjugate would be to make the imaginary part negative, so 1/(x-iy), which clearly is equal to 1/z*.
You are assuming what you are trying to prove here.
I would recommend algebraically showing that ## \frac {1}{x-iy}=\left(\frac {1}{x+iy}\right)^* ## by separating the latter into its real and imaginary parts.
 
  • #4
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
15,228
1,830
To elaborate on what RUber said, all you can really assume as far as conjugation goes is ##(u+iv)^* = u-iv##. ##1/(x+iy)## isn't of the form ##u+iv##, so you can't simply replace ##iy## with ##-iy## and claim it's equal to the conjugate.
 

Related Threads on Proving various properties of complex numbers

  • Last Post
Replies
4
Views
996
Replies
1
Views
955
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
15
Views
2K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
10
Views
4K
  • Last Post
Replies
6
Views
47
Replies
3
Views
1K
Replies
43
Views
4K
  • Last Post
Replies
10
Views
5K
Top