Proving various properties of complex numbers

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.

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.

SteamKing
Staff Emeritus
Homework Helper

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.

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.

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.

RUber
Homework Helper
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.

vela
Staff Emeritus