Proving various properties of complex numbers

[Yosty22]

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.

 

[SteamKing]

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.

 

[RUber]

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]

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.