Complex conjugate clarification. So I understand that something like $4-i$ is a complex conjugate of $4+i$, however are things like; (a) $-4+i = \overline{-4-i}$, (b) $-4+i = \overline{4+i}$, and (c) $-4-i = \overline{4+i}$.
Generally am I allowed to say an reflection of a complex number about the x, y, or both axis is a conjugate of the original number?
Thanks.
 A: Only (a) $-4+i = \overline{-4-i}$ is correct. Others are not. When we take conjugate we only change the sign of imaginary part. Real part not changed at all.
A: Conjugate of a complex number
Geometric definition
In the Argand Plane representation of a complex number $z$, its conjugate $\bar{z}$ is the reflection of $z$ about the x-axis. 

Algebraic definition
For a complex number $z=x+iy$, the complex conjugate $\bar{z}$ is defined as $\bar{z}=x-iy$. 
You should be able to show that both definitions are equivalent, i.e. reflection about x-axis causes sign of $y$ to flip. 
An alternate definition
The conjugate of a complex number $\bar{z}$ of $z$ has two properties


*

*$z+\bar{z}\in\mathbb{R}$

*$z.\bar{z}\in\mathbb{R}$
In fact, if $\bar{z}$ is defined to be a number having the above mentioned properties, it can be shown that such a number is unique. You can do it yourself. I'll get you started. 


*

*Take $z=a+ib$

*Let $\bar{z}=x+iy$, where $x$ and $y$ are unknowns

*Using both equations mentioned for conjugates above, solve for $x$ and $y$. 


You will get a unique value for $x$ and $y$. So, the formal definition statement could be this:
Definition: For a complex number $z$, its conjugate is defined to be the number $\bar{z}$ such that


*

*$z+\bar{z}\in\mathbb{R}$

*$z.\bar{z}\in\mathbb{R}$ 

A: Your choices (b) and (c) are incorrect. "overline" stands for the "complex conjugate." So for any real number, complex conjugate is itself.
A: In terms of reflection, reflection about x(real-axis) is conjugation. So (a) is true, (b) and (c) are false.
A: For a general case we have:
$$z=a+bi \Rightarrow \overline{z}=a-bi$$
where $a,b \in \Bbb R$. 
So just $a)$ is correct.
A: 
Conjugate of a complex number of the form $a+ib$ is defined as $a-ib$. Conjugate of a complex number is just the mirror image of that complex number along $x-axis$ or so called real axis.
I think this is sufficieent information.
