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.
6 Answers
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
- $z+\bar{z}\in\mathbb{R}$
- $z.\bar{z}\in\mathbb{R}$
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.
Your choices (b) and (c) are incorrect. "overline" stands for the "complex conjugate." So for any real number, complex conjugate is itself.
In terms of reflection, reflection about x(real-axis) is conjugation. So (a) is true, (b) and (c) are false.
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.
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.
