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