19
$\begingroup$

Just as $$\{x \in \mathbb{R}: a \leq x \leq b\}$$ can be written in the more-compact form $[a,b],$ is there an analogous notation for $$\{z \in \mathbb{C}:z=x+yi, x \in[a,b], y \in[c,d]\} \quad ?$$

Pictorially, the set of all $z \in \mathbb{C}$ lying in the green area is the set that I'd like to express in a more concise form: enter image description here

$\endgroup$

9 Answers 9

17
$\begingroup$

Maybe just define something as $[a,b]+[c,d]i$ ?

$\endgroup$
10
  • 5
    $\begingroup$ @RokKralj Why not? There exists interval arithmetic, which we can do on real numbers, why not to do it on complex? $\endgroup$
    – Somnium
    Aug 16, 2014 at 14:07
  • 2
    $\begingroup$ Because you can do it in the real numbers, it wouldn't be obvious that you mean this to be complex numbers. $\endgroup$
    – Teepeemm
    Aug 16, 2014 at 18:53
  • 4
    $\begingroup$ @Teepeemm We choose one real value from $[c,d]$, multiplying it by $i$ we get imaginary number, then adding any real from interval $[a,b]$ we get complex number. Quite obvious, isn't it? $\endgroup$
    – Somnium
    Aug 16, 2014 at 19:40
  • 2
    $\begingroup$ @alexqwx it's not a matter of being sensible, they mean different things! The former (probably needing round brackets) makes $z$ a complex number, the latter makes it an interval. $\endgroup$
    – Sean D
    Aug 16, 2014 at 20:09
  • 2
    $\begingroup$ @JimmyK4542 It is common to define $X+Y = \{x+y:x\in X, y\in Y\}$ IMHO. There's no need to introduce an extra symbol for that. $\endgroup$
    – yo'
    Aug 17, 2014 at 17:41
17
$\begingroup$

Perhaps $\{z \in \mathbb{C}: \operatorname{Re}(z) \in [a,b], \; \operatorname{Im}(z) \in [c,d]\}$.

The complex numbers have no inherent order, so unless you invent something like $[[a+ci, b+di]]$ I know no more compact way to write this.

$\endgroup$
1
  • $\begingroup$ @Alizter: thanks for improving the format of my answer. I didn't know how to get Re() and Im() in normal style, and I suppose I was too lazy to look it up. $\endgroup$ Aug 16, 2014 at 19:14
14
$\begingroup$

As far as I know there is no widely recognized standard notation. Define one in your paper/book/essay if you need it often.

$\endgroup$
2
  • 7
    $\begingroup$ +1. Do not just use a notation and expect the reader to know what it means. $\endgroup$
    – GEdgar
    Aug 16, 2014 at 13:15
  • 3
    $\begingroup$ @GEdgar Better one: Do not introduce notation unless you are 111% sure you need it and it makes your paper easier to follow. $\endgroup$
    – yo'
    Aug 17, 2014 at 17:38
9
$\begingroup$

Since $\mathbb C$ is just $\mathbb R^2$ with specific operations, the notation $[a,b]\times[c,d]$ obviously do the trick.

$\endgroup$
2
  • 4
    $\begingroup$ This is correct in a sense, but likely to be misleading unless explained. Whether $\mathbb{C}$ "is" $\mathbb{R}^2$ depends on your personal view of the foundations, and readers who prefer to define $\mathbb{C}$ in a different way may be confused. $\endgroup$ Aug 17, 2014 at 21:41
  • 2
    $\begingroup$ I admit that the "just" in my answer was a bit provocative... $\endgroup$
    – Taladris
    Aug 18, 2014 at 15:42
5
$\begingroup$

During the Complex Analysis course I took we used $$[z,w]:=\{(1-t)z+tw:t\in[0,1]\},\quad z,w\in\mathbb{C},$$ which coincides with normal intervals when $z,w\in\mathbb{R}$. But you should define this yourself, because people won't know what you mean if you don't.

$\endgroup$
3
  • 2
    $\begingroup$ This is a line segment connecting $z$ and $w$, not the rectangle. $\endgroup$
    – Teepeemm
    Aug 16, 2014 at 18:54
  • 1
    $\begingroup$ This is a pretty common notation, at least in analytic number theory. $\endgroup$ Aug 17, 2014 at 5:53
  • 3
    $\begingroup$ +1 This just shows that defining a complex interval to be a rectangle doesn't sound like a great idea... $\endgroup$
    – yo'
    Aug 17, 2014 at 17:39
4
$\begingroup$

Well, Cartesian product could work here, $\{z\in\mathbb{C}:\Re(z)\in[a,b],\Im(z)\in[c,d]\}$ as $[a,b]\times[ic,id]$.

But I've never seen a standardized notation for something like this, except for circular regions, such as $|z+1|<4$.

$\endgroup$
3
  • 1
    $\begingroup$ Similar to denoting a set by $|z+1|<4$, you could write $a<Re z<b, c<Im z<d$. $\endgroup$ Aug 16, 2014 at 11:54
  • 1
    $\begingroup$ I do not think it works. First of all, an element of the set $[a,b]\times [ic,id]$ is a pair $(x,iy)$, which is not usually canonically identified with a complex number. Then, $[ic,id]$ means nothing in my opinion since you can't define an ordering for imaginary numbers; $i[c,d]$ would look better, but still dubious. Overall, I'd say do not use it unless you define it in advance. $\endgroup$ Aug 16, 2014 at 12:34
  • $\begingroup$ Yeah, that was something I was having trouble with, figuring out how to define that the second set is imaginary. $\endgroup$
    – Silynn
    Aug 16, 2014 at 12:39
4
$\begingroup$

You could write it like this:

$\{(x + yi) \in \mathbb{C}: x \in [a,b], \; y \in [c,d]\}$.

Alternatively, choose one of these:

$[a,b] \times i[c,d]$

$[a,b] \times [ic,id]$

$\endgroup$
1
  • $\begingroup$ You could even drop the $\in \mathbb{C}$, since the inclusion could be inferred from the fact $x$ and $y$ are reals. $\endgroup$
    – Rok Kralj
    Aug 16, 2014 at 12:56
3
$\begingroup$

A teacher of mine once used an "interval notation" for boxes in $\Bbb R^n$. If $\vec{a} = (a_1, \ldots, a_n)$ and $\vec{b} = (b_1, \cdots, b_n)$, then: $$]\vec{a}, \vec{b}[ = ]a_1, b_1[ \times \cdots \times ]a_n, b_n[ = \prod_{i = 1}^n ]a_i, b_i[$$ Maybe you can adapt it for what you need.

$\endgroup$
3
$\begingroup$

Maybe not the answer you're looking for, but you can include the picture and then write:

X

Let $X$ be a rectangle in the complex plane, as depicted.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.