0

In $\mathbb{R}$, we usually denote $\{x \in \mathbb{R} : 0<x<1 \}$ by $(0,1)$. In $\mathbb{R^2}$, the corresponding set of points on the $x$-axis would be $(0,1) \times \{0\}$. However, if I identify $\mathbb{R^2}$ and $\mathbb{C}$, the points will only have one coordinate. So how would I denote the same set of points as a subset of $\mathbb{C}$? Is $(0,1)$ correct?

Davide Masi
  • 1,677
  • In $\Bbb C$ you cannot write $0<z<1$, see here. – Dietrich Burde Sep 02 '24 at 09:29
  • 3
    The notation $[z_1,z_2]$ is often used for the line segment from $z_1$ to $z_2$. – Kavi Rama Murthy Sep 02 '24 at 09:29
  • @DietrichBurde so what should I write? – Davide Masi Sep 02 '24 at 09:30
  • 1
    I think it is fine to write $(0,1)$ to mean ${x\in\mathbb R\mid 0<x<1}$, even if you are doing complex analysis. It's true that there is no total order $<$ that gives $\mathbb C$ the structure of an ordered field, but $\mathbb C$ has a subfield, namely $\mathbb R$, that does have an ordering, and this is the order that you are referring to. – Joe Sep 02 '24 at 09:36
  • @Joe great, thanks a lot – Davide Masi Sep 02 '24 at 09:38
  • 1
    While I support @geetha290krm's comment, one should point out that you use this exact notation. Maybe it would be both easy and clear to just write it out, so in your case ${z\in\mathbb{C}: 0<\text{Re}z<1 \text{ and } \text{Im}z=0}$. – Mathemann Sep 02 '24 at 09:39
  • 1
    If you have defined $\mathbb C$ as $\mathbb R^2$, then the subfield $S={(x,0)\mid x\in\mathbb R}$ is canonically isomorphic to $\mathbb R$, and so unless you are being hyper-formal, you can just identity $\mathbb R$ with $S$, and think of $\mathbb R$ as being a subfield of $\mathbb C$ via this identification. So $(0,1)$ would officially mean ${(x,y)\in\mathbb R^2\mid\text{$y=0$ and $0<x<1$}}$ in the context of complex analysis, but I can't think of many instances where this level of precision is necessary. – Joe Sep 02 '24 at 09:42
  • 1
    @Joe Why not an official answer? – Paul Frost Sep 02 '24 at 14:57
  • 1
    Also see https://math.stackexchange.com/q/3970520/349785. – Paul Frost Sep 02 '24 at 15:01

1 Answers1

3

Even if you are doing complex analysis, I think it is fine to write $(0,1)$ to mean $\{x\in\mathbb R\mid 0<x<1\}$. You might want to write "the interval $(0,1$)" if you feel the reader could mistake $(0,1)$ as referring to the ordered pair $(0,1)$.

It's mentioned in the comments that there is no way to order the complex numbers. More precisely, there is no total order $\le$ on the complex field that endows it with the structure of an ordered field. While this is true, I don't see why it is relevant: the "$<$" in the notation $\{x\in\mathbb R\mid 0<x<1\}$ refers to the usual (strict) total order on $\mathbb R$, not any kind of ordering on $\mathbb C$. There is only one way I think one could reasonably interpret the statement that $0<x<1$, even in the context of complex analysis.

Joe
  • 22,603