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Does $x>0$ suggest that $x\in\mathbb R$?

For numbers not in $\,\mathbb R\,$ (e.g. in $\mathbb C\setminus \mathbb R$), their sizes can't be compared.

So can I omit "$\,x\in\mathbb R\,$" and just write $\,x>0\,$?

Thank you.

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5  
Pretty sure similar binary relations $\cdot>\cdot$ are defined on $\mathbb{N}$ and $\mathbb{Z}$, too. How are you going to distinguish which might be meant? –  oldrinb May 19 '13 at 3:22
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Well, it is in $N$ and $Z$ it is also in $R$. –  nayrb May 19 '13 at 3:55
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@nayrb Umm, 2.5 is in $\mathbb{R}$ but not in $\mathbb{N}$ nor $\mathbb{Z}$. Yet $2.5 > 0$. So the point still stands, out of context it could mean either a positive integer or a position real (or a positive rational, or a positive.. anything in any ordered field) –  Thomas May 19 '13 at 4:09
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One could also write $\omega>0$, and $\omega\notin\mathbb{R}$ since it is the first infinite ordinal. –  Baby Dragon May 19 '13 at 4:12
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Authors of complex analysis textbooks do often take this shortcut (e.g. Eli Stein and Rami Shakarchi), and within complex analysis I feel this is fine. Within this subject, there should be no confusion, especially once readers know that $\mathbb{C}$ is not an ordered field. –  kigen May 20 '13 at 2:10

7 Answers 7

up vote 18 down vote accepted

It really depends on context. But be safe; just say $x > 0, x\in \mathbb R$.

Omitting the clarification can lead to misunderstanding it. Including the clarification takes up less than a centimeter of space. Benefits of clarifying the domain greatly outweigh the consequences of omitting the clarification.

Besides one might want to know about rationals greater than $0$, or integers greater than $0$, and we would like to use $x \gt 0$ in those contexts, as well.

ADDED NOTE: That doesn't mean that after having clarified the context, and/or defined the domain, you should still use the qualification "$x\in \mathbb R$" every time you subsequently write $x \gt 0$, in a proof, for example. But if there's any question in your mind about whether or not to include it, error on the side of inclusion.

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Deserves another thumbs up! +1 –  Amzoti May 19 '13 at 4:02
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It looks like 1.5cm to me... –  Zhen Lin May 19 '13 at 7:43
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@ZhenLin You might be right: I read the question/typed my answer on my iPod ;-) –  amWhy May 19 '13 at 12:20
    
The clarification takes up some number (1 or 1.5 seems large) of square centimeters unless you are measuring the inline space in your font. But the point is good. +1 –  Ross Millikan May 20 '13 at 3:59

One might be able to decode that notation, but why force this puzzle on the reader?

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Easiest solution is to just say $$ x\in\mathbb{R}^+ $$ Expresses both conditions in one hit.

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2  
From my experience, more often than not, $0\in {\bf R}^+$. ${\bf R}_{>0}$ is less amibguous. –  tomasz May 20 '13 at 1:34
    
$\mathbb{R}^+$ means positive reals, hence the symbol. Zero isn't positive. Note that $0\ni\mathbb{Z}^+$. –  Glen O May 20 '13 at 1:55
    
Well, the fact that the same symbol means different things to different people seems like a clear indication that it should be avoided. –  bubba May 20 '13 at 3:47
    
Oops - I just realised that I used $\ni$ when I meant it to be $\not\in$. Sorry about that. –  Glen O May 20 '13 at 5:57
    
@GlenO: I just said that in practice, ${\bf R}^+$ (or, more often, ${\bf R}_+$) often denotes nonnegative reals. It's commonplace, for instance, when discussing manifolds with boundary. –  tomasz May 20 '13 at 7:30

There are ordered fields which strictly extend the real numbers, there $x>0$ is meaningful, but need not imply $x\in\Bbb R$.

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1  
Not just ordered fields by any means. To go in one direction, $0$ could be taken to be the ordinal $0$, and it could also be the cardinal $0$ (which may or may not be the same thing, depending on the definitions in play, but these are generally not considered real numbers). In an overlapping context, $0$ could be the bottom element of an ordered set/class. In a different context, it could be the zero of an ordered ring. Altogether, assuming that $0 \in \mathbb R$ is just not a good plan. –  dfeuer May 19 '13 at 5:31
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BTW, I know that Asaf Karagila is perfectly aware of this; my comment was for the benefit of the asker. –  dfeuer May 19 '13 at 5:33

Of course, everything depends on context. I usually prefer to say things like

For any $x\in\mathbb{R}$ with $x>0$, ...

instead of

For any $x>0$, ...

to remove ambiguity, but I'm not insistent on it; I might be willing to sacrifice the "$x\in\mathbb{R}$" if it's making an orphan at the end of a paragraph, for example.

In contrast, everyone knows what

For any $\epsilon>0$, ...

almost always means, and it doesn't really add anything to say $\epsilon\in\mathbb{R}$. Of course, if you want to write $\epsilon>0$ and $\epsilon$ is not an element of $\mathbb{R}$, then it is all the more incumbent upon you to warn the reader of this non-standard use.

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My favorite would be "$x$ is a positive real number". Simple, clear, unambiguous, and no strange symbols to decipher.

It's a bit long, maybe, but who cares -- the point of writing is clarity, not brevity.

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Well, no, it's still ambiguous. You've said $x$ is a positive real number. But will it remain one in the future? See it's those edge cases you gotta look out for... :p –  Thomas May 19 '13 at 10:12
    
@Thomas So it depends on what the meaning of the word "is" is? :-) –  Trevor Wilson May 19 '13 at 15:18
    
In my view "$x$ is a positive real number" means exactly the same as "$x \in \mathbb R$ and $x > 0$". I either case, we omit the part that says "and will always remain so". I would think that this is implied and understood. But, anyway, I'm willing to withdraw my claim of "unambiguous" and replace it with "no less ambiguous than the other symbol-laden versions". –  bubba May 19 '13 at 23:05
    
And, if you're not careful, the fretting about the edge cases can easily overwhelm and obscure the central argument. Actually, this is true both in mathematics and programming. The edge cases do need to be handled, but they should be on the edges of the discussion. I like Donald Knuth's idea of "telling lies" (ignoring the edge cases, at least temporarily). –  bubba May 19 '13 at 23:14
    
@bubba It was a joke. You are of course right that it is implied and that too much symbolism can often make things worse when a simple english sentence is perfectly adequate. –  Thomas May 20 '13 at 3:26

It's probably best to say something like "there is a real number $x > 0$ such that...," or "let $x > 0$ be a real number," etc. depending on what you want to say about $x$. This avoids ambiguity, and also using words to describe the new symbol being introduced makes life easier for the reader. The goal of writing (even mathematical writing) should not be to avoid redundancy.

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