[-0, 0] = 0 ?

[mmm4444bot]

Until recently, I have never seen the number zero expressed as the closed "interval" [-0, 0].

This use of interval notation seems nonsensical, to me, as I do not think of a Real number as an interval.

Is there some other interpretation for [-0, 0] ?

 

[tkhunny]

Fell asleep on the keyboard?

Really, there may have been some need to express the interval in Set Notation?

 

[Denis]

Is there some other interpretation for [-0, 0] ?

It's a typo; it's really Subhotosh's "Charlie Brown glasses" : [0 - 0]

 

[JeffM]

Well it is a sort of clever way of stating succinctly that zero is the only real that can be treated as either a positive or negative. That is, [- r, r] is definable for every real, but only for zero is it a point on the real number line rather than a segment of the line.

Also [and this is a question, not a suggestion, because I never studied non-standard analysis], what does [- 0, 0] define in the system of hyperreal numbers, or is it even meaningful?

I hope this response is not too inane.

 

[mmm4444bot]

zero is the only real that can be treated as either a positive or negative

This is an interesting point-of-view (for me) because I've always considered zero to be the only Real that is neither positive nor negative. Can anybody explain a scenario where there is use for implementing -0 or +0?



[- r, r] is definable for every real, but only for zero is it a point on the real number line rather than a segment

This is where I'm hung up. Can a single point be considered an interval? It seems to me that the very notion of interval requires at least two points.



I hope this response is not too inane

Not at all! One of the reasons we have this odd-and-ends board is to discuss weirdness.

I believe that succinctness is probably why [-0, 0] was written, but I'm not yet sure that [-0, 0] is mathematically cogent.

 

[daon]

There is a theorem that states the intersection of a nested sequence of compact sets in R^n is non-empty. In some cases, this arises. For instance the intersection of all sets [-1/n,1/n] is [-0,0], or just [0,0]. I see no reason in having "-0" but some classes which rely strictly on pure rigor might require one to prove that -0=0.

 

[JeffM]

This is an interesting point-of-view (for me) because I've always considered zero to be the only Real that is neither positive nor negative. Can anybody explain a scenario where there is use for implementing -0 or +0?

Although I know very little math (my academic training was in history and the so-called social sciences), I suspect that it is true in math (as I am certain that it is true in other fields) that definitions are chosen rathen than compelled. Perhaps in mathematics, there are some definitions that are necessary to produce any consistent theorems. But most definitions even in mathematics are carefully chosen to enhance communication and to simplify thought. That is, most definitions are conventional and convenient rather than necessary. So, mathematics could probably be developed on the basis that zero is the only real number that is both positive and negative or on the basis that zero is the only real number that is neither positive nor negative. The second definition is conventional and probably more convenient, but mathematics is hardly threatened by the proposition that zero is both non-negative and non-positive. "A rose by any other name would smell as sweet."

Can a single point be considered an interval? It seems to me that the very notion of interval requires at least two points.

The same thought process applies to notation: does anything depend on defining "x belongs to [a, b]" as meaning "a \(\displaystyle \leq\\) x \(\displaystyle \leq\\) b"
or as "a \(\displaystyle \leq\\) x \(\displaystyle \leq\\) b and a \(\displaystyle \neq\\) b." One definition may lead to much more convenient exposition and proof, but would the other definition result in a different or more impoverished mathematics?

Of course, you also asked whether there was any advantage in using the notation [a, a] to describe the point a. There you have me. I am still wondering in my ignorance about the hyperreals, but that is just handwaving.

 

[mmm4444bot]

For instance the intersection of all sets [-1/n,1/n] is [-0,0], or just [0,0]. I see no reason in having "-0" but some classes which rely strictly on pure rigor might require one to prove that -0=0.

I'm glad to learn that the notation [-0, 0] is actually used. That knowledge makes it easier for me to "swallow" it when borrowed for zero.