2 questions about the exact geometry of a 1 unit line segment

[Mates]

I am a little confused about what a single line segment of 1 unit (from say an x-y graph in an Euclidean space) would have to entail.

Let's take a line segment of 1 unit, from 0 to 1, for example. When I first thought about this, I thought that the notation [0,1] would suffice. But when I thought about 2 units, from 0 to 2, there seems to be a small inconsistency when deconstructing the two segments. The first segment would still be [0,1] (as I had already defined it that way), but the second segment would have to be (1,2]. Cleary the first unit is not the same as the second. Don't we want them both to be same?

The 1 unit segment seems to have to either include 0 or 1, but not both. So the unit would be either [0,1) or (0,1] in segment form. Or in general, for every unit to be the same, each unit would be in the form, [n,m) or (n,m]. Is this correct?

 

[lev888]

The length of the segment is 1 unit, not the segment itself. Not sure what the inconsistency is. If you see it in [0,2] segment, then it exists in the [0,1] segment, because it also can be split into 2 segments at 0.5.

 

[Dr.Peterson]

I am a little confused about what a single line segment of 1 unit (from say an x-y graph in an Euclidean space) would have to entail.

Let's take a line segment of 1 unit, from 0 to 1, for example. When I first thought about this, I thought that the notation [0,1] would suffice. But when I thought about 2 units, from 0 to 2, there seems to be a small inconsistency when deconstructing the two segments. The first segment would still be [0,1] (as I had already defined it that way), but the second segment would have to be (1,2]. Cleary the first unit is not the same as the second. Don't we want them both to be same?

The 1 unit segment seems to have to either include 0 or 1, but not both. So the unit would be either [0,1) or (0,1] in segment form. Or in general, for every unit to be the same, each unit would be in the form, [n,m) or (n,m]. Is this correct?

You're restricting yourself to segments on the number line (one dimension), not the Euclidean plane, right?

Segments can be open, closed, or half open (e.g. (0,1), [0,1], [0,1), (0,1]). These are all unit segments; they all have the same length.

There is no reason two segments can't share an endpoint; I'm pretty sure that's how Euclid himself would have seen it. It probably makes most sense to think in terms of closed segments.

But what is the context of your question? Why do you think this matters?

 

[Mates]

The length of the segment is 1 unit, not the segment itself. Not sure what the inconsistency is. If you see it in [0,2] segment, then it exists in the [0,1] segment, because it also can be split into 2 segments at 0.5.

Let's say we want choose units to be [n,m] as you have. How can every natural number have the same form? Don't we want each unit to be the same?

 

[Mates]

You're restricting yourself to segments on the number line (one dimension), not the Euclidean plane, right?

Yes (I should have made that clear).

Segments can be open, closed, or half open (e.g. (0,1), [0,1], [0,1), (0,1]). These are all unit segments; they all have the same length.

There is no reason two segments can't share an endpoint; I'm pretty sure that's how Euclid himself would have seen it. It probably makes most sense to think in terms of closed segments.

But what is the context of your question? Why do you think this matters?

I just want to know how we can define a unit. What form should it have to take?

 

[lev888]

Let's say we want choose units to be [n,m] as you have. How can every natural number have the same form? Don't we want each unit to be the same?

Please post the specific definition of the word "unit" so we know what you are referring to. And let us know what exactly are you interested in measuring.

 

[Mates]

Please post the specific definition of the word "unit" so we know what you are referring to. And let us know what exactly are you interested in measuring.

By "unit", I meant a Euclidean distance. Wikipedia has a one-dimensional Euclidean distance as "d(p,q) = |p - q|. In this case the distance is 1.

 

[Mates]

Segments can be open, closed, or half open (e.g. (0,1), [0,1], [0,1), (0,1]). These are all unit segments; they all have the same length.

I don't see how they are the same length. For example, you can fit the segment (8,4) in an opening between [1,3] and [4,5]. But you can't fit [8,9] through the same way. It is somehow infinitesimally too long. Isn't this correct?

 

[Dr.Peterson]

I don't see how they are the same length. For example, you can fit the segment (8,4) in an opening between [1,3] and [4,5]. But you can't fit [8,9] through the same way. It is somehow infinitesimally too long. Isn't this correct?

What??? I assume you didn't proofread, and you meant (3,4), not (8,4)], which is not even proper interval notation, and [3,4], not [8,9].

You seem to be confusing length of a segment with the points of which it is composed. We don't measure length by counting points; the length of a single point is zero.

By "unit", I meant a Euclidean distance. Wikipedia has a one-dimensional Euclidean distance as "d(p,q) = |p - q|. In this case the distance is 1.

Did you notice that Euclidean distance doesn't care whether the endpoints are included or not? The distance is 1 whether you talk about (0,1), [0,1], [0,1), or (0,1].

 

[Steven G]

The distance of a point is 0, that is the distance for [3, 3] is 0. This is why the distances [0, 1], [0,1), (0, 1], (0, 1) are all the same.

 

[Mates]

The distance of a point is 0, that is the distance for [3, 3] is 0. This is why the distances [0, 1], [0,1), (0, 1], (0, 1) are all the same.

I just want to ask you something about that. Is it accurate to say that 0 distance is "longer" than no distance? For example, the point at [3,3] has zero distance, but no point at all would seem to have less distance.

 

[lev888]

I just want to ask you something about that. Is it accurate to say that 0 distance is "longer" than no distance? For example, the point at [3,3] has zero distance, but no point at all would seem to have less distance.

What distance would be less than 0?

 

[blamocur]

I just want to ask you something about that. Is it accurate to say that 0 distance is "longer" than no distance? For example, the point at [3,3] has zero distance, but no point at all would seem to have less distance.

To me this sounds more like a matter for philosophers (or linguists?). As far as math is concerned you can believe that it is longer, or that it is the same -- I don's see any practical or theoretical difference since this discussion is not in terms of any formal theory. In math questions like this would most likely belong to measure theories, and if I remember correctly most measures of an empty set would be 0, so the answer to your question would be "the same".

 

[Steven G]

Suppose someone ask you how far it is from your home to school and you respond with 10 miles. If someone were to ask you how far is it from your school to home would you say -10 miles. I suspect not, as distance is never negative.

 

[lev888]

By "unit", I meant a Euclidean distance. Wikipedia has a one-dimensional Euclidean distance as "d(p,q) = |p - q|. In this case the distance is 1.

Then, again, where is the inconsistency? The distance from 0 to 1 is 1. From 0 to 2 is 2.
Earlier you wrote: "Let's say we want choose units to be [n,m] as you have". I think you need to decide what units you want to discuss: distance or something else.

 

[pka]

I am a little confused about what a single line segment of 1 unit (from say an x-y graph in an Euclidean space) would have to entail.
Let's take a line segment of 1 unit, from 0 to 1, for example. When I first thought about this, I thought that the notation [0,1] would suffice. But when I thought about 2 units, from 0 to 2, there seems to be a small inconsistency when deconstructing the two segments. The first segment would still be [0,1] (as I had already defined it that way), but the second segment would have to be (1,2]. Cleary the first unit is not the same as the second. Don't we want them both to be same?
The 1 unit segment seems to have to either include 0 or 1, but not both. So the unit would be either [0,1) or (0,1] in segment form. Or in general, for every unit to be the same, each unit would be in the form, [n,m) or (n,m]. Is this correct?

Hello again Mates,
Having read through this thread, it is clear that you do not understand that the idea of distance in not a property of space in general. But rather distance is a synthetic notion introduced by way of axioms, definitions, or postulates. HERE in a great reference Elementary Geometry from an Advanced Point of View by Edwin Moise, Moise was a student of R L Moore who arguably had the most impact on the teaching of mathematics of any one in twentieth century. In fact, Keith Devlin has written that Moore was the greatest Maths teacher: SEE HERE.(1999, May & Jun)
If you have access to a good mathematics library surely one would find the text there. But were I you, I would buy a good used copy,