Concerns about the geometry of the real number line

[Lorentz Jr]

8 minutes ago, Boltzmannbrain said:

This example has always been my intuition when thinking about a "next real number".  However, this only shows that this particular method to find a next number does not work.

because there is no "next" real number. The word "next" only has meaning in the context of countable sequences.

Intuition won't help you understand the real number line, because your intuition (and everyone else's) is based on finite numbers and countable sequences.

Edited March 3, 2023 by Lorentz Jr

[Boltzmannbrain]

1 hour ago, Genady said:

I don't think there are such concepts as "interval of points" and "interval ends at ..." in math.

Yeah, I realize now that "end" should be "boundary".  I see now that this kills my argument in its literal form.  

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I have no idea what revolving, breaking, and reattaching segments have to do with numbers and interval lengths. The former are not mathematical concepts, AFAIK.

Right, I am not using formal language again.  The words that I used have an intended meaning that may not exist in math, but then again maybe they do exist.  I will have to figure this out.

 

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@Boltzmannbrain, I start to suspect that the root of confusion is here: you are talking about an actual physical segment, while the "real number line" is a mathematical concept.

The "real" in the latter does not refer to "line", i.e., it is not a "real line."

It refers to "number", i.e., they are "real numbers."

Well, again, I do don't know what I can say or not say "mechanically" regarding mathematical concepts.  I am hoping to keep this discussion in the mathematical "realm".  I do, however, think that we can think about the real number line geometrically.  For example, we say that we have a line segment in a coordinate plane in R¹.

 

33 minutes ago, Lorentz Jr said:

That doesn't mean it's finite. The density of points is infinity divided by a finite number, which is also infinite.

If the length of the closed interval is L, then you can think of the length of the open interval as the limit of L - dx as dx approaches zero. The difference is a finite number divided by infinity.

I meant it can have a finite distance.

1 hour ago, studiot said:

Sometimes it is just a question of getting the right words as using popular general ones in their non scientific sense can lead to misunderstandings.

🙂

Yes, I agree.  I absolutely need to use the proper terms.

Edited March 3, 2023 by Boltzmannbrain

[studiot]

1 hour ago, Boltzmannbrain said:

This example has always been my intuition when thinking about a "next real number".  However, this only shows that this particular method to find a next number does not work. 

What if there were an infinite number of 0's after the decimal.  If we can do this, (although I can't imagine there is a function that would allow this) then we could theoretically get to a smallest real number, specifically, 0.000 ... 0001.  And it would be defined to be the smallest like the natural numbers define 1 (at least in the more common definition of the naturals) to be its smallest.  So you couldn't divide 0.000 ... 0001 by 2 to get a real number, just like you can't divide 1 by 2 to get a natural number. 

The difficulty with answering your queries is how much you do or do not remember from the past.

What you need to know here is this minor theorem.

Every real number has a decimal representation ( note it is not necessarily unique)

So any decimal you can write will be some real number.

So is 0.0 repeating a real number.

Yes 

But is 0.0 repeating the smallest real number ?

No

-1 < 0

 

So to your worry over an infinite count of zeros after the decimal point, so what ?

Pi is well known as and infinite string of digits, as is 0.3 recurring.

The process never ends that is what infinity means!

Since it never ends, there is no last digit in an infinite string.

So the situation you envisage cannot arise.

No matter how many noughts you have after the decimal point you can always add another one.

45 minutes ago, Boltzmannbrain said:

For example, we say that we have a line segment in a coordinate plane in R¹.

This is a fine example of using the wrong words, resulting in nonsense.

 

47 minutes ago, Boltzmannbrain said:

meant it can have a finite distance.

As is this leading to a statement that is just plain wrong

48 minutes ago, Boltzmannbrain said:

I do, however, think that we can think about the real number line geometrically. 

Only some aspects can be thought of geometrically, and only in some circumstances.

And distance is not really one of them.

 

Consider this

Every number has a square. (theorem)

So let us lay out a line of numbers against a line of squares

 

1,2,3,4,5,6,7,8,9

1,4,,9,16,25,36,49,64,81

Every number on the first line is in one-to-one correspondence with a number on the second line.

but the 'lines' have quite a different character.

[Boltzmannbrain]

1 hour ago, Lorentz Jr said:

because there is no "next" real number. The word "next" only has meaning in the context of countable sequences.

Intuition won't help you understand the real number line, because your intuition (and everyone else's) is based on finite numbers and countable sequences.

By "this" I was talking about the example that studiot gave.

23 minutes ago, studiot said:

This is a fine example of using the wrong words, resulting in nonsense.

I meant to put R² not R¹

 

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As is this leading to a statement that is just plain wrong

I can't see how it is wrong.  Loosely speaking, it seems fine to say that there is an infinite number of 0 dimensional points in a centimeter.  How can this possibly be wrong?

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Only some aspects can be thought of geometrically, and only in some circumstances.

And distance is not really one of them.

Yeah, I think the term is unit instead of distance.

Quote

 

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Consider this

Every number has a square. (theorem)

So let us lay out a line of numbers against a line of squares

 

1,2,3,4,5,6,7,8,9

1,4,,9,16,25,36,49,64,81

Every number on the first line is in one-to-one correspondence with a number on the second line.

but the 'lines' have quite a different character

 

I am not sure what this has to do with what we are discussing.

[Genady]

1 hour ago, Boltzmannbrain said:

If we can do this, (although I can't imagine there is a function that would allow this) then we could theoretically get to a smallest real number, specifically, 0.000 ... 0001.  And it would be defined to be the smallest like the natural numbers define 1 (at least in the more common definition of the naturals) to be its smallest.  So you couldn't divide 0.000 ... 0001 by 2 to get a real number, just like you can't divide 1 by 2 to get a natural number. 

To find out if such a smallest real number exists, we need to use a definition of real numbers. There are several equivalent ones, so let's pick one. It can be formulated rigorously, but here is the idea. 

Assume we know what rational numbers are (fractions of integers, k/n.) We then define convergent sequences of rational numbers. Then, we discover that not all such sequences, in spite of being convergent, have limits which are themselves rational numbers. Then, we extend the set of numbers by including all such limits, and define this extended set, real numbers.  

Now we can answer the question: can any real number be divided by 2 to get a real number?

Let's take a real number, Q. By definition, it is a limit of some convergent sequence of rational numbers, let's say, the sequence q₁, q₂, q₃, ...  Now let's take another sequence of rational numbers: q₁/2, q₂/2, q₃/2, ...  It can be shown to be a convergent sequence. Then, by definition, its limit is a real number.

Thus, any real number can be divided by 2 to get a real number. Hence, there is no smallest real number.

[Lorentz Jr]

40 minutes ago, Boltzmannbrain said:

By "this" I was talking about the example that studiot gave.

I knew that. It was obvious.

1 hour ago, Boltzmannbrain said:

I meant it can have a finite distance.

It's also obvious that ranges can be finite.

40 minutes ago, Boltzmannbrain said:

I am not sure what this has to do with what we are discussing.

I'm not sure what either of your responses have to do with my comments.

Edited March 3, 2023 by Lorentz Jr

[studiot]

2 hours ago, Boltzmannbrain said:

By "this" I was talking about the example that studiot gave.

I meant to put R² not R¹

 

I can't see how it is wrong.  Loosely speaking, it seems fine to say that there is an infinite number of 0 dimensional points in a centimeter.  How can this possibly be wrong?

Yeah, I think the term is unit instead of distance.

I am not sure what this has to do with what we are discussing.

 

If you want to discuss numbers (or anything else) mathematically you need to only use properties that numbers have, not introduce new properties they do not have more especially not when you name those properties something whuch already has a specific mathematical definition as something other than your new properties.

I won't both again to say that numbers are not lines and lines are not numbers.

You will not find a single number as an element of a set of lines or even the set of all lines.
Similarly you will not find a single line as an element of a set of numbers or even the set of all numbers.

You keep mentioning a well define property of lines notably distance.

How do you define distance ?

Numbers do not have this property , they have the property of > < or =, and you must use these to couch you arguments in.

Equally, lines do not have the property of + - x or / as well defined operations nor do they follow the normal rules of arithmetic.

 

The conventional way of presenting the notion of a limit as getting closer and closer to something is called the epsilon-delta argument, which involves the normal arithmetic operations along with the less than, greater than or equals notation.

Have you heard of this ?

I can show you this if you are prepared to listen

Are you prepared to listen becasue at the moment you do not seem to be.

 

@Genady has offered you an alternative method via sequences, you don't seem interested in these either.

I should warn you that this only works with certain types of sequence called null or Cauchy sequences.

 

1 hour ago, Lorentz Jr said:

I'm not sure what either of your responses have to do with my comments.

They don't.

But there is nothing wrong with your comments.

A finite range is just the difference between two selected numbers.

[Boltzmannbrain]

3 hours ago, Genady said:

To find out if such a smallest real number exists, we need to use a definition of real numbers. There are several equivalent ones, so let's pick one. It can be formulated rigorously, but here is the idea. 

Assume we know what rational numbers are (fractions of integers, k/n.) We then define convergent sequences of rational numbers. Then, we discover that not all such sequences, in spite of being convergent, have limits which are themselves rational numbers. Then, we extend the set of numbers by including all such limits, and define this extended set, real numbers.  

Now we can answer the question: can any real number be divided by 2 to get a real number?

Let's take a real number, Q. By definition, it is a limit of some convergent sequence of rational numbers, let's say, the sequence q₁, q₂, q₃, ...  Now let's take another sequence of rational numbers: q₁/2, q₂/2, q₃/2, ...  It can be shown to be a convergent sequence. Then, by definition, its limit is a real number.

Thus, any real number can be divided by 2 to get a real number. Hence, there is no smallest real number.

That makes sense, but I feel like there is more to the story.  For example, a segment can always be divided by 2 in the same sense that a real number can always be divided by 2 to get a real number.  To finish cutting a line segment in half may seem like an impossible task.  But if you introduce infinity into the problem, then you can exhaust all divisions of a line segment.

Sometimes there are ways around what seems impossible.  I enjoy trying to find these things.

 

[Genady]

31 minutes ago, Boltzmannbrain said:

That makes sense, but I feel like there is more to the story.  For example, a segment can always be divided by 2 in the same sense that a real number can always be divided by 2 to get a real number.  To finish cutting a line segment in half may seem like an impossible task.  But if you introduce infinity into the problem, then you can exhaust all divisions of a line segment.

Sometimes there are ways around what seems impossible.  I enjoy trying to find these things.

 

It is quite clear with the numbers, but not with the segments:

- What is a segment?

- What is a segment division?

- What is same sense?

- What is introduction infinity?

If I knew these definitions, I might be able to figure out if it is or it is not possible to exhaust all divisions of a line segment.

As of now, there is no relation between segment and real numbers. The latter are built on rational numbers and their converging sequences. I don't see anything like that in the former.

[Boltzmannbrain]

13 hours ago, Genady said:

It is quite clear with the numbers, but not with the segments:

- What is a segment?

- What is a segment division?

- What is same sense?

- What is introduction infinity?

If I knew these definitions, I might be able to figure out if it is or it is not possible to exhaust all divisions of a line segment.

As of now, there is no relation between segment and real numbers. The latter are built on rational numbers and their converging sequences. I don't see anything like that in the former.

I don't know how it came to be, but in linear algebra the parameters of lines and vectors were always elements of the reals. 

19 hours ago, studiot said:

 

If you want to discuss numbers (or anything else) mathematically you need to only use properties that numbers have, not introduce new properties they do not have more especially not when you name those properties something whuch already has a specific mathematical definition as something other than your new properties.

I won't both again to say that numbers are not lines and lines are not numbers.

You will not find a single number as an element of a set of lines or even the set of all lines.
Similarly you will not find a single line as an element of a set of numbers or even the set of all numbers.

You keep mentioning a well define property of lines notably distance.

How do you define distance ?

Numbers do not have this property , they have the property of > < or =, and you must use these to couch you arguments in.

Equally, lines do not have the property of + - x or / as well defined operations nor do they follow the normal rules of arithmetic.

 

I believe distance can be defined by Euclidean distance which uses Pythagorean theorem.  

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The conventional way of presenting the notion of a limit as getting closer and closer to something is called the epsilon-delta argument, which involves the normal arithmetic operations along with the less than, greater than or equals notation.

Have you heard of this ?

 

Yes, the definition of a limit.  That was a very interesting topic that we had to learn.  

[Genady]

8 minutes ago, Boltzmannbrain said:

I don't know how it came to be, but in linear algebra the parameters of lines and vectors were always elements of the reals. 

Vectors in linear algebra may be elements of a real vector space, a complex vector space, a rational vector space, or any other field vector space.

I don't think there are 'lines' in linear algebra.

[Boltzmannbrain]

8 minutes ago, Genady said:

Vectors in linear algebra may be elements of a real vector space, a complex vector space, a rational vector space, or any other field vector space.

 

Not just the spaces, but the parameters too.

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I don't think there are 'lines' in linear algebra.

Lines are also taught in linear algebra.

[Genady]

3 minutes ago, Boltzmannbrain said:

Not just the spaces, but the parameters too.

Lines are also taught in linear algebra.

What do you call "parameters" of vectors?

What is 'lines' in linear algebra?

[Boltzmannbrain]

1 minute ago, Genady said:

What do you call "parameters" of vectors?

What is 'lines' in linear algebra?

Do you not know these terms, or are you wondering if I know them?

[Genady]

Just now, Boltzmannbrain said:

Do you not know these terms, or are you wondering if I know them?

I don't think they are defined in linear algebra.

[Boltzmannbrain]

4 minutes ago, Genady said:

I don't think they are defined in linear algebra.

Here are both terms wrapped up in one video, from Khan Academy 

 

[Genady]

27 minutes ago, Boltzmannbrain said:

Here are both terms wrapped up in one video, from Khan Academy 

 

OK. First, there still no such thing as "parameters of vectors".

Second, I see what they define as a "line" and its parameters. Now, to your original question,

1 hour ago, Boltzmannbrain said:

I don't know how it came to be, but in linear algebra the parameters of lines and vectors were always elements of the reals.

the answer is, a) not always, b) they are real numbers when you consider vectors in a real vector space.

If the vectors space is real vector space, the parameters are real by definition. If the vector space is some other kind of vector space, the parameters will belong to a different field as well.

[Boltzmannbrain]

3 hours ago, Genady said:

OK. First, there still no such thing as "parameters of vectors".

Second, I see what they define as a "line" and its parameters. Now, to your original question,

the answer is, a) not always, b) they are real numbers when you consider vectors in a real vector space.

If the vectors space is real vector space, the parameters are real by definition. If the vector space is some other kind of vector space, the parameters will belong to a different field as well.

I will try to explain formally.  But since I haven't studied this stuff in 10 years, I might need a little help if I use the wrong terms.

22 hours ago, Genady said:

It is quite clear with the numbers, but not with the segments:

- What is a segment?

- What is a segment division?

- What is same sense?

- What is introduction infinity?

If I knew these definitions, I might be able to figure out if it is or it is not possible to exhaust all divisions of a line segment.

As of now, there is no relation between segment and real numbers. The latter are built on rational numbers and their converging sequences. I don't see anything like that in the former.

Are you saying this because I forgot to put "line segment" in one of my sentences?  Or is there really no such thing as dividing a line segement?

[Genady]

50 minutes ago, Boltzmannbrain said:

Are you saying this because I forgot to put "line segment" in one of my sentences?

No.

 

50 minutes ago, Boltzmannbrain said:

Or is there really no such thing as dividing a line segement?

To the contrary, there are different things that can be called this, and I need to know what you mean.

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a segment can always be divided by 2 in the same sense that a real number can always be divided by 2 to get a real number.

I don't understand this statement without your definitions of segment (line segment) and its division. I don't know how they relate to real numbers and can't figure what "the same sense" means.

Also, I don't know what you try to accomplish. I guess, you try to get some contradiction. There are no contradictions in real numbers, it is a mathematical fact. If your definitions regarding segments establish correspondence with real numbers, then automatically, there will be no contradictions in the segments as well.

For example, following the definitions in the video you've linked, we can define a segment as part of a parametrized line, which is covered by the parameter t being in an interval [a,b], IOW, a ≤ t ≤ b. Then, we define segment division. Etc. After everything is consistently defined, there will be no contradictions.

50 minutes ago, Boltzmannbrain said:

I might need a little help if I use the wrong terms.

I'll do my best.

 

* This definition of segment assumes that the vector space is real. If it is complex, then a ≤ t ≤ b is undefined.

Edited March 5, 2023 by Genady

[Mc2509]

On 2/28/2023 at 4:20 AM, Boltzmannbrain said:

The real numbers cannot have a next number, but I don't understand how that can be logical.

For example, consider the segment inclusively from 1 to 2, so there are the numbers 1 and 2 at each end of the segment.  We can take off a number like 1.3 or 2 from it.  If we take the number 2 away, we are left with something like the segment 1 to the limit 2 - 1/x as x goes to infinity (or whatever it is). 

So my ultimate question is, why can we take off the end of the segment if it is something that we call 2, but we can't take off another number?  The segment only has real numbers; what makes 2 so special that it can end a segment and be removable?  

 

Real numbers are the backbone of mathematics, and they always exist along the real number line. This line is made up of infinitely many segments that stretch from negative infinity to positive infinity.

Each segment contains an infinite amount of numbers, making it impossible for us to ever run out! So no matter how much we explore math and its applications, real numbers will always be there waiting for us on the real number line.

Taking the real numbers off the line doesn't mean they all disappear.  They're still right there on the line.  And the real numbers are, in actual fact, not the line. Yeah, they're all numbers, not geometric shape.   

You can imagine yourself deleting any real number on the line( or deleting anything in the universe ).

But...don't let imaginations play tricks on you!  No one in the universe can actually delete or remove any real number.  Real numbers always exist in the mathematical realm of nature!  

[studiot]

8 hours ago, Mc2509 said:

Real numbers are the backbone of mathematics, and they always exist along the real number line. This line is made up of infinitely many segments that stretch from negative infinity to positive infinity.

Each segment contains an infinite amount of numbers, making it impossible for us to ever run out! So no matter how much we explore math and its applications, real numbers will always be there waiting for us on the real number line.

Taking the real numbers off the line doesn't mean they all disappear.  They're still right there on the line.  And the real numbers are, in actual fact, not the line. Yeah, they're all numbers, not geometric shape.   

You can imagine yourself deleting any real number on the line( or deleting anything in the universe ).

But...don't let imaginations play tricks on you!  No one in the universe can actually delete or remove any real number.  Real numbers always exist in the mathematical realm of nature!  

One or two debatable points there but your heart is in the right place.

+1 for encouragement

@Boltzmannbrain

Whilst @Genady is doing such a good job offering straightforward textbook details I think it is worth looking deeper into the number line concept.

I will readily grant you that the number line is a very attractive way of presenting aspects of the properties of numbers.

But what we are really doing is as in fig1 putting numbers in one-to-one correspondence with a straight line and hoping that other properties are then transferable.

But also as in fig 1 it is easy to see that the actual length of the line is irrelevant - it works for all straight lines, regardless of theier length.

Radical conclusion : You can establish one-to-one correspondence for the whole set of numbers with the shortest possible straight line, as well as the longest possible.

Question : How does this affect you proposed distance function ?

Surely the distances apart of any two numbers will depend upon the length of you line.

 

But who says the line must be straight ?

Fig 2 shows how to use a spiral to place the numbers.

This has the interesting property of being able to cover the entire plane.

Question : How does this affect you proposed distance function ?

As can be seen in  fig 2   - the numbers 2,4 and 6 are all equally close to the number 1, whereas the numbers 3 and 5 are further away.

 

But does the line have to have a definite geoemtric shape ?  So far I have discussed straight and spiral.

Surely all that is required is that it must not cross itself.

Question : How does this affect you proposed distance function ?  Can you see why it must not cross ?

Fig3 shows one such semi random arrangement. There are an infinite count of such non crossing arangements.

 

What about the numbers themselves. There is nothing in basic set theory to say that they have to appear in the set in any particular configuration.

Fig 4 shows a random positioning of the numbers which is perfectly permissible.

Question : How does this affect you proposed distance function ?

Clearly 4 is the closest number to 1.

 

So that you can have them is value order you have to add further structure to your set of numbers so that you can introduce a 'well ordering' of the set.

 

Your thoughts on this ?

[Boltzmannbrain]

16 hours ago, Genady said:

To the contrary, there are different things that can be called this, and I need to know what you mean.

I want to take a line that is in a real space and divide it until there is nothing left, sort of like we do with Reiman integrals as n goes to infinity.  Is this possible? 

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I don't understand this statement without your definitions of segment (line segment) and its division. I don't know how they relate to real numbers

Wikipedia says, "In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints." from Line segment - Wikipedia

It says that line segments contain every point on the line.  Can those "points" be real numbers? 

 

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and can't figure what "the same sense" means.

It is becoming a common phrase that people in Canada and the U.S. use.  It generally means "the same way".  I found a page of synonyms Same Sense synonyms - 35 Words and Phrases for Same Sense (powerthesaurus.org)

 

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Also, I don't know what you try to accomplish. I guess, you try to get some contradiction. There are no contradictions in real numbers, it is a mathematical fact. If your definitions regarding segments establish correspondence with real numbers, then automatically, there will be no contradictions in the segments as well.

I am just really interested in this subject.  I don't actually think that I will find a contradiction, but I want to just go on the endeavour anyways.  If I don't find one, then I am just happy to learn more about infinity and its properties.

 

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For example, following the definitions in the video you've linked, we can define a segment as part of a parametrized line, which is covered by the parameter t being in an interval [a,b], IOW, a ≤ t ≤ b. Then, we define segment division. Etc. After everything is consistently defined, there will be no contradictions.

So the point of dividing a line in R was to give an example of something similar to running out of real numbers.  Of course infinity will be used.  But dividing is not how I want to look for the smallest real number.  

[Genady]

4 minutes ago, Boltzmannbrain said:

I want to take a line that is in a real space and divide it until there is nothing left, sort of like we do with Reiman integrals as n goes to infinity.  Is this possible? 

So, now you are talking about a different 'line', not the one we have defined earlier in real vector space as per the linked video? If so, we need a new definition.

Riemann integral does not deal with lines, it deals with numbers.

 

9 minutes ago, Boltzmannbrain said:

"In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints."

Again, a different 'line', this time, geometric one.

 

10 minutes ago, Boltzmannbrain said:

Can those "points" be real numbers? 

No, they cannot be real numbers. Points are not numbers and numbers are not points.

 

10 minutes ago, Boltzmannbrain said:

It is becoming a common phrase that people in Canada and the U.S. use.  It generally means "the same way".

My not understanding was not that of an English language. It was a way to say, that saying "the same sense", or "the same way" in this case is meaningless. You are comparing different animals, lines and numbers. There is no obvious "same" between them.

 

22 minutes ago, Boltzmannbrain said:

the point of dividing a line in R

Another 'line'.

R is a set of real numbers. There are no lines in this set, only numbers.

[joigus]

On 3/3/2023 at 8:18 PM, Lorentz Jr said:

because there is no "next" real number. The word "next" only has meaning in the context of countable sequences.

This is probably the best starting point for this question --no pun intended.

If x, y are in R, and x \( \neq\) y. Then,

Either x>y or x<y. Assume x<y.

A property of the real numbers is there always exists a z in R such that x<z<y if x and y are different. So there isn't such a thing as 'next number' in R.

x-posted with @Genady

[Genady]

35 minutes ago, Boltzmannbrain said:

But dividing is not how I want to look for the smallest real number.  

I thought that I have already shown to you earlier that there is no 'smallest real number', haven't I?

Edited March 5, 2023 by Genady