Non unicity of decimal expansion and extremes of intervals

In summary: It really doesn't make sense to talk about uniqueness being partial. A number either has a unique decimal expansion, or it doesn't. There is no in-between.As a final remark in this post, I would say that if we are adopting an identification between two near but different points on the real axis, the fact that whether these numbers are expanded with nines or not seems to be of less importance. If I imagine an infinite line of standing dominoes and call this line 0.99999..., being each nine one domino standing, I would not accept an identification of this system with an infinite line of fallen dominoes, which I would call
  • #36
Samy_A said:
And the only correct answer to these questions is: there are no such real numbers.

So I must conclude that you disaprove the bijection between the points in a straight line and the real set of numbers.
 
Mathematics news on Phys.org
  • #37
DaTario said:
So I must conclude that you disaprove the bijection between the points in a straight line and the real set of numbers.
How do you arrive at that conclusion?
Of course, it depends on how do you define these concepts. What exactly do you mean by straight line?
 
  • #38
DaTario said:
But real numbers offer several examples showing us that there are situations concerning representation which are hard to deal with (for example, where is x = e?). I see no problem in having on the number line, a number whose representation provokes disconfort. And I would not use this disconfort to propose that this number must be equal to an eventually confortable close neighbor of it.
No.
As for your question, "where is e?" It's somewhere between 2.7 and 2.8. If that's not close enough for you, I'll say it's between 2.71 and 2.72, and so on. I can make the interval that contains e as small as you like, but the interval will always have some finite, positive length. I can say that e is approximately equal to 2.71828, but it would be erroneous to say that e is equal to that number. The vast majority of the numbers on the real number line are irrational, so their decimal representations go on endlessly, with no repeating patterns. But so what?

I hope that you aren't still asking your students, "what is the smallest number that is greater than 1" or "what number follows 1?"
 
  • #39
DaTario said:
So I must conclude that you disaprove the bijection between the points in a straight line and the real set of numbers.

Samy_A said:
How do you arrive at that conclusion?
Of course, it depends on how do you define these concepts. What exactly do you mean by straight line?

I'm going to go get some popcorn... :oldbiggrin:
 
  • Like
Likes fresh_42
  • #40
Mark44 said:
I hope that you aren't still asking your students, "what is the smallest number that is greater than 1" or "what number follows 1?"

What are your arguments against such provocative questions? I am not saying that I give grades to the students for the presented answers.
 
  • #41
Samy_A said:
How do you arrive at that conclusion?

Let me answer your question with another question. The continuous structure of points in a straight line (which I define as being the shortest path between two point infinitely apart) can still be thought of as having order, in such a way that at the right of a small number must be a number which is greater than the first?
Observe that I am on purpose mixing " at the right" = geometry with " greater than " = real analysis.
 
  • #42
DaTario said:
What are your arguments against such provocative questions? I am not saying that I give grades to the students for the presented answers.
The argument is based on the fact that the real numbers are dense; that is, between any two real numbers, there are an infinite number of other real numbers.

If the student asserts that 1.1 is the smallest number that is greater than 1, I will point out that 1.01 is greater than 1 and smaller than 1.1.
If the student thinks awhile, and comes back with 1.01 as the smallest number larger than 1, I will point out that 1.001 is greater than 1 and smaller than 1.01.

And so on...

A little more mathematically, the sequence ##a_n = \{1 + \frac 1 {10^n}\}## is monotonically decreasing and bounded below, so it converges (to 1). Yet each member of the sequence is strictly larger than 1.
 
  • #43
Mark44 said:
I'm going to go get some popcorn... :oldbiggrin:

I am doing with fried chicken.
 
  • #44
Mark44 said:
The argument is based on the fact that the real numbers are dense; that is, between any two real numbers, there are an infinite number of other real numbers.

If the student asserts that 1.1 is the smallest number that is greater than 1, I will point out that 1.01 is greater than 1 and smaller than 1.1.
If the student thinks awhile, and comes back with 1.01 as the smallest number larger than 1, I will point out that 1.001 is greater than 1 and smaller than 1.01.

And so on...

A little more mathematically, the sequence ##a_n = \{1 + \frac 1 {10^n}\}## is monotonically decreasing and bounded below, so it converges (to 1). Yet each member of the sequence is strictly larger than 1.

But this is exactly the conclusion I would like my students to arrive at. I see no reason to stop asking this. As they have familiarity with integers, this question poses a lot of new stuff for them to think about.
 
  • #45
DaTario said:
But this is exactly the conclusion I would like my students to arrive at. I see no reason to stop asking this. As they have familiarity with integers, this question poses a lot of new stuff for them to think about.
And yet you seem to want to point to 0.999... as a real number which you believe is strictly less than one.
 
  • #46
DaTario said:
But this is exactly the conclusion I would like my students to arrive at. I see no reason to stop asking this. As they have familiarity with integers, this question poses a lot of new stuff for them to think about.

jbriggs444 said:
And yet you seem to want to point to 0.999... as a real number which you believe is strictly less than one.

The point that jbriggs444 brings up makes it difficult for us to tell whether you (@DaTario) are asking the questions in a spirit of Platonic dialogue or are just confused about the properties of the real numbers.

In addition, you still haven't answered Samy's question asking how you reached the following conclusion.
DaTario said:
So I must conclude that you disaprove the bijection between the points in a straight line and the real set of numbers.

Samy_A said:
How do you arrive at that conclusion?
Of course, it depends on how do you define these concepts. What exactly do you mean by straight line?
 
  • #47
DaTario said:
Let me answer your question with another question. The continuous structure of points in a straight line (which I define as being the shortest path between two point infinitely apart) can still be thought of as having order, in such a way that at the right of a small number must be a number which is greater than the first?
Observe that I am on purpose mixing " at the right" = geometry with " greater than " = real analysis.
I have no idea what "the shortest path between two points infinitely apart" is supposed to mean.
"At the right" of a number there will be a number greater than the first number. Everyone will agree with that. But so what? As Mark44 wrote above, between these two numbers there will be infinitely many other real numbers: "to the right" of the first one and "to the left" of the second one.

Frankly, I don't understand what point exactly you are trying to make.
 
  • #48
jbriggs444 said:
And yet you seem to want to point to 0.999... as a real number which you believe is strictly less than one.

No, I am not that courageous.
 
  • #49
Samy_A said:
I have no idea what "the shortest path between two points infinitely apart" is supposed to mean.
"At the right" of a number there will be a number greater than the first number. Everyone will agree with that. But so what? As Mark44 wrote above, between these two numbers there will be infinitely many other real numbers: "to the right" of the first one and "to the left" of the second one.

Frankly, I don't understand what point exactly you are trying to make.

I understand your difficulty. Frankly I don´t expect us to solve this in a different manner that the traditional books do.
 
  • #50
Mark44 said:
The point that jbriggs444 brings up makes it difficult for us to tell whether you (@DaTario) are asking the questions in a spirit of Platonic dialogue or are just confused about the properties of the real numbers.

In addition, you still haven't answered Samy's question asking how you reached the following conclusion.

More like Platonic.

I would refrain from sustaining my conclusion. He believes in the bijection "geometry of straight line" --- "real numbers".

I wolud like to thank you all for the contributions, and I hope we keep on learning about such complexities.
 
Last edited:
  • #51
DaTario said:
I understand your difficulty. Frankly I don´t expect us to solve this in a different manner that the traditional books do.
My only difficulty here is that I have no idea of what we are supposed to solve in the first place.
 
  • #52
Samy_A said:
My only difficulty here is that I have no idea of what we are supposed to solve in the first place.

You are lucky, I have more difficulties.
 
  • #53
Samy_A said:
Frankly, I don't understand what point exactly you are trying to make.

DaTario said:
I understand your difficulty. Frankly I don´t expect us to solve this in a different manner that the traditional books do.
The difficulty is entirely on your part. This is not an insurmountable problem that needs to be "solved." Throughout this thread you have made a number of statements that are just flat wrong.

"There is no uniqueness in decimal expansions of real numbers, specially if one wishes to compare numbers (and their decimal expansions) extremely close of one another."
Is this is correct, shouldn't this imply that it is kind of useless to assign to intervals adjectives as closed or opened?
perhaps we could demonstrate that each point in the real axis may be seen as a number having in its decimal expansion an infinite number of whatever the highest digit of the system used in that turn. This creates a system of infinite number (Aleph one) of demonstrations for each point in the real axis. In these demonstrations, each point would accept two decimal expansions.
If your intent was to start a Platonic dialogue, that was not something that you made clear. Without such a disclaimer, your statements come across either as someone who is confused or someone who is trolling.
 
  • #54
Mark44 said:
The difficulty is entirely on your part. This is not an insurmountable problem that needs to be "solved." Throughout this thread you have made a number of statements that are just flat wrong.

If your intent was to start a Platonic dialogue, that was not something that you made clear. Without such a disclaimer, your statements come across either as someone who is confused or someone who is trolling.
Ok, time to stop.
 
  • #55
Thread closed.
 
Back
Top