Non unicity of decimal expansion and extremes of intervals

[DaTario]

Do you feel (yes, I will use feel) that it seems a little contradictory to agree with the following two statements:

1) in a strictly monotonic convergent sequence none of its terms are equal to the limit.

2) $1 = 0.9999999...$

?

 

[Samy_A]

Do you feel (yes, I will use feel) that it seems a little contradictory to agree with the following two statements:

1) in a strictly monotonic convergent sequence none of its terms are equal to the limit.

2) $1 = 0.9999999...$

?

No, I see no contradiction.
As I explained above, 0.999999... is shorthand for a sequence, and that sequence converges to 1.
The "..." is indicative of an infinite process, and mathematically that process is the limit of the sequence.

But, if you want to interpret 0.99999... as a number, you have to define exactly what number you mean. And then you will have no other (reasonable) choice but to define it as the number that is the limit of a sequence, as I did above.

 

[DaTario]

No, I see no contradiction.
As I explained above, 0.999999... is shorthand for a sequence, and that sequence converges to 1.
The "..." is indicative of an infinite process, and mathematically that process is the limit of the sequence.

But, if you want to interpret 0.99999... as a number, you have to define exactly what number you mean. And then you will have no other (reasonable) choice but to define it as the number that is the limit of a sequence, as I did above.

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.

 

[DaTario]

One question I use to make to my students was precisely "what is the smallest number which is greater than one?" or " In the real sequence, what number follows one?"

Of course there is disconfort. But addressing the question of the mathematical truth, it seems that the number which is the answer of those questions above mentioned are not equal to one, in principle.

 

[Samy_A]

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.

And we are back to where we started: no real number is equal to a close neighbor of it. I don't even know what a close neighbor is.

One question I use to make to my students was precisely "what is the smallest number which is greater than one?" or " In the real sequence, what number follows one?"

And the only correct answer to these questions is: there are no such real numbers (assuming you use the usual order on the real numbers).

 

[DaTario]

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.

 

[Samy_A]

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?

 

[Mark44]

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?"

 

[Mark44]

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?

I'm going to go get some popcorn...

 

[DaTario]

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.

 

[DaTario]

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.

 

[Mark44]

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.

 

[DaTario]

I'm going to go get some popcorn...

I am doing with fried chicken.

 

[DaTario]

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.

 

[jbriggs444]

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.

 

[Mark44]

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.

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.

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?

 

[Samy_A]

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.

 

[DaTario]

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.

 

[DaTario]

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.

 

[DaTario]

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.

 

[Samy_A]

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.

 

[DaTario]

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.

 

[Mark44]

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.

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.

 

[DaTario]

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.

 

[Mark44]

Thread closed.