Non unicity of decimal expansion and extremes of intervals

[DaTario]

Hi All,

The famous proof of the theorem: ## 1 = 0.9999999...## seems to point to a statement more or less like this:
"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?

It seems that one of the extremes of an interval may be proved to be equal to anyone of its two immediate neighbors.

Best wishes,

DaTario

 

[Samy_A]

Hi All,

The famous proof of the theorem: ## 1 = 0.9999999...## seems to point to a statement more or less like this:
"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?

It seems that one of the extremes of an interval may be proved to be equal to anyone of its two immediate neighbors.

Best wishes,

DaTario

Can you give an example of "one of its two immediate neighbors" for some real number?
Take 1 as an example. What are the "immediate neighbors of 1?

 

[Mark44]

Hi All,

The famous proof of the theorem: ## 1 = 0.9999999...## seems to point to a statement more or less like this:
"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."

This isn't a theorem -- it's just a statement about two representations of a single number.

1 and 0.999... are not merely extremely close -- they are the same number

Is this is correct, shouldn't this imply that it is kind of useless to assign to intervals adjectives as closed or opened?

No. For example, the half-open, half-closed interval [0, 1) contains .9, .99, .999, and other numbers very close to 1, but does not contain 1.

It seems that one of the extremes of an interval may be proved to be equal to anyone of its two immediate neighbors.

No that's not true at all. The interval of my example has a least upper bound of 1, but 1 is not a member of this interval. Unlike integers, real numbers do not have immediate neighbors. Between any two real numbers, there are an infinite number of other real numbers.

 

[HallsofIvy]

Your basic mistake is in thinking that there is such a thing as an "immediate neighbor" for a number.

 

[jbriggs444]

Yet another take on the matter...

"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."

There is uniqueness in the decimal expansions of real numbers in the sense that almost all real numbers have unique decimal expansions. The ones that do not are those whose decimal expansions end in an infinite string of zeros or nines. For instance, 1/3 has a unique decimal expansion.

There are only countably many real numbers whose expansions are not unique. There are uncountably many real numbers with unique expansions.

 

[DaTario]

Thank you all,

I would like to express my gratitude for the answers.

Is there a real problem with calling the problem of proving the statement ## 1= 0.9999999...## a theorem?

I guess I understand the problems with the concept of immediate neighbors in real numbers. But this famous problem evokes an apparent identity between decimal expansions (or numbers !?) that are good candidates to be immediate neighbors. I (although smiling here now) would not expect any other number to live between 1 and 0.9999999...

There is uniqueness in the decimal expansions of real numbers in the sense that almost all real numbers have unique decimal expansions. The ones that do not are those whose decimal expansions end in an infinite string of zeros or nines. For instance, 1/3 has a unique decimal expansion.

Because it involves the "almost all" clause, do you think we should treat this uniqueness as partial?

There are only countably many real numbers whose expansions are not unique. There are uncountably many real numbers with unique expansions.

Even so, it seems to me that it may be a sufficient reason to treat the uniqueness as partial.

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 1.000000..., being each zero one domino fallen (and the 1 would be the unique domino lying on a perfect horizontal).

I am sorry, dear coleagues, if I am pushing too far in thinking in a non rigorous manner. This is a very interesting and misterious subject.

Best wishes,

DaTario

 

[Mark44]

Thank you all,

I would like to express my gratitude for the answers.

Is there a real problem with calling the problem of proving the statement ## 1= 0.9999999...## a theorem?

Why would you think it should be a theorem? If it should, then I guess we would need similar "theorems" to show that 2 = 1.999..., 3 = 2.999..., .5 = .4999..., and on and on.

I guess I understand the problems with the concept of immediate neighbors in real numbers. But this famous problem evokes an apparent identity between decimal expansions (or numbers !?) that are good candidates to be immediate neighbors. I (although smiling here now) would not expect any other number to live between 1 and 0.9999999...

Good, because there aren't any.

Because it involves the "almost all" clause, do you think we should treat this uniqueness as partial?

If something is unique, there is only one of it. An attribute either is unique or isn't -- it can't be partially unique, any more than someone can be "partially pregnant."

Even so, it seems to me that it may be a sufficient reason to treat the uniqueness as partial.

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 1.000000..., being each zero one domino fallen (and the 1 would be the unique domino lying on a perfect horizontal).

Your analogy is inapt. There is no difference between .999... and 1.000... None.
If you subtract .999... from 1.000... you are apparently thinking that there is a 1 digit out there somewhere.
Can you tell me its exact position?

I am sorry, dear coleagues, if I am pushing too far in thinking in a non rigorous manner. This is a very interesting and misterious subject.

 

[WWGD]

Thank you all,

I would like to express my gratitude for the answers.

Is there a real problem with calling the problem of proving the statement ## 1= 0.9999999...## a theorem?

I guess I understand the problems with the concept of immediate neighbors in real numbers. But this famous problem evokes an apparent identity between decimal expansions (or numbers !?) that are good candidates to be immediate neighbors. I (although smiling here now) would not expect any other number to live between 1 and 0.9999999...
Because it involves the "almost all" clause, do you think we should treat this uniqueness as partial?
Even so, it seems to me that it may be a sufficient reason to treat the uniqueness as partial.

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 1.000000..., being each zero one domino fallen (and the 1 would be the unique domino lying on a perfect horizontal).

I am sorry, dear coleagues, if I am pushing too far in thinking in a non rigorous manner. This is a very interesting and misterious subject.

Best wishes,

DaTario

But the rule whereby two Real numbers are considered equivalent to each other is not necessarily the same rule you would use to consider arrangements of dominos to be equivalent to each other.

 

[Mark44]

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.

The endless repetition of nines is related to our commonly used decimal system. The same situation exists in other number systems. For example, in binary (base-2), .111...₂ = 1₂. The number on the left is ##\frac 1 2 + \frac 1 4 + \frac 1 8 + \dots##, an infinite sum that can be shown to converge to 1.

Similarly, in base-3, 1.222...₃ = 2.000...₃. In any given number system, if the fractional part contains an endless string of whatever the highest digit is in that number system, that number has another representation that has an endless string of 0 digits.

 

[DaTario]

Why would you think it should be a theorem? If it should, then I guess we would need similar "theorems" to show that 2 = 1.999..., 3 = 2.999..., .5 = .4999..., and on and on.

Sorry, I was not aware that the number of theorems above the inverted pyramid of an axiomatic structure should be small.

Theorem may be defined as "a formula, proposition, or statement in mathematics or logic deduced or to be deduced from other formulas or propositions"
(http://www.merriam-webster.com/dictionary/theorem)
So, I will not take your saying, in this specific point, as a serious contribution, sorry.

If something is unique, there is only one of it. An attribute either is unique or isn't -- it can't be partially unique, any more than someone can be "partially pregnant."

I understand this professorial tone of yours, but from what jbriggs444 has said the question of this uniqueness was disputable in some sense.

Your analogy is inapt. There is no difference between .999... and 1.000... None.

I guess we both believe in that famous demonstration of equality.

 

[fresh_42]

Sorry, I was not aware that the number of theorems above the inverted pyramid of an axiomatic structure should be small.

Theorem may be defined as "a formula, proposition, or statement in mathematics or logic deduced or to be deduced from other formulas or propositions"
(http://www.merriam-webster.com/dictionary/theorem)
So, I will not take your saying, in this specific point, as a serious contribution, sorry.

You may take this serious. Your claim to call the equality that simply arises from the fact that our descriptions with decimals are not perfect a theorem is not serious.

I understand this professorial tone of yours, but from what jbriggs444 has said the question of this uniqueness was disputable in some sense.

No, it is not.

 

[jbriggs444]

I understand this professorial tone of yours, but from what jbriggs444 has said the question of this uniqueness was disputable in some sense.

Do all real numbers have unique decimal representations? Certainly not. Do some real numbers have unique decimal representations? Assuredly so.

The point that I think Mark44 is trying to make is that there is not a one to one correspondence between real numbers and decimal expansions. If you look at one decimal expansion and note that there is another decimal expansion that is the next one, that does not demonstrate that given one real number you can find a next one. Nor does it serve to somehow make the real numbers fuzzy so that open and closed intervals become meaningless.

 

[DaTario]

The endless repetition of nines is related to our commonly used decimal system. The same situation exists in other number systems. For example, in binary (base-2), .111...₂ = 1₂. The number on the left is ##\frac 1 2 + \frac 1 4 + \frac 1 8 + \dots##, an infinite sum that can be shown to converge to 1.

Similarly, in base-3, 1.222...₃ = 2.000...₃. In any given number system, if the fractional part contains an endless string of whatever the highest digit is in that number system, that number has another representation that has an endless string of 0 digits.

So, Mark44, if we are allowed to use different and "well chosen" basis for different sets of numbers in the real axis in such a way that, after several sessions of demonstrations, 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.

My point (which I am suffering here to defend ) is not only to put this notion in check, but to connect it with our language in mathematics that uses the terms open and closed to intervals.

Best wishes.

 

[DaTario]

You may take this serious. Your claim to call the equality that simply arises from the fact that our descriptions with decimals are not perfect a theorem is not serious.No, it is not.

If you demand students to prove that between 1 and 2 there is no integer then it is a theorem. Don´t you agree?

 

[jbriggs444]

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.

There only countably many possible bases and in each one, only countably many real numbers with dual expansions in that base. That is not Aleph one. That is Aleph null.

In any case, the number of representations that exist for a real number has nothing at all to do with whether or not that real number is distinct from every other real number , whether it has adjacent neighbors or whether it can be the endpoint of an open interval.

 

[fresh_42]

If you demand students to prove that between 1 and 2 there is no integer then it is a theorem. Don´t you agree?

No, it is by construction. You can write 1 in decimal digits in two ways. So what? It only means that decimal digits aren't a perfect way to write them. That's all. You cannot write π at all in decimal digits. Does it mean it doesn't exist?

 

[DaTario]

There only countably many possible bases and in each one, only countably many real numbers with dual expansions in that base. That is not Aleph one. That is Aleph null.

Sorry, I guess I haven´t made myself clear, I was also referring to choose ##\sqrt{2}## for example as unity.

 

[DaTario]

No, it is by construction. You can write 1 in decimal digits in two ways. So what? It only means that decimal digits aren't a perfect way to write them. That's all. You cannot write π at all in decimal digits. Does it mean it doesn't exist?

I am not addressing anything like this. I am not arguing that a number, expressed in some manner, doesn´t exist.
There are representations that, used with some numbers, provide an infinite string of digits or provide a much less compressible package of information in that basis.

 

[DaTario]

No, it is by construction. You can write 1 in decimal digits in two ways. So what? It only means that decimal digits aren't a perfect way to write them. That's all. You cannot write π at all in decimal digits. Does it mean it doesn't exist?

Where did you find that if some statement admit proof by construction it cannot be called a theorem?

 

[fresh_42]

Where did you find that if some statement admit proof by construction it cannot be called a theorem?

Ok, let's pretend, that "There is no integer between 1 and 2." is a theorem.
Please start and define the term "integer" as well as "1" and "2". That is needed before we can think of a proof.
Go ahead.

 

[DaTario]

Ok, let's pretend, that "There is no integer between 1 and 2." is a theorem.
Please start and define the term "integer" as well as "1" and "2". That is needed before we can think of a proof.
Go ahead.

I guess you could do the same joke with the theorem of Pythagoras.

But in the case in context, it seems that I would have to recall Peano´s axioms.

 

[DaTario]

This world of infinite numbers seems to have these somewhat strange properties. In dealing with probabilities in a real number context, that game in which one randomly chooses a real number and the other has to guess what number was that, the probability of matching the chosen number is zero.
This shows to me that the concept of infinitely unlikely is identical to the concept of impossible although the semantics seems to leave a tiny space between them.

I would like to appologize for any caused disturbance, and I would like say thank you all for the discussions.

Best Regards,

DaTario

 

[fresh_42]

The point is: real numbers have nothing to do with the way we write them. By the decision to write them in decimal digits we make a compromise. It is not a unique way. But that has nothing to do with the reals itself. Only with our chosen representation. So if at all you could have such a theorem in the theory of languages, not in the theory about reals.

 

[Mark44]

So, Mark44, if we are allowed to use different and "well chosen" basis for different sets of numbers in the real axis in such a way that, after several sessions of demonstrations, 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.

As already stated by another member, an arbitrarily chosen point on the number line does NOT have an infinite number of representations.

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.

This too, has been shown to be untrue in general, as there are only a countable number (i.e., of cardinality Aleph-null) of points with more than one representation.

My point (which I am suffering here to defend ) is not only to put this notion in check, but to connect it with our language in mathematics that uses the terms open and closed to intervals.

This is the point you started off attempting to make. The interval (0, 1) is an open interval, as it does not contain its two accumulation points (or endpoints, if you like). The right endpoint, whether called 1 of .999... is not included in the interval. Is that so difficult to understand?

If you demand students to prove that between 1 and 2 there is no integer then it is a theorem. Don´t you agree?

Seems too trivial to me to be classed as a "theorem." As is the statement that 1 and .999... are the same number. Possibly this is of great concern to philosophers, but mathematicians wouldn't dignify this by calling it a theorem.

 

[DaTario]

Thank you, Mark44 and all participants.

It is not the case of something being hard to understand. As a teacher I assume the positions you are denfending. It is more like trying to attack a notion to see if it is strong enough. It seemed to me that what ties the "validity" of a proven statement to the "truth" of this same statement, in this case, is a somewhat fragile argument, based on our decision to believe that if two things are different by an infinitely small amount, then they are equal.

But I agree with you in that this discussion tends to the field of philosophy.

I have some doubts if you all are in fact convinced of these statements, but this is none of my business, I admit.
Anyway, I think the "debate" was a good one and it seems you have won it , as I was not able to produced anything close to a solid argument.

Best Regards,

Dario

 

[Samy_A]

Thank you, Mark44 and all participants.

It is not the case of something being hard to understand. As a teacher I assume the positions you are denfending. It is more like trying to attack a notion to see if it is strong enough. It seemed to me that what ties the "validity" of a proven statement to the "truth" of this same statement, in this case, is a somewhat fragile argument, based on our decision to believe that if two things are different by an infinitely small amount, then they are equal.

A statement about Mathematics is sometimes difficult to translate into usual language.
Standard real analysis doesn't claim that "if two things are different by an infinitely small amount, then they are equal".
Take the 0.9999... =1 claim.
What we claim is the following:
Set ##a_n=0.9999 \dots 99##, where there are exactly ##n## nines after the decimal point.
With 0.9999... =1, we mean that ##\displaystyle \lim_{n\rightarrow +\infty} a_n=1##
And that limit statement is shorthand for:
##\forall \epsilon>0\ \ \exists N \in \mathbb N## such that ##\forall n>N: |a_n-1|<\epsilon##.

There are no "two things different by an infinitely small amount" in this case. There is a sequence whose elements are as close to 1 as we want them to be, provided we go "far enough" in the sequence.
But none of the ##a_n## is equal to 1, or "infinitesimally close" to 1, or different from 1 by "an infinitely small amount".

 

[DaTario]

A statement about Mathematics is sometimes difficult to translate into usual language.
Standard real analysis doesn't claim that "if two things are different by an infinitely small amount, then they are equal".
Take the 0.9999... =1 claim.
What we claim is the following:
Set ##a_n=0.9999 \dots 99##, where there are exactly ##n## nines after the decimal point.
With 0.9999... =1, we mean that ##\displaystyle \lim_{n\rightarrow +\infty} a_n=1##
And that limit statement is shorthand for:
##\forall \epsilon>0\ \ \exists N \in \mathbb N## such that ##\forall n>N: |a_n-1|<\epsilon##.

There are no "two things different by an infinitely small amount" in this case. There is a sequence whose elements are as close to 1 as we want them to be, provided we go "far enough" in the sequence.
But none of the ##a_n## is equal to 1, or "infinitesimally close" to 1.

Great! But every time we ask our class to take a limit ## x \to 4 ## we stress the fact that this is different from simply substituting ##x ## by ##4 ##. So a typical convergent sequence, in principle, do not " stand on " its limit (apart from the existing exceptions to this statement, which I denoted by non-typical)

OBS.: What I am referring to as "typical convergent sequence" may be reexpressed as strictly monotonic convergent sequence.

 

[Samy_A]

Great! But every time we ask our class to take a limit ## x \to 4 ## we stress the fact that this is different from simply substituting ##x ## by ##4 ##. So a typical convergent sequence, in principle, do not " stand on " its limit (apart from the existing exceptions to this statement, which I denoted by non-typical)

A convergent sequence is indeed not equal to its limit, it converges to it (equality would mean that sequence and limit are in the same set, which they quite obviously are not). I don't know what you mean with "do no stand on its limit".

 

[DaTario]

A convergent sequence is indeed not equal to its limit, it converges to it (equality would mean that sequence and limit are in the same set, which they quite obviously are not). I don't know what you mean with "do no stand on its limit".

Sorry for the english. I was trying to say that in a strictly monotonic convergent sequence none of this terms are equal to the limit.

 

[Samy_A]

Sorry for the english. I was trying to say that in a strictly monotonic convergent sequence none of this terms are equal to the limit.

Yes, that is correct.

 

[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).