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Paul
guest
Apr-07-04, 00:43 AM (EST)
 
"A question"
 
   Hello everyone.

I'm not a big student of math, but ever since I learn that the following number:

0.123123123...

could be represented this way:
` ___
0.123

I thought of a problem which no one I know could answer:


` 1 _
- 0.9
-------

could the answer be represented as
` _
0.01 ?

Thanks


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  Subject     Author     Message Date     ID  
  RE: A question JJ Apr-07-04 1
  RE: A question Graham C Apr-07-04 2
     RE: A question CliveT Mar-19-05 9
         RE: A question alexbadmin Mar-19-05 10
             RE: A question CliveT Mar-19-05 11
                 RE: A question alexbadmin Mar-19-05 12
  RE: A question alexbadmin Apr-07-04 3
     RE: A question Paul Apr-07-04 4
         RE: A question alexbadmin Apr-07-04 5
         RE: A question alexbadmin Apr-07-04 6
             RE: A question Paul Apr-07-04 7
                 RE: A question alexbadmin Apr-07-04 8
                     RE: A question tomgfp Mar-20-05 13
  RE: A question mr_homm Jun-10-05 14
     RE: A question alexbadmin Jun-10-05 15
         RE: A question mr_homm Jun-11-05 17
             RE: A question alexbadmin Jun-11-05 18
                 RE: A question sfwc Jun-12-05 24
                 RE: A question mr_homm Jun-12-05 25
                     RE: A question sfwc Jun-13-05 26
                         RE: A question alexbadmin Jun-13-05 27
                             RE: A question sfwc Jun-13-05 28
                                 RE: A question alexbadmin Jun-13-05 29
     RE: A question sfwc Jun-11-05 16
         RE: A question mr_homm Jun-11-05 19
             RE: A question alexbadmin Jun-11-05 20
                 RE: A question mr_homm Jun-12-05 22
         RE: A question alexbadmin Jun-11-05 21
             RE: A question sfwc Jun-12-05 23
  RE: A question shij Dec-03-06 30
     RE: A question alexbadmin Dec-04-06 31

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JJ
guest
Apr-07-04, 09:33 AM (EST)
 
1. "RE: A question"
In response to message #0
 
   In this kind of representation, so is the rule that it isn't possible to add extra digit on the right side, because the number of the repeated sequence is not defined (infinite, in fact).
Since there is an infinite series of 9, the result is an infinite series of 0 (no exta "1" on the right side)


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Graham C
Member since Feb-5-03
Apr-07-04, 09:33 AM (EST)
Click to EMail Graham%20C Click to send private message to Graham%20C Click to view user profileClick to add this user to your buddy list  
2. "RE: A question"
In response to message #0
 
   >Hello everyone.
>
>I'm not a big student of math, but ever since I learn that
>the following number:
>
>0.123123123...
>
>could be represented this way:
>` ___
>0.123
>
>I thought of a problem which no one I know could answer:
>
>
>` 1 _
>- 0.9
>-------
>
>could the answer be represented as
>` _
>0.01 ?
>
>Thanks

Try
''' _
0.0

After the sign for *infinite* repetition you can't have anything else.


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CliveT
Member since Mar-18-05
Mar-19-05, 01:17 AM (EST)
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9. "RE: A question"
In response to message #2
 
   >>
>>could the answer be represented as
>>` _
>>0.01 ?
>>
>>Thanks
>
>Try
>''' _
>0.0
>
>After the sign for *infinite* repetition you can't have
>anything else.

I've also wondered about this - it'seems like a legitimate question...
The best answer I can think of is

1/infinity

as presumably there are an infinite number of zeros before you can add a 1.
Not very neat (or useful) i suspect.

Is there a 'correct' way to write and/or use numbers like this?


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alexbadmin
Charter Member
1925 posts
Mar-19-05, 01:30 AM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
10. "RE: A question"
In response to message #9
 
   >I've also wondered about this - it'seems like a legitimate
>question...

The question is legitimate, but the answer is not unless explained. Notations like .(9)1, where (9) stands for "9 bar", do not make sense, because .(9) is just a shorthand for .999... In the latter, every 9 has a meaning: 9/10some power. But what an appended 1 could possibly mean?

The problem is that you not only give a name to something, your notation seems to attempt to imbue it with a meaning. In mathematics, it is different. Mathemmatician first find a meaningful something and then give it a name.

>The best answer I can think of is
>
>1/infinity

But no one knows what this means either.

>as presumably there are an infinite number of zeros before
>you can add a 1.

There is an infinite number of zeros, but there is no room to add 1.

>Not very neat (or useful) i suspect.

>Is there a 'correct' way to write and/or use numbers like
>this?

You see, this is a problem. You talk of "numbers like this", but there is none that satisfies your criteria.


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CliveT
Member since Mar-18-05
Mar-19-05, 07:34 AM (EST)
Click to EMail CliveT Click to send private message to CliveT Click to view user profileClick to add this user to your buddy list  
11. "RE: A question"
In response to message #10
 
   Appreciate your reply alexb, and I think I see your point...
I think I read numbers and don't fully understand what they mean in mathematical terms, so attribute them with properties they probably don't have.

I assume 1/3 = 0.(3)
so you can do this
0.(3) * 3 = 0.(9)
which means
0.(9) = 1

I relate this to using computers for calculations, where precision is lost due to not having enough bits to store some numbers.
And that's probably where I go wrong - is it possible to multiply 0.(3) by 3? Or are they different 'types' of numbers?

If it IS allowed, doesn't this cause problems where mathematical formulae lose accuracy?
Or do you have a way of using fractions as if you were working on paper (where 1/3 is always a third, as opposed to 0.333 approximately)?

Hope that made sense!


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alexbadmin
Charter Member
1925 posts
Mar-19-05, 07:51 AM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
12. "RE: A question"
In response to message #11
 
   >I assume 1/3 = 0.(3)

"1/3" and "0.(3)" are notations that stand for the same number. There are two of them (and in fact more) because the number they stand for can be arrived at in different ways. The first notaion "1/3" is suggestive of the fact that this number taken three times gives 1, which is why it is called "one third."

The second notation tells us that the number can be obtained by an infinite addition of terms like 10-(some power).

The fact you assumed is a provable result: the two notations stand for the same number.

>so you can do this
>0.(3) * 3 = 0.(9)
>which means
>0.(9) = 1

Yes.

>I relate this to using computers for calculations, where
>precision is lost due to not having enough bits to store
>some numbers.

I do not exactly understand the problem with calculators. I can imagine that when a calculator says

1/3 = 0.33333333333 or 1/3 = 0.33333333334

that this may be confusing. But do they nowadays work with .(3)?

>And that's probably where I go wrong - is it possible to
>multiply 0.(3) by 3?

Yes, of course.

>Or are they different 'types' of numbers?

They are just notations, not a different or, for that matter, the same kind of numbers. Just notations that stand for the same kind of numbers, viz., rational numbers.

>If it IS allowed, doesn't this cause problems where
>mathematical formulae lose accuracy?

I fail to see how possibly this might be.

>Or do you have a way of using fractions as if you were
>working on paper (where 1/3 is always a third, as opposed to
>0.333 approximately)?

You can use .333 on paper. Just do not forget that this is an approximation (in case you mean it as such.)

>Hope that made sense!


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alexbadmin
Charter Member
1925 posts
Apr-07-04, 09:45 AM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
3. "RE: A question"
In response to message #0
 
   >Hello everyone.
>
>I'm not a big student of math, but ever since I learn that
>the following number:
>
>0.123123123...
>
>could be represented this way:
>` ___
>0.123
>
>I thought of a problem which no one I know could answer:

I should give a thought to the company you keep.

>` 1 _
>- 0.9
>-------
>
>could the answer be represented as
>` _
>0.01 ?

If I just replied "No", you would probably have an impulse to ask "Why?", right? So, too, when I saw your question, my first impulse was to ask "Why?" and add after an interval of meaningful silence "... for god's sake?"

Can you explain how you arrived from one thing to the other and why you thought that the latter might make sense at all.


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Paul
guest
Apr-07-04, 12:53 PM (EST)
 
4. "RE: A question"
In response to message #3
 
   Does that mean the answer can't be represented in math?

Since 1 > 0.999... shouldn't the answer be greater than 0.000...?


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alexbadmin
Charter Member
1925 posts
Apr-07-04, 12:56 PM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
5. "RE: A question"
In response to message #4
 
   >Since 1 > 0.999...

But 1 = 0.999...

https://www.cut-the-knot.org/arithmetic/999999.shtml

>shouldn't the answer be greater than
>0.000...?

It's not a question of whether it'should or should not be something, but rather of whether it is or is not. The answer is "is not".


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alexbadmin
Charter Member
1925 posts
Apr-07-04, 12:57 PM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
6. "RE: A question"
In response to message #4
 
   There is a problem with your response. You do not even attempt to answer the question "Why ...?"


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Paul
guest
Apr-07-04, 09:01 PM (EST)
 
7. "RE: A question"
In response to message #6
 
   I understand the logic in saying 1 = 0.99999...

Then how about 1 - .88888...?


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alexbadmin
Charter Member
1925 posts
Apr-07-04, 09:03 PM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
8. "RE: A question"
In response to message #7
 
   >I understand the logic in saying 1 = 0.99999...

Logic? What is it?

>Then how about 1 - .88888...?

1 - .88888... = .1111...


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tomgfp
guest
Mar-20-05, 08:34 AM (EST)
 
13. "RE: A question"
In response to message #8
 
   I think, perhaps, that 0.(0)1 means a number that approaches zero.


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mr_homm
Member since May-22-05
Jun-10-05, 09:03 PM (EST)
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14. "RE: A question"
In response to message #0
 
   >` 1 _
>- 0.9
>-------
>
>could the answer be represented as
>` _
>0.01 ?
>
Strangely enough, I don't think that this question is as nonsensical as it first appears. There is a real possibility of making this notation meaningful, although it goes far beyond what I think the original poster had in mind:

When we construct the real numbers, we can start from the integers, then construct the rationals, then the reals, in each case using sets of the previous class of numbers to define the next type. In so doing, we have first whole numbers, then finite (or repeating) decimals, and then non-repeating decimals. Normally the process stops here, but there have been efforts to go further. For example, there is the set of hyperreal numbers associated with the so-called Dean plane, and the set of surreal numbers of J. H. Conway. Both of these can be considered as further refinements of the standard real numbers.

To my knowledge, there has been no investigation as to whether these classes of numbers can be represented in something similar to the decimal format we use for real numbers. Furthermore, the question arises, since we are already using all possible infinitely long sequences of digits, where is there to go from here? What else is there that we could use to extend the representation?

There is Cantor's theory of transfinite ordinals, that's what. In this theory, the ordinary sequence of whole numbers (which can be used to number the digits of an infinite decimal) is merely the smallest type of infinity. There are transfinite ordinals that have the property that the sequence can trail off to infinity and then start over, only to trail off again. For example, the sequence

1,2,3,...;1,2,3,... sets all the integers in order, and then defines another copy of this order, and states that every element of the second copy comes after every element of the first copy. Surprisingly, you do not get into trouble doing this, and this whole theory is logically consistent mathematics. See, for example, "Contrbutions to the Foundation of the Theory of Transfinite Numbers" by Georg Cantor.

Therefore, it is at least possible to construct transfinite ordinal decimals. However, once you do that, you have extended the number system. You are no longer dealing with the real numbers, but with something larger, and therefore, you must rethink all the rules. Does "plus" still mean what it used to mean? How does the extended meaning of "equals" relate to the meaning it has for real numbers. And so on. This would be a lot of work, though interesting work.

Cantor's theory is rather advanced stuff compared to the level at which the question was asked, but it does provide a possible interpretation of the symbol 0.(0)1. However, (and this is just a hunch, since I haven't gone to the trouble of constructing the theory) the answer to the original poster's query is probably "no" because the proposed number 0.(0)1 is of a different ordinal type from the numbers 1 and .(9).

By the way, I have actually used this idea of transfinite ordinals in a computer program. My task was to fit an n-dimensional ellipsoid inside a convex figure bounded by planes, and to do this, I started with degenerate ellipses (essentially line segments) and constructed a nondegenerate ellipse from them. In the process, it was necessary to invert some matrices that had determinant zero, but I could show on geometric grounds that the final answer must exist. So I used a pair of floating point numbers to represent 0.(0);0.a where 0.a was treated as an infinit'simal error term. To multiply such numbers, I used the rule (a;b)(c;d) = ac;ad plus bc, based on the idea that b and d were acting like differentials. The program worked, the matrices inverted as expected, and all the calculation artifacts produced by the infinitesimals cancelled at the end of the calculation, leaving a nondegenerate ellipsoid.

So, in short, you CAN work with stuff like 0.(0)1, even in something as practical as a computer program.

--Stuart Anderson


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alexbadmin
Charter Member
1925 posts
Jun-10-05, 11:14 PM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
15. "RE: A question"
In response to message #14
 
   >Strangely enough, I don't think that this question is as
>nonsensical as it first appears. There is a real
>possibility of making this notation meaningful, although it
>goes far beyond what I think the original poster had in
>mind:

Yesterday I had a conversation with ny soon to be 6 son. He was primed at the kindergarten to think of the coming father's day. He showed a face he just drew and said: This is the face I'd like to kiss. (Below the picture he wrote "My dad.")

It was nice. To surprise him, I remarked, "No, this is a drawing of the face you would like to kiss." To make a stronger case, I framed the face on his drawing into a rectangle and said, "This is a picture of the picture of the face you'd like to kiss."

You had to see him wonder. For him, the drawing was the face. He eventually got the point and visibly enjoyed it, but it was a complete news for him.

Why I am telling the story? Symbols in mathematics have an endowed meaning. Much confusion may and does arise if the symbols are used outside their intended meaning. For many, the symbol for, say, integral, is the integral, and the symbol for the decimal fraction is a decimal fraction.

.(0)1 has no defined meaning. Period.

The meaning sought by the original poster would have created confusion.

>To my knowledge, there has been no investigation as to
>whether these classes of numbers can be represented in
>something similar to the decimal format we use for real
>numbers.

Never saw it. As you correctly note later, were it possible, the rules would change. A notation alone, without sensible usage rules makes no sense. Thus you can't just start writing .(0)1 without basing it on some kind of a theory.

>Furthermore, the question arises, since we are
>already using all possible infinitely long sequences of
>digits, where is there to go from here? What else is there
>that we could use to extend the representation?

>There is Cantor's theory of transfinite ordinals, that's
>what.

I absolutely disagree with this and what followed. One has nothing to do with the other, i.e., as long as you did not formalize the link. Neither Cantor in the theory of the transfinite ordinals, not Robinson in the theory of the hyper-reals, nor Conway in his numbers and games made any attempt to extend a decimal representation. In fact, Conway does consider infinite sums of the transfinte powers of ordinals with transfinite coefficients, and I believe Polish mathematicians (Kuratowski, Mostowski) did something similar with the ordinals, but none of them, i.e., as far as I know, ever tried to incorporate those sums in the decimal framework.

>Therefore, it is at least possible to construct transfinite
>ordinal decimals.

This does not follow from the argument you put out. Nope, is does not.

>However, once you do that, you have
>extended the number system. You are no longer dealing with
>the real numbers, but with something larger, and therefore,
>you must rethink all the rules.

With this, as I mentioned before, I do agree, with the caveat that to go any further with that agreement, there need to be something more specific to agree on.

>Does "plus" still mean what
>it used to mean? How does the extended meaning of "equals"
>relate to the meaning it has for real numbers. And so on.
>This would be a lot of work, though interesting work.

It may also be fruitless or useless.

>Cantor's theory is rather advanced stuff compared to the
>level at which the question was asked, but it does provide a
>possible interpretation of the symbol 0.(0)1.

I disagree. Moreover, I disagree with the need to seek an interpretation to an undefined symbol.

>However, (and
>this is just a hunch, since I haven't gone to the trouble of
>constructing the theory) the answer to the original poster's
>query is probably "no" because the proposed number 0.(0)1 is
>of a different ordinal type from the numbers 1 and .(9).

You see how easy it is to start treating an a priori nonsense as something very real. Without even defining it, you already claim that 0.(0)1 has an ordinal type, which has a property of being different from that of 1 or .(9). Also, I am not sure that .(9) has an ordinal type. The ordinal type of 1 is 1. What is the ordinal type of .(9)? Do you mean w? I do not accept that.

>
>By the way, I have actually used this idea of transfinite
>ordinals in a computer program.

What you describe looks more like hyper-real numbers than transfinite. They are essentially different.

>So, in short, you CAN work with stuff like 0.(0)1, even in
>something as practical as a computer program.

I do not see how this conclusion follows from what you wrote.

Alex


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mr_homm
Member since May-22-05
Jun-11-05, 08:26 PM (EST)
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17. "RE: A question"
In response to message #15
 
   >Why I am telling the story? Symbols in mathematics have an
>endowed meaning. Much confusion may and does arise if the
>symbols are used outside their intended meaning. For many,
>the symbol for, say, integral, is the integral, and the
>symbol for the decimal fraction is a decimal fraction.

I agree. I have seen this far too often in students who try
to study only the "surface" of mathematics and have no grasp
of any real meaning behind the symbols.

>
>.(0)1 has no defined meaning. Period.

I disagree. .(0)1 has no defined meaning. Currently.

What the original poster wanted was to know if the "true meaning"
of the expression was what he thought. This of course is utter nonsense. There is no "true meaning," ever; there are only defined meanings, within the context of a theoretical structure.

What I was more interested in, was the question as to whether one could set up a theoretical structure, into which the real numbers could be embedded, and within which one could define notation in such a way that

1: symbols such as .(0)1 had a meaning;
2: symbols such as .(9) have their usual meaning as real numbers;
3: the meaning assigned to .(0)1 was intuitively compatible with the meanings assigned to symbols such as .(9), in the sense that one could interpret the new symbols as denoting members of an extension of the real number system.

>
>The meaning sought by the original poster would have created
>confusion.

I completely agree here. The original question was very naive; the poster did not appreciate that .(0)1 had no meaning within the real number system notational conventions, nor that meaning was conventional, nor that the number system would need to be expanded if such a notation were to be included.


>
>>To my knowledge, there has been no investigation as to
>>whether these classes of numbers can be represented in
>>something similar to the decimal format we use for real
>>numbers.
>
>Never saw it. As you correctly note later, were it possible,
>the rules would change. A notation alone, without sensible
>usage rules makes no sense. Thus you can't just start
>writing .(0)1 without basing it on some kind of a theory.

While I was writing this, post number #16 from swfc came in.
I see that Conway has done something along these lines.
Interesting reading ahead for me.

>
>>Furthermore, the question arises, since we are
>>already using all possible infinitely long sequences of
>>digits, where is there to go from here? What else is there
>>that we could use to extend the representation?
>
>>There is Cantor's theory of transfinite ordinals, that's
>>what.
>
>I absolutely disagree with this and what followed. One has
>nothing to do with the other, i.e., as long as you did not
>formalize the link. Neither Cantor in the theory of the
>transfinite ordinals, not Robinson in the theory of the
>hyper-reals, nor Conway in his numbers and games made any
>attempt to extend a decimal representation. In fact, Conway
>does consider infinite sums of the transfinte powers of
>ordinals with transfinite coefficients, and I believe Polish
> mathematicians (Kuratowski, Mostowski) did something
>similar with the ordinals, but none of them, i.e., as far as
>I know, ever tried to incorporate those sums in the decimal
>framework.

Here I think we have a misunderstanding, and I apologize for not making myself clearer. The hyperreals, and surreals were mentioned as examples of number systems that subsume the reals and therefore, at least potentially, might call for notations beyond the standard decimal notation for real numbers. In other words, since decimal notation is adequate for the reals, there would be no point in extending it unless there was a more extensive system of numbers for it to denote. Therefore I mentioned some extensions of the reals merely to keep open the possibility that there might be a motivation for extending the standard decimal notation.

In no way did I intend to make a direct connection between the transfinite ordinals and the surreals or hyperreals. I was using the transfinite ordinals to discuss the notation, not the mathematical entities it denoted. There are actually two levels of denotation here. The symbol .(9) denotes a mapping M from the natural numbers to the set of digits. In this particular case, it maps every natural number to the digit 9. This is a purely typographical structure, the sort of object that is studied in the context-free grammar theory of computer science. Therefore, .(9) denotes an infinite digit'string, which in turn denotes a real number.

You can therefore talk about the ordinality of the domain of the mapping M. This is where I was applying the transfinite ordinal idea, and it should be perfectly clear that from this standpoint the ordinal type of .(9) is in fact w (Cantor's omega). I should perhaps have said ".(9)" rather than .(9), to indicate that I was referring to the typographic symbol denoted by .(9), rather than the number denoted by that symbol.

>
>>Therefore, it is at least possible to construct transfinite
>>ordinal decimals.
>
>This does not follow from the argument you put out. Nope, is
>does not.
>

I did not mean to say that you could construct transfinite ordinal decimal numbers, so I fully agree that no such thing follows from my argument. However, I was talking about notation here, and it certainly is possible to extend the notation in this way. Whether it denotes anything is a separate question. This discussion of notation is mostly in response to posts #1, 2, and 3, who were concerned that there was somehow "no room" to put a final 1 after an infinite string of zeroes. I am sorry now that I did not tag this comment as responding to those concerns, since in the context of the rest of my post, it was certainly easy to interpret it as saying that you could construct the numbers, not the notation.

Anyway, my main point in this paragraph is that infinite decimals are mappings into the digits, whose domain has ordinality omega, while the original poster was proposing a typographic structure that was a mapping into the digits, whose domain has ordinality omega plus 1. This addresses the (I think purely typographic) concerns of responses #1, 2, and 3.

>>However, once you do that, you have
>>extended the number system. You are no longer dealing with
>>the real numbers, but with something larger, and therefore,
>>you must rethink all the rules.
>
>With this, as I mentioned before, I do agree, with the
>caveat that to go any further with that agreement, there
>need to be something more specific to agree on.
>
>>Does "plus" still mean what
>>it used to mean? How does the extended meaning of "equals"
>>relate to the meaning it has for real numbers. And so on.
>>This would be a lot of work, though interesting work.
>
>It may also be fruitless or useless.
>
Of course. We are discussing the possibility of a mathematical construction. The best we can do, prior to attempting the actual creation of a new structure, is to ask whether any obvious logical impediments present themselves. If not, the next step is to attempt to construct our new structure, and at any time during the process, an insurmountable impediment may reveal itself, or the construction may be possible but not very interesting or useful.

>>Cantor's theory is rather advanced stuff compared to the
>>level at which the question was asked, but it does provide a
>>possible interpretation of the symbol 0.(0)1.
>
>I disagree.

When I said "a possible interpretation" perhaps I should have said instead "the possibility of constructing an interpretation," as I did not mean that an interpretation had been found, only that the way appears open to constructing one.

Moreover, I disagree with the need to seek an
>interpretation to an undefined symbol.
>
Here I must flatly disagree with you. It is occasionally true that the symbolism devised to fit a particular theory is suggestive of extensions, which can drive the search for an extended set of mathematical objects to which the extended symbols may be attached. Consider splitting fields in general, or the complex numbers in particular. At a certain stage the desire to factor something (a polynomial, or -1) results in a purely formal solution. There was no meaning attached to the symbol sqrt(-1) within the then-existing structure, which simply had a rule saying that this symbol was not well formed. The construction of the complex number field can be viewed as an attempt to make sense of a disallowed operation and its corresponding undefined symbol, an attempt which you will agree was both fruitful and useful.

I do not assert that this present case is necessarily of the same type or will have such a successful outcome, but I do assert that in general an argument can be made for allowing notation to suggest extensions of the objects denoted. On a purely personal level, I make a hobby of seeing if I can rehabilitate crank ideas by placing them in a context that the originator did not know about. This does not mean that I think the idea is necessarily a productive one, or that the originator was insightful. It is merely a game I enjoy.

>>However, (and
>>this is just a hunch, since I haven't gone to the trouble of
>>constructing the theory) the answer to the original poster's
>>query is probably "no" because the proposed number 0.(0)1 is
>>of a different ordinal type from the numbers 1 and .(9).
>
>You see how easy it is to start treating an a priori
>nonsense as something very real. Without even defining it,
>you already claim that 0.(0)1 has an ordinal type, which has
>a property of being different from that of 1 or .(9). Also,
>I am not sure that .(9) has an ordinal type. The ordinal
>type of 1 is 1. What is the ordinal type of .(9)? Do you
>mean w? I do not accept that.
>
The ordinal type of the decimal notation string .(9) and 1.(0) is
w. The ordinal type of the string .(0)1 is w plus 1. See above.

>>
>>By the way, I have actually used this idea of transfinite
>>ordinals in a computer program.
>
>What you describe looks more like hyper-real numbers than
>transfinite. They are essentially different.

The numerical values are hyperreal; the positional notation expressions have ordinal type w plus w.
>
>>So, in short, you CAN work with stuff like 0.(0)1, even in
>>something as practical as a computer program.
>
>I do not see how this conclusion follows from what you
>wrote.
>
>Alex

I hope this clarifies my position. This is a productive discussion for me, because you have forced me to clarify my presentation and therefore my thinking. If at this point you still have objections, they are likely to be substantive instead of arising from my somewhat ambiguous presentation, and I can benefit from them. I would like to hear your further comments, as they are interesting to me.

--Stuart


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alexbadmin
Charter Member
1925 posts
Jun-11-05, 09:32 PM (EST)
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18. "RE: A question"
In response to message #17
 
   >>.(0)1 has no defined meaning. Period.
>
>I disagree. .(0)1 has no defined meaning. Currently.

Of course you agree. I have not said "may not have a defined meaning." Currently, if you wish, it has no defined meaning.

>What I was more interested in, was the question as to
>whether one could set up a theoretical structure, into which
>the real numbers could be embedded, and within which one
>could define notation in such a way that
>
>1: symbols such as .(0)1 had a meaning;
>2: symbols such as .(9) have their usual meaning as real
>numbers;
>3: the meaning assigned to .(0)1 was intuitively compatible
>with the meanings assigned to symbols such as .(9), in the
>sense that one could interpret the new symbols as denoting
>members of an extension of the real number system.

I absolutely understand your intentions.

>While I was writing this, post number #16 from swfc came in.
>I see that Conway has done something along these lines.
>Interesting reading ahead for me.

No, I do not believe Conway ever did anything of the sort. swfc nearly built a theory on Conway's idea of numbers having birthdays. The problem with his construction is manifold.

First of all, I do not see how his binary or decimal representations are extensions of the regular positional systems. Second, there are plenty of numbers with the same birthday. Thus I do not see how his representation may be unique to a number. Third, there is no hint that those representations fit in any way into any algebraic structure.

>Here I think we have a misunderstanding, and I apologize for
>not making myself clearer.

Believe me, I absolutely understood what it was you were after.

>>>Therefore, it is at least possible to construct transfinite
>>>ordinal decimals.
>>
>>This does not follow from the argument you put out. Nope, is
>>does not.
>>
>
>I did not mean to say that you could construct transfinite
>ordinal decimal numbers, so I fully agree that no such thing
>follows from my argument. However, I was talking about
>notation here, and it certainly is possible to extend the
>notation in this way.

This is too strong an assertion. When you say "notations" you not only mean the mapping M but also the potential association of its image set with the set of real numbers. So I think that the assertion that "it certainly is possible to extend the notation in this way" should have covered somehow this association with, perhaps, an extended number system.

>Whether it denotes anything is a
>separate question.

Not at all. There is no question that you can use strings of symbols (integers or anything else) whose ordinality goes beyond w's. Of course you can. But whether it may denote anything is a real question.

>This discussion of notation is mostly in
>response to posts #1, 2, and 3, who were concerned that
>there was somehow "no room" to put a final 1 after an
>infinite string of zeroes.

... and get a number. Unless this is what you eventually get, the discussion will have diverged radically from the posts ##1-3. Do you realize that the basis for the original question was the doubts that .(9) = 1?

>I am sorry now that I did not
>tag this comment as responding to those concerns, since in
>the context of the rest of my post, it was certainly easy to
>interpret it as saying that you could construct the numbers,
>not the notation.

This was indeed the impression.

>Anyway, my main point in this paragraph is that infinite
>decimals are mappings into the digits, whose domain has
>ordinality omega, while the original poster was proposing a
>typographic structure that was a mapping into the digits,
>whose domain has ordinality omega plus 1.

I know what you mean. I doubt however that this was the proposal of the original poster. It might have been on an intuitive level at best.

>>It may also be fruitless or useless.
>>
>Of course.

Sure.

>>>Cantor's theory is rather advanced stuff compared to the
>>>level at which the question was asked, but it does provide a
>>>possible interpretation of the symbol 0.(0)1.
>>
>>I disagree.
>
>When I said "a possible interpretation" perhaps I should
>have said instead "the possibility of constructing an
>interpretation,"

I just had to react to that to clear things up. If anybody ever reads this discussion, there should not be misunderstanding of what has been claimed, discussed, or agreed upon.

>Moreover, I disagree with the need to seek an
>>interpretation to an undefined symbol.
>>
>Here I must flatly disagree with you.

I do not think you disagree so categorically. sqrt(-1) was not just a meaningless symbol, even from the very beginning. It always had the meaning in conjunction with

sqrt(-1)·sqrt(-1) = -1.

However, for the generation that had not yet accepted negative numbers, sqrt(-1) was a monstrosity, however meaningful.


>The construction of the
>complex number field can be viewed as an attempt to make
>sense of a disallowed operation and its corresponding
>undefined symbol, an attempt which you will agree was both
>fruitful and useful.

Very fruitful and extermely useful. I agree. (With the above reservation.)

>I do not assert that this present case is necessarily of the
>same type or will have such a successful outcome, but I do
>assert that in general an argument can be made for allowing
>notation to suggest extensions of the objects denoted.

Usually symbols pop up along the way, not from the very beginning.

>On a
>purely personal level, I make a hobby of seeing if I can
>rehabilitate crank ideas by placing them in a context that
>the originator did not know about.

Ah, now you said that. But I abhor even a possibility of getting involved in making sense into some absurd ideas which did not have any sense to start with.

My father was a Fermatist: a human calculator who got swayed by the simplicity of the FLT formulation and Fermat's claim to a solution. He was convinced he had a proof, albeit with a few nonsignificant flaws, and, for this reason, was perpetually sore with me because of my refusal to patch the latter.

>>What you describe looks more like hyper-real numbers than
>>transfinite. They are essentially different.
>
>The numerical values are hyperreal; the positional notation
>expressions have ordinal type w plus w.

I am not sure of that. (a;b) is not exactly of type w plus w. But this is not important.

>>>So, in short, you CAN work with stuff like 0.(0)1, even in
>>>something as practical as a computer program.

Here you go again. By saying that you endow the original 0.(0)1 with validity it does not desrve. To boot, the stuff you used in your program had an a priori meaning. Granted, it was something one may not expect to find in a computer program, but a meaning nonetheless. 0.(0)1 is still waiting to be made a sense of.

>I hope this clarifies my position. This is a productive
>discussion for me, because you have forced me to clarify my
>presentation and therefore my thinking. If at this point
>you still have objections, they are likely to be substantive
>instead of arising from my somewhat ambiguous presentation,
>and I can benefit from them. I would like to hear your
>further comments, as they are interesting to me.

Whatever I had to say is up there. To sum up, the apparent disagreement was not solely about notations. I shall be pleasantly surprised to learn of any extension of the decimal system that would meaningfully included .(9)1.


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sfwc
Member since Jun-19-03
Jun-12-05, 10:59 AM (EST)
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24. "RE: A question"
In response to message #18
 
   >No, I do not believe Conway ever did anything of the sort.
>swfc nearly built a theory on Conway's idea of numbers
>having birthdays. The problem with his construction is
>manifold.
>
>First of all, I do not see how his binary or decimal
>representations are extensions of the regular positional
>systems.
In the following way: Take any normal positional binary expansion which does not terminate with infinitely many 1s. Then use the rules I specified to make it into a sign-expansion. Then the surreal number with that sign-expansion is precisely that real number.

>Second, there are plenty of numbers with the same
>birthday. Thus I do not see how his representation may be
>unique to a number.
I suspect this is due to my unclear explanation of the construction. My apologies. I assure you that, just as the sign-expansion is unique, so is the construction I gave.

>Third, there is no hint that those
>representations fit in any way into any algebraic structure.
I believe the Surreal numbers are a suitable algebraic structure.

I would like to take this opportunity to point out a flaw in my original post. This is in the final decimal construction. Due to the 'binary' nature of simplicity, this does not work for finite decimal expansions; even a number like 0.1 has an infinite birthday. However, it does work if we first specify that any expansion must have length some limit ordinal, rather than just any old ordinal, where the padding is provided by 0s. So 0.1 would be considered as 0.1(0)

Thankyou

sfwc
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mr_homm
Member since May-22-05
Jun-12-05, 10:14 PM (EST)
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25. "RE: A question"
In response to message #18
 
   (snip stuff we agree about now)

>No, I do not believe Conway ever did anything of the sort.
>swfc nearly built a theory on Conway's idea of numbers
>having birthdays. The problem with his construction is
>manifold.
>
It'seemed to me that swfc was saying that Conway had done this. Since I have not read Conway's book yet (only Knuth's) I assumed that what (I thought) he said was true.

>First of all, I do not see how his binary or decimal
>representations are extensions of the regular positional
>systems. Second, there are plenty of numbers with the same
>birthday. Thus I do not see how his representation may be
>unique to a number. Third, there is no hint that those
>representations fit in any way into any algebraic structure.
>
As to the first comment, I had assumed from swfc's remarks that Conway had done this and details had been omitted in the post for brevity. It seems to me that if your second and third concerns are satisfied, the first would cease to have any force, because then we would have a positional notation for surreal numbers that embedded the positional notation for real numbers within it, and which had a proper algebraic structure. (More on this under my response to your third comment.)

As to the second, this is unclear to me also.

Pure speculation: Presumably either there is a canonical choice of which number to use for each birthday k (this seems less likely to me) or there is a theorem to the effect that the bit pattern produced by this method is independent of choices of numbers for each birthday k less than n (this seems more likely). I do not presently know whether either of these assertions is true or if so whether it is proved by Conway. I will look at what Conway has to say and see for myself.

As to the third, this is the most critical problem. As far as I am aware, the surreals are a large, ordered set of objects, with an additional partial ordering by birthdays, but to call them numbers requires much more than this. They should have some kind of algebraic structure where we can at least add, subtract, and check equality. Furthermore, this structure should be compatible with that of the reals, so that when restricted to them it agrees with the usual algebraic operations there. It's pretty obvious that we could arbitrarily embed the SET of real numbers in some larger set, but that would be pointless unless the algebraic rules somehow are extended to the larger set. As far as I know, there is no such algebraic structure in place for the surreal numbers.

>>I did not mean to say that you could construct transfinite
>>ordinal decimal numbers, so I fully agree that no such thing
>>follows from my argument. However, I was talking about
>>notation here, and it certainly is possible to extend the
>>notation in this way.
>
>This is too strong an assertion. When you say "notations"
>you not only mean the mapping M but also the potential
>association of its image set with the set of real numbers.
>So I think that the assertion that "it certainly is possible
>to extend the notation in this way" should have covered
>somehow this association with, perhaps, an extended number
>system.
>

OK I agree with this. Without anything to denote, it is a misnomer to call a string of digits a "notation." Furthermore, there is no motivation for considering extended strings of digits unless one is looking forward to eventually using them to denote something, such as members of some (as yet unspecified, or merely hoped for) number system that embeds the reals. However...(see next paragraph)

>>Whether it denotes anything is a
>>separate question.
>
>Not at all. There is no question that you can use strings of
>symbols (integers or anything else) whose ordinality goes
>beyond w's. Of course you can.

... I do not think that posters #1,2,and 3 knew this. I am new to these discussions, and I have never encountered these people before. Perhaps you have, and know their levels of understanding better than I do. It appears to me that, while the original poster's idea was absurd as it'stood, their objections to it were also absurd, inasmuch as they seemed to be objecting to the existence of a string with ordinality exceeding w. I felt that this clouded the whole issue, and it was necessary to remove the absurd objection first, so as to reveal the real difficulty.

>But whether it may denote anything is a real question.
>
Yes, this is the important question here.

>>This discussion of notation is mostly in
>>response to posts #1, 2, and 3, who were concerned that
>>there was somehow "no room" to put a final 1 after an
>>infinite string of zeroes.
>
>... and get a number. Unless this is what you eventually
>get, the discussion will have diverged radically from the
>posts ##1-3.

I do not think these posters had got this far. The objections seemed to be to the typographic string itself, rather than to what it denotes.

(I just realized, with much embarrassment, that when I have been referring to posters #1 2 and 3, I was going by the order they appeared in the thread, not the topic number. I mean of course TOP ID #1,2, and 9, the posters in the top sub-thread. Sorry.)

Do you realize that the basis for the original
>question was the doubts that .(9) = 1?

Yes, that was abundantly clear from the start. This is such a common idea among people who don't understand about limits or about the Archimedean property of the reals that I've seen lots of people who think this. What surprised me about the original poster was that he tried to think of a way out of his difficulty, instead of just flatly denying that .(9) = 1?. Not that it was a good way out.

>>Anyway, my main point in this paragraph is that infinite
>>decimals are mappings into the digits, whose domain has
>>ordinality omega, while the original poster was proposing a
>>typographic structure that was a mapping into the digits,
>>whose domain has ordinality omega plus 1.
>
>I know what you mean. I doubt however that this was the
>proposal of the original poster. It might have been on an
>intuitive level at best.
>

I doubt that he had any clear idea that he was out of his depth.
>
>>Moreover, I disagree with the need to seek an
>>>interpretation to an undefined symbol.
>>>
>>Here I must flatly disagree with you.
>
>I do not think you disagree so categorically. sqrt(-1) was
>not just a meaningless symbol, even from the very beginning.
>It always had the meaning in conjunction with
>
>sqrt(-1)·sqrt(-1) = -1.
>
>However, for the generation that had not yet accepted
>negative numbers, sqrt(-1) was a monstrosity, however
>meaningful.
>

I think it is impossible to consider a symbol meaningful when one denies that its referent exists. There is certainly a meaning to the sqrt operation, but if one disbelieves in the complex numbers (referring to the generation who had not yet accepted them), one would be forced to regard sqrt(-1) as an abuse of notation, and the formula sqrt(-1)·sqrt(-1) = -1 as an absurdity. Only after the complex numbers had been constructed, proven consistent, shown to be useful, and shown to subsume the reals, would the symbol sqrt(-1) and its associated equation acquire a meaning.

In the present case, however, we all agree that the computation 1-.(9) already has a solution within the real numbers, namely 0, and so there is not an operation crying out for the creation of a solution. This is an important dissimilarity between the cases, and I admit that in the present case I have no particular reason to try to extend the number system, other than just "let's see what happens."
>
>>On a
>>purely personal level, I make a hobby of seeing if I can
>>rehabilitate crank ideas by placing them in a context that
>>the originator did not know about.
>
>Ah, now you said that. But I abhor even a possibility of
>getting involved in making sense into some absurd ideas
>which did not have any sense to start with.
>
>My father was a Fermatist: a human calculator who got swayed
>by the simplicity of the FLT formulation and Fermat's claim
>to a solution. He was convinced he had a proof, albeit with
>a few nonsignificant flaws, and, for this reason, was
>perpetually sore with me because of my refusal to patch the
>latter.
>

This explains much about the tenor of the discussion. We are approaching this question from diametrically opposite viewpoints, because we have opposite backgrounds. I was a child mathematical prodigy in a rural community where no one knew any significant mathematics. When I was seven, I discovered the successive difference method for constructing patterns in finite sequences of integers, realized that any finite pattern could be extended in many ways, invented linear recursive sequences and classified them by degree of recursion. When I tried to explain this to my second grade teacher, she didn't understand a word I said, nor did any other adults I tried to talk to about it. It was not until I went away to college that I met anyone who even know what a sequence was. My town did not have a library.

Now I am not saying this to try to win your sympathy, but just to explain my perspective. I was happy working on my own, not bitter. However, I still have a feeling of being an outsider looking in on the ongoing conversation that is mathematical research, and this necessarily colors my attitude towards crank ideas. While you see your father, I see myself at seven.

That's enough biography, I think, as it is wandering far from the topic. Given the differing perspectives we have on topics such as this, I think we have had all the productive discussion possible for now. I will continue to post comments on other threads, of course, but as for this thread, there is not much further to say. I will take one of the following courses of action:

(1) Go refresh my reading on hyperreals, surreals, and ordinals, and try to construct in detail a number field-and-notation combination that renders .(9)1 meaningful;

(2) Lose interest and move on.

>Whatever I had to say is up there. To sum up, the apparent
>disagreement was not solely about notations. I shall be
>pleasantly surprised to learn of any extension of the
>decimal system that would meaningfully included .(9)1.

If I elect course (1) I will post my results here.

--Stuart


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sfwc
Member since Jun-19-03
Jun-13-05, 06:58 AM (EST)
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26. "RE: A question"
In response to message #25
 
   >As to the first comment, I had assumed from swfc's remarks
>that Conway had done this and details had been omitted in
>the post for brevity.
I apologise for the fact that this was unclear. As I say, Conway had built up a way to represent finite binary expansions which has a natural generalisation.

>Pure speculation: Presumably either there is a canonical
>choice of which number to use for each birthday k (this
>seems less likely to me)
This is in fact the case.

>I will look at what Conway has to say and see for
>myself.
An excellent idea; I intend to refresh my memory by doing the same.

>As to the third, this is the most critical problem. As far
>as I am aware, the surreals are a large, ordered set of
>objects, with an additional partial ordering by birthdays,
>but to call them numbers requires much more than this. They
>should have some kind of algebraic structure where we can at
>least add, subtract, and check equality. Furthermore, this
>structure should be compatible with that of the reals, so
>that when restricted to them it agrees with the usual
>algebraic operations there.
In fact, the following are all well defined notions for the surreals:

1. equality
2. ordering
3. addition (which is commutative and associative)
4. multiplication (which is also commutative and associative, and which is distributive over the addition, as you would expect)
5. reals
Since you may not believe me on this last one, I shall give more details. We define a surreal r to be real iff:
a. There exists a natural number n (eg. one of the form 1+1+1+1+...+1) with -n < r < n
b. r = {{r-1/n: n natural}|{r+1/n: n natural}}

The set of reals so defined behaves exactly as the conventional reals.

>It's pretty obvious that we
>could arbitrarily embed the SET of real numbers in some
>larger set, but that would be pointless unless the algebraic
>rules somehow are extended to the larger set. As far as I
>know, there is no such algebraic structure in place for the
>surreal numbers.
There is; this is the beauty of Conway's construction.

Thankyou (and my apologies once more for the lack of clarity in my original post)

sfwc
<><


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alexbadmin
Charter Member
1925 posts
Jun-13-05, 07:40 AM (EST)
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27. "RE: A question"
In response to message #26
 
   >>As to the first comment, I had assumed from swfc's remarks
>>that Conway had done this and details had been omitted in
>>the post for brevity.
>I apologise for the fact that this was unclear. As I say,
>Conway had built up a way to represent finite binary
>expansions which has a natural generalisation.

With this epidemic of apologies, I should apologize for making my remark in reposnse to mr_homm's post and not yours. I realized the mistake after the fact.

Non-uniqueness is also a mistake - sorry.

As I said, I am aware of a very close construction by Conway: sign-expansions.

As to my questioning of the algebraic properties, it was related to the expansion itself, not the insuing embedding. Must think of that.


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sfwc
Member since Jun-19-03
Jun-13-05, 10:30 AM (EST)
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28. "RE: A question"
In response to message #27
 
   >As I said, I am aware of a very close construction by
>Conway: sign-expansions.
I have now checked how far Conway goes in the book 'On Numbers and Games.' The relevant material is in the first 4 chapters of the book. Chapter 0 (sic) gives the details of the construction of the surreals. Chapter 1 demonstrates that they form an ordered field. The beginning of chapter 2 includes a demonstration that the usual reals are isomorphic to a subfield of this field. The first section of chapter 3 consists of the definition and demonstration of uniqueness of sign expansions and the rules for converting the binary expansion of any real number into a sign expansion.

Since this rule may be easily extended to convert any binary expansion to a unique sign expansion, and so to a unique surreal number, in such a way that every bounded surreal has such a representation, I think this is a fairly natural interpretation of the notation.

>As to my questioning of the algebraic properties, it was
>related to the expansion itself, not the insuing embedding.
>Must think of that.
Please explain more clearly the distinction you are making here. What sort of algebraic properties does a notation have? Do you, for example, mean 'Is there a natural way to add or multiply two numbers represented in this way, which doesn't involve converting explicitly back and forth to the surreals?'; this is an interesting question which I will consider when I next find some free time.

Thankyou

sfwc
<><


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alexbadmin
Charter Member
1925 posts
Jun-13-05, 06:18 PM (EST)
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29. "RE: A question"
In response to message #28
 
   >The first section of chapter 3
>consists of the definition and demonstration of uniqueness
>of sign expansions and the rules for converting the binary
>expansion of any real number into a sign expansion.

Found it, thank you.
>
>>As to my questioning of the algebraic properties, it was
>>related to the expansion itself, not the insuing embedding.
>>Must think of that.
>Please explain more clearly the distinction you are making
>here. What sort of algebraic properties does a notation
>have? Do you, for example, mean 'Is there a natural way to
>add or multiply two numbers represented in this way, which
>doesn't involve converting explicitly back and forth to the
>surreals?'; this is an interesting question which I will
>consider when I next find some free time.

Right. You can put it this way. I've been thinking in terms of having positional notations. Since we are talking of expanding the decimal system, this would be natural to expect.


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sfwc
Member since Jun-19-03
Jun-11-05, 06:55 AM (EST)
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16. "RE: A question"
In response to message #14
 
   >For example,
>there is the set of hyperreal numbers associated with the
>so-called Dean plane, and the set of surreal numbers of J.
>H. Conway. Both of these can be considered as further
>refinements of the standard real numbers.
It is on the second of these that I wish to focus: the first I am unfamiliar with. I would be interested to hear the details of the construction.

>To my knowledge, there has been no investigation as to
>whether these classes of numbers can be represented in
>something similar to the decimal format we use for real
>numbers.
Decimal is not a very natural system, being based, supposedly, on something as arbitrary as the number of fingers on each hand. However, the binary system is sufficiently similar and sufficiently nice that a binary format seems like something it would be reasonable to explore. Surprisingly enough, there is a pleasant way to describe the surreal numbers in a very similar manner to that which you describe, using binary expansions. Those already familiar with the construction of the surreal numbers may wish to skip the next two paragraphs.

A surreal number is an ordered pair of sets of surreal numbers such that all the numbers in the first set are strictly less than all the numbers in the second set. Initially this definition seems absurdly self referential, but it can be made to work. Firstly, there is at least one surreal number, obtained by taking both sets to be empty. Using that we can build more and more numbers from the numbers we have at any time. We have to define the notion of order, and the notion of equality, as we go along (the details of how this is done are not important).

One way to make this more rigorous is to divide the surreal numbers into generations, with one generation corresponding to each ordinal number. A generation k surreal number is an ordered pair of sets of surreal numbers of generation less than k, where all the numbers in the first set are less than all those in the second set and we define order and equality as we go along. So for example the only generation 0 number would be the one I mentioned before, in which both sets are empty. (It turns out that by the time we reach generation omega we already have all the real numbers). The birthday of a number is then defined to be the least k such that the number is a generation k number.

Now, it is necessary to define the equality as we go along; that is, not all the numbers we construct are unequal and there is a simple rule for deciding which pairs are equal. However, it turns out that each number also has a unique simplest representation. The system Conway proposed assigns to each number n a sequence of numbers, one of birthday k for each k less than the birthday of n. These are divided into two sets, those less than n and those greater than n and the canonical representation of n is then the ordered pair consisting of those two sets.

A key point to note here is that the actual sets in the canonical representation of n need not be listed. It is sufficient to specify, for each k less than the birthday of n, which of the two sets the generation k number in the representation is put into. So each surreal number corresponds to a sequence of bits of length some ordinal. This is hopeful. Because it works out more nicely, I shall use the slightly odd convention of having the kth digit equal to 0 if the generation k number is greater than n, and 1 otherwise.

However, this most natural sequence of bits does not, for example, include a binary point or a specification of sign, so it is clearly different to our standard method of binary expansion. All hope is not lost, however. Things work out nicely inside the interval (0, 1). Any dyadic rational in this interval is usually represented as a finite string of 0s and 1s. For uniqueness of representation, we have the convention that the final digit is always 1. Now for the remarkable fact: If we take such a sequence, add 10 to the start (to specify that we wish to take a number in (0, 1)) and remove the redundant final 1 then we get exactly the canonical representation of that number as a surreal number.

Now it is possible (if a little boring) to specify conventions to extend this notation and so produce rules for representing any dyadic rational as a finite sequence of 0s and 1s corresponding to it's canonical representation. By reverse-engineering these rules it is then possible to represent many surreal numbers as binary expansions. For example, 0.1111... corresponds to 1011111... Working within this system we have the following facts, which somehow seem to represent people's intuition about how things ought to be:
_
0.1 < 1
_ _
1 - 0.1 = 0.01

I apologise for skipping over so much detail, but there is a lot of theory behind this. A very clear exposition can be found in Conway's book 'On Numbers and Games' (I can't specify exactly where in the book since I do not have it to hand). I am sure these ideas could be extended to decimals too, but it would be uglier and would take a little more work. In essence, it would work like this:

Let S be the set of extended decimal expansions. That is, any s in S is a function with domain some ordinal and codomain the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Then define a function f: S -> S which takes any s with last digit n < 9 to the same s with last digit n + 1, and such that for any s with last digit 9, where k is the least ordinal with all digits of s from k onwards equal to 9, we have f(s) = f(s | k). Define d(s) from S to the set of ordinals to be the number of digits of s. Then define g from S to the surreal numbers by g(s) = {{s|k: k<d(s)} U {0} | {f(s|k): k<d(s) U {1}}. g(s) is the surreal number with that decimal expansion.

I believe it could then be proved that every surreal number in (0, 1) has a unique decimal expansion.

Thankyou

sfwc
<><


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mr_homm
Member since May-22-05
Jun-11-05, 09:36 PM (EST)
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19. "RE: A question"
In response to message #16
 
   >>For example,
>>there is the set of hyperreal numbers associated with the
>>so-called Dean plane, and the set of surreal numbers of J.
>>H. Conway. Both of these can be considered as further
>>refinements of the standard real numbers.
>It is on the second of these that I wish to focus: the first
>I am unfamiliar with. I would be interested to hear the
>details of the construction.

There is more than one construction of the hyperreals. The more usual one is from Abraham Robinson; these numbers are a freature of his Nonstandard Analyss. A later constuction is due largely to Edward Nelson, who, having constructed his system, went on to use it as a revised foundation for probability theory, essentially replacing Kolmogorov's measure theory based foundation with a nonstandard equivalent. See his book "Radically Elementary Probability Theory" for both a good exposition of the hyperreals and a lot of illuminating applications of these numbers to known results in probability theory.

While Conway provides a beautifully simple recursion for developing the surreals, the hyperreals are based on some rather subtle reasoning about the foundations of arithmetic. Essentially, as a corollary to Goedel incompleteness, the set of integers is not fully determined by the axioms of arithmetic. Therefore, it is possible to introduce an extension of the integers to include the non-standard integers, which are regarded intuitively as fulfilling the role of "infinite numbers," and their reciprocals, which behave as you would expect "infinit'simals" to behave.

Nelson differs from Robinson in that he chooses to separate the integers into two classes (note! NOT sets. classes), standard and nonstandard. This allows him to regard every integer as finite, including the nonstandard integers. Although Robinson appends the nonstandards to the integers and Nelson separates them by a division within the integers, both approaches result in a two-fold structure of standard and nonstandard elements. Formally, they appear different, but logically they are equivalent (proved by Nelson in his foundational paper on this topic -- title escapes me at the moment, but he references it from within his book cited above.)

Personally, I prefer Nelson's formulation, but this is merely a matter of taste.

>
>>To my knowledge, there has been no investigation as to
>>whether these classes of numbers can be represented in
>>something similar to the decimal format we use for real
>>numbers.
>Decimal is not a very natural system, being based,
>supposedly, on something as arbitrary as the number of
>fingers on each hand. However, the binary system is
>sufficiently similar and sufficiently nice that a binary
>format seems like something it would be reasonable to
>explore. Surprisingly enough, there is a pleasant way to
>describe the surreal numbers in a very similar manner to
>that which you describe, using binary expansions. Those
>already familiar with the construction of the surreal
>numbers may wish to skip the next two paragraphs.

Although I used the term "decimal notation" I should really have said "positional notation," because of course the fact that we use base 10 is not really relevant. The central issue is the positional structure of this string of digits and how it differs from the standard positional notation. I stayed with the term "decimal expansion" largely because the original poster and the first few responders did not seem to me to have the background to enter the more general discussion, and I wanted to stay within their "comfort zone" as much as possible. Of course, these posts are very old, and it is unlikely that these people will ever return to see how this thread has grown, so perhaps that was a wasted consideration.

I am more than happy to look at binary rather than decimal digit strings. They are much cleaner and easier to work with, as well as having a close connection with the dyadic rationals, which as you point out below, are easy to construct within the surreal framework.

(Large snip of very interesting information)

>Working within
>this system we have the following facts, which somehow seem
>to represent people's intuition about how things ought to
>be:
> _
>0.1 < 1
> _ _
>1 - 0.1 = 0.01
>

I did not know when this discussion started that this would actually work for the surreal numbers, I just kind of had a hunch that it would. I will now go study the surreals in more depth. Long ago, I had read Donald Knuth's book on the construction of the surreals, a very silly book indeed, with hip mathematical teenagers in the '60's deriving the surreal numbers based on clues that are miraculously carved on rocks on the island where they have gone to "drop out". You can almost hear the "love beads" rattle and smell the patchouli -- it is most embarassingly dated sounding, but it does give the actual construction of the surreals. Since Knuth also wrote TeX, which I cannot live without, I will forgive him for this book. I am sure that Conway himself gives a cleaner exposition and more details.

>I apologise for skipping over so much detail, but there is a
>lot of theory behind this. A very clear exposition can be
>found in Conway's book 'On Numbers and Games' (I can't
>specify exactly where in the book since I do not have it to
>hand).

No need to apologize. I look forward to reading the details for myself.

>
>Thankyou
>
>sfwc

No, I should thank you for this interesting information.

--Stuart Anderson


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alexbadmin
Charter Member
1925 posts
Jun-11-05, 09:42 PM (EST)
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20. "RE: A question"
In response to message #19
 
   >the actual construction of the surreals. Since Knuth also
>wrote TeX, which I cannot live without, I will forgive him
>for this book.

Well, Knuth deserves a credit for the nomenclature. It's due to this book that the numbers got their "surreal" adjective.


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mr_homm
Member since May-22-05
Jun-12-05, 10:22 AM (EST)
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22. "RE: A question"
In response to message #20
 
   >
>Well, Knuth deserves a credit for the nomenclature. It's due
>to this book that the numbers got their "surreal" adjective.

I did not know this interesting tidbit. Thank you.

(Also, of course, I was semi-joking about the book. I actually did enjoy it, although the hippies made me wince.)

--Stuart


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alexbadmin
Charter Member
1925 posts
Jun-11-05, 10:07 PM (EST)
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21. "RE: A question"
In response to message #16
 
   > _
>0.1 < 1
> _ _
>1 - 0.1 = 0.01
>
>I apologise for skipping over so much detail, but there is a
>lot of theory behind this. A very clear exposition can be
>found in Conway's book 'On Numbers and Games' (I can't
>specify exactly where in the book since I do not have it to
>hand).

I believe I saw "sign-expansions" in the book, but not the explicit binary form. Are you sure it is in there?

Alex


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sfwc
Member since Jun-19-03
Jun-12-05, 10:39 AM (EST)
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23. "RE: A question"
In response to message #21
 
   It may be my memory failing, but I believe Conway gives rules in His book for converting back and forth between the digits of dyadic rationals and finite sign-expansions. As far as I know, he does not go on to extend this to infinite sign-expansions. I shall find a copy and check sometime this week.

Thankyou

sfwc
<><


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shij
guest
Dec-03-06, 11:56 PM (EST)
 
30. "RE: A question"
In response to message #0
 
   what is the difference between a polynomial and an algebraic expression


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alexbadmin
Charter Member
1925 posts
Dec-04-06, 00:24 AM (EST)
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31. "RE: A question"
In response to message #30
 
   Algebraic expression is any valid combination of constants, variables and algebraic operations. E.g., these are algebraic expressions:


  1. a + b,
  2. a/b,
  3. ab
  4. many other.

A polynomial is an algebraic expression of a special form. If you ask about polynomials in one variable then the expression in question must be of the form

a + bx + cx2 + ... + exn,

where a, b, c, ..., e are real constants; n is an integer; x a variable.

However, note that, a + b may be looked at as a polynomial in two variables (a and b) with coefficients 1 by each. (a + b) may be also looked at as the value P(1, 1) of the polynomial ax + by in two variables x and y with coefficients a and b.


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