Re: Corrective interpretation of real numbers
From: Eckard Blumschein (blumschein_at_et.uni-magdeburg.de)
Date: 02/11/05
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Date: Fri, 11 Feb 2005 15:07:14 +0100
Thank you, Bernd, for guiding me.
On 2/9/2005 9:35 PM, Bernd Funke provided links to three replies outside
what I immediately found in succession to the following posting of mine:
http://www.google.de/groups?selm=4203A4CF.2070405%40et.uni-magdeburg.de
_______________________________________
> http://www.google.de/groups?selm=1107535303.254665.110260@g14g2000cwa.googlegroups.com
> (Randy Poe, 4 Feb 2005 17:41:43)
Randy Poe wrote:
> Eckard Blumschein wrote:
>> I would not like to quarrel about that. I meant pi is exactly the
>> circumference belonging to the mathematical notion of a circle with a
>> diameter of one. So it is an exact number without exact representation
>> in terms of numerals.
>
> It has an infinite non-repeating decimal expansion.
Yes, and therefore it cannot be exactly represented.
> If you think that the numerical expansion we associate
> with pi is not exactly equal to pi, please tell me the
> magnitude of the difference.
It does not matter what you abstractly associate. A representation is
something tangible, and it is definitely incomplete.
After all, if x != y, then
> |x-y| is a real number of some finite magnitude.
>> >> What made me curious was Cantor's second diagonal proof. It was
>> >> performed with "reals" in decimal representation of numbers between zero
>> >> and one.
>> >
>> > That's the standard setup, yes.
>> >
>> >
>> >> Naively, I imagine any positive decimal representation a ratio between
>> >> two natural numbers, e.g. 0.2345 = 2345 divided by 10000
>
> That's only true when the the decimal representation
> terminates with a period sequence of digits (in your
> case 000...).
No. In this case both numerator and denominator are natural numbers. So
they are not subject to any limitation. We merely have to exclude the
case oo / oo.
>
>> > That's *too* naive. How do you represent 0.10110011100011110000... (the
>> > meaning of the "..." being unambiguous, I hope) as a ratio of two natural
>> > numbers?
>>
>> Simply as follows:
>> 10110011100011110000... diveded by 10000000000000000000...
>
> Neither of those is a natural number.
Why not. With ... I did not denote infinite periodic repetition but I
simply indicated that natural numbers do not have a limitation.
>
>> >> Where is the error in my reasoning?
>> >
>> > Irrational numbers can *not* be represented as a ratio of two natural
>> > numbers.
>>
>> Yes, however it was not the idea of mine but Cantor's idea to take the
>> ratio of two natural numbers for reals.
>
> Where did he do that?
Cantor did so when he presented evidence concerning the reals.
I got aware of this when I looked at his second diagonal proof.
>
>> If I recall correctly,
>
> It is pretty clear you don't.
>
> - Randy
This is what I wrote:
>> If I recall correctly, this "famous" and often attacked second
>> diagonal evidence was the fundamental for Cantor's conclusion that
>> there is no bijection IR<->IN.
Everybody might correct me. I am curious to learn a different proof.
_______________________________
(Daniel Grubb, 4 Feb 2005 19:03:18)
>>> That's *too* naive. How do you represent 0.10110011100011110000... (the
>>> meaning of the "..." being unambiguous, I hope) as a ratio of two natural
>>> numbers?
>>Simply as follows:
>>10110011100011110000... diveded by 10000000000000000000...
>
> This betrays a misunderstanding about natural numbers. The point (well,
> one point anyway) is that natural numbers have *finite* decimal expansions.
>
> So, for example,
>
> 482648753876098379865
>
> is a natural number, but
>
> 19373000000....
>
> is NOT a natural number.
The points usually indicate infinite continuation. With this definitual
precondition one could actually write 123000... = 567000....
because oo = oo + anything.
> This distinction is, at least in part,
> definitional.
Entirely, and I do not use the point in that sense.
> One way of seeing the distinction is to imagine
> that you start at 1 and add 1 again and again and again. At no point
> do you ever come up with anything that has an infinite decimal
> expansion. So even if your intuition about natural numbers doesn't
> correspond to this, it *is* what mathematicians use for the concept.
Even if I am just an electrical engineer, I do not need such lecturing.
Do not insinuate that I was following my intuition.
>
> One reason for making this distinction is that there are ways
> of adding two natural numbers or multiplying two natural numbers
> and getting natural numbers back. Simple things like cancellation
> work for natural numbers. On the other hand, if you let x=1000.....,
> what would x times x be? Would it be the same as x? If so, x^2 =x, which
> by algebra would mean that x=0 or x=1. Neither seems likely. If it
> is not the same as x, what *would* it be? Lots and lots of difficulties
> arise when you allow such things in your system. In the same way, there are
> problems with allowing things like .000.....1, where there are supposed
> to be infinitely many 0's and then a 1.
Perhaps you used to teach rather stupid students. I am 62 years old.
The number one after infinitely many 0's? What nonsens! You are at the
same unbelievable level with Weierstrass who spoke of infinite numbers
and of course Cantor who wrote oo, oo+1, oo+2,... as if his actual
infinity was reasonable.
> So, to say that .12643 is rational is simply to say that it can
> be written as a ration of two natural numbers, say 12643/100000.
You got it.
> But to say that sqrt(2) is not rational means that it cannot be
> written as a ratio m/n where both m and n have finite decimal
> expensions.
That is in the line of my argumentation. One cannot give a finite
decimal list including the irrational reals. Cantor did so.
> So, any real number with a finite decimal expansion is rational.
This is what I am about. The numbers in Cantor's 'evidence' were
rational. They did not include the irrationals.
> But there are real numbers without a finite decimal expansion which
> are still rational. For example, .33333.... which can be written as 1/3.
> One fact that comes out of this is that rational numbers are exactly
> those that have decimal expansions that eventually repeat.
> So, for
> example, .12342323232323232... will be rational (where the 2 and 3
> alternate from then on) but .101001000100001000001.... will not
> be rational (where the number of 0's in the expansion grows by one
> between each 1). But it is also important to remember that *both*
> of these represent real numbers.
Does not matter.
>
> Another quite common mistake is to say that something like
> .101001000100001..... 'approaches' a real number.
>
> Again, that is
> a mistake because that expression actually represents a *single*
> real number. There is a natural sequence, i.e. {.1, .10, .101, ...}
> whose limit is this real number, but the sequence and the real number
> are different things.
I did not make a mistake of this sort.
> I hope this helps.
>
> --Dan Grubb
Thank you Dan, for illustrating my argumentation.
Eckard
________________________________
>> On 2/4/2005 8:34 AM, Bernd Funke wrote:
>>
>>
>>>> Incidentally, I recon the exact mathematical notion of pi a PRN,
>>>
>>> What the heck...? What *is* "the exact mathematical notion" of \pi? Could
>>> you *please* write it down for us?
>>
>> I would not like to quarrel about that. I meant pi is exactly the
>> circumference belonging to the mathematical notion of a circle with a
>> diameter of one. So it is an exact number without exact representation
>> in terms of numerals.
>
> Geez...
>
> You'll find my answer over there at dsm: news:36i06oF520tifU1@individual.net
>
> Do I have to apologize to the readers here?
Maybe for words like heck and Geez.
What about your answer in German, I am not sure if I already replied to
it. If so, then I perhaps did so in German. At least I do not recall any
serious argument that refuted my argumentation.
Eckard
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