Random odd digit generators

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Joseph K. Horn

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Nov 25, 2002, 2:47:11 PM11/25/02
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How to Generate Random Numbers with All Odd Digits:
Another Matter of Consequence from the Recreational Mathematics Asylum
by Joe Horn, lifer.

Truly random numbers between 0 and 1 contain an infinite number of digits.
Therefore, the probability that *all* of the digits in any one random number
are odd, is zero. (Question: "But what about 1/3? It's .33333333..., all of
which are odd!" Answer: "1/3 is not a random number. The probability of a
random number being 1/3 is 1/infinity which is zero.") This is no surprise
to anybody with half a brain, although it often startles those with a
complete brain.

However, HP-calculator "random" numbers DO have a chance of consisting of
only odd digits, since they are only 12 digits long. The probability of an
HP random number having all odd digits (I think) is 5/18432; slim but real.

On any *real* 12-digit HP calculator, all of the following RAND commands
generate results containing 12 odd digits. Use whatever commands are
equivalent in your machine (e.g. in the HP-32SII use SEED instead of RDZ,
and RANDOM instead of RAND).

4198 LN RDZ RAND --> .795513171959
1081 ACOSH RDZ RAND --> .951357993155
440 3 XROOT RDZ RAND --> .535599331197
217 8 XROOT RDZ RAND --> .117151399579
188 33 XROOT RDZ RAND --> .915153717955
174 35 XROOT RDZ RAND --> .359197357153
109 50 XROOT RDZ RAND --> .175137795713

Finding more, and finding random *even* digit generators, is left as an
exercise for the student in dire need of diversion.

-Joe- -getting even by staying odd-


Joseph K. Horn

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Nov 25, 2002, 3:49:23 PM11/25/02
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Gene Wright emailed me to say:

> If you're going to call 1/3 a zero probability event,
> then ALL random numbers of infinite length
> are zero probability events.

Correct. Another way of putting that is: There is no such thing as a random
number. There are only *sequences* that are "close enough to random" to
satisfy one's own particular needs and/or favorite definition of "random".
Extract any one number from that sequence, however, and by itself it's just
a number, not a "random number".

> After all, you can never get an infinite set of
> decimal digits whether they are all 3's or not.
> 1/3 is not a special case as you have defined it. :-)

Exactly.

> Wish I could have posted this to comp.sys.hp48,
> but [my place of work] frowns on that lately.
> Feel free to put this out there yourself...

There are no random numbers. There are only random sequences. The
probability that the Nth element of that sequence is X is zero, where N is a
number I have sealed in this envelope and X is any number you wish.

-Joe-


Rodger Rosenbaum

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Nov 25, 2002, 8:20:25 PM11/25/02
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On Mon, 25 Nov 2002 20:49:23 GMT, "Joseph K. Horn" <Joe...@jps.net> wrote:

>Gene Wright emailed me to say:
>
>> If you're going to call 1/3 a zero probability event,
>> then ALL random numbers of infinite length
>> are zero probability events.
>
>Correct. Another way of putting that is: There is no such thing as a random
>number. There are only *sequences* that are "close enough to random" to
>satisfy one's own particular needs and/or favorite definition of "random".
>Extract any one number from that sequence, however, and by itself it's just
>a number, not a "random number".
>
>> After all, you can never get an infinite set of
>> decimal digits whether they are all 3's or not.
>> 1/3 is not a special case as you have defined it. :-)
>
>Exactly.
>
>> Wish I could have posted this to comp.sys.hp48,
>> but [my place of work] frowns on that lately.
>> Feel free to put this out there yourself...
>
>There are no random numbers.

If there are no random numbers, then how much less are there random digits? Less than
none? The Rand Corporation thought they had found some random digits a while ago.
http://www.rand.org/publications/classics/randomdigits/

And what about the 100,000 normal deviates? Have some of them found their way to this
newsgroup? And, are they still normal?

Joseph K. Horn

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Nov 25, 2002, 10:55:48 PM11/25/02
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Rodger Rosenbaum wrote:

>> There are no random numbers.

>> There are only random sequences.
>

> If there are no random numbers,
> then how much less are there random digits?

There is no such thing as a random digit.
There are only random *sequences* of digits.

Ah, but your point is well taken; it refines my statement to this:

There is no such thing as a random number.

There are only random *sequences* of numbers.

Can we agree on that?

> The Rand Corporation thought they had found some random digits a while
ago.
> http://www.rand.org/publications/classics/randomdigits/

They found some very large random *sequences* of digits. But if you take
any one of those digits out of its table, and write it down by itself on a
scrap of paper, is it still a random digit? No.

Further proof: Is 0 a random digit? No. Is 1? No. Etc, up to 9. And
those are the only digits there are. Therefore, by process of brute
elimination, there is no such thing as a random digit.

Ah, but is { 0 8 6 1 1 1 2 9 1 8 8 9 3 0 2 4 6 } a random sequence of
digits? It is, according to the HP32SII, which generates those after
clearing the random seed and running RANDOM 10 * IP repeatedly. Now, I see
a 4 in that list of "random digits". Does that mean that 4 is a random
digit? No. There is one 4 in that list, but there are four 1's. Does that
mean that 1 is "more random" than 4? There is not a single 5 or 7 in the
list; does that mean that they are not random digits?

The sequence is random. The digits are not.

In a random sequence of infinitely long numbers, the digits of each number
form a sequence that is also random. But each number by itself, just like
each digit by itself, cannot be called random.

> And what about the 100,000 normal deviates?

Sounds like an oxymoron to me. ;-)

> Have some of them found their way to this
> newsgroup? And, are they still normal?

The best oxymorons in the galaxy can be found here, but watch your step;
this place can get a little rough.

-Joe- -having another levulose rush-


Rodger Rosenbaum

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Nov 26, 2002, 12:23:56 AM11/26/02
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On Tue, 26 Nov 2002 03:55:48 GMT, "Joseph K. Horn" <Joe...@jps.net> wrote:

>Rodger Rosenbaum wrote:
>
>>> There are no random numbers.
>>> There are only random sequences.
>>
>> If there are no random numbers,
>> then how much less are there random digits?
>
>There is no such thing as a random digit.
>There are only random *sequences* of digits.
>
>Ah, but your point is well taken; it refines my statement to this:
>
>There is no such thing as a random number.
>There are only random *sequences* of numbers.
>
>Can we agree on that?
>
>> The Rand Corporation thought they had found some random digits a while
>ago.
>> http://www.rand.org/publications/classics/randomdigits/
>
>They found some very large random *sequences* of digits. But if you take
>any one of those digits out of its table, and write it down by itself on a
>scrap of paper, is it still a random digit? No.

How many elements must the sequence have before it is a sequence of "random" numbers?
20? 10? 3? 2? Why not one?


>
>Further proof: Is 0 a random digit? No. Is 1? No. Etc, up to 9. And
>those are the only digits there are. Therefore, by process of brute
>elimination, there is no such thing as a random digit.

If I write any of the digits inside curly brackets, it becomes a sequence, doesn't it?
Why can't that be a sequence with randomness properties? I think it is the process of
selecting the digits that makes them random. If I select a single digit by throwing a
dart at a page of the Rand volume mentioned above, while blindfolded, isn't that sequence
consisting of a single digit entitled to be called random just as much as a sequence of 17
digits selected by the same method? The problem here seems to be that the things we
associate with randomness such as lack of obvious patterns, non-predictability of some
elements of the sequence from others, can't be tested for in really short sequences; it is
only the relation of the elements of the sequence to one another that makes for
randomness. So, a single digit might be randomly generated; you just can't give a
probability to the likelyhood that it is or isn't. Knuth, in Volume 2 of "The Art of
Computer Programming", discusses these and more difficulties with the attempt to
mathematically describe what we mean by "random" sequences.


>
>Ah, but is { 0 8 6 1 1 1 2 9 1 8 8 9 3 0 2 4 6 } a random sequence of
>digits? It is, according to the HP32SII, which generates those after
>clearing the random seed and running RANDOM 10 * IP repeatedly. Now, I see
>a 4 in that list of "random digits". Does that mean that 4 is a random
>digit? No. There is one 4 in that list, but there are four 1's. Does that
>mean that 1 is "more random" than 4? There is not a single 5 or 7 in the
>list; does that mean that they are not random digits?
>
>The sequence is random. The digits are not.
>
>In a random sequence of infinitely long numbers,

By this, do you mean that each number in the sequence has a non-terminating decimal
representation, and is less than 1; the sort of numbers returned by the random number
generator in the 49?

the digits of each number
>form a sequence that is also random.

Well then, since the exact number 1/3 could be returned by a random number generator,
and has a decimal representation of .33333333333333333333..., we seem to get the result
that the sequence {3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3...} is a sequence of random
numbers, eh?

Veli-Pekka Nousiainen

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Nov 26, 2002, 12:28:32 AM11/26/02
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"Rodger Rosenbaum" <rodg...@siteconnect.com> wrote in message
news:7dv5uu0d2e063vhav...@4ax.com...

> How many elements must the sequence have before it is a sequence of
"random" numbers?
> 20? 10? 3? 2? Why not one?
How about two? Then at least I can calculate the standard deviation.
:-D

> Well then, since the exact number 1/3 could be returned by a random
number generator,
> and has a decimal representation of .33333333333333333333..., we seem to
get the result
> that the sequence {3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3...} is a sequence
of random
> numbers, eh?
You did not seem to get the Joe's point here...
:-(
VPN


Rodger Rosenbaum

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Nov 26, 2002, 1:16:37 AM11/26/02
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On Mon, 25 Nov 2002 19:47:11 GMT, "Joseph K. Horn" <Joe...@jps.net> wrote:

>How to Generate Random Numbers with All Odd Digits:
>Another Matter of Consequence from the Recreational Mathematics Asylum
>by Joe Horn, lifer.
>
>Truly random numbers between 0 and 1 contain an infinite number of digits.

But there is reason to say that not ALL numbers between 0 and 1 that have an infinite
number of digits are random (see Knuth's discussion of infinity-distributed sequences).
In this sense, only transcendental numbers are truly random.

>Therefore, the probability that *all* of the digits in any one random number
>are odd, is zero.
(Question: "But what about 1/3? It's .33333333..., all of
>which are odd!" Answer: "1/3 is not a random number. The probability of a
>random number being 1/3 is 1/infinity which is zero.")

It would seem that since there are an infinite number of transcendentals (more than
uncountably infinite, in fact), the probability of any one being selected by a random
process is zero. Therefore, there are no random numbers.

Rodger Rosenbaum

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Nov 26, 2002, 1:19:55 AM11/26/02
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The issue I am raising is whether Joe's point is valid. The question of what
constitutes a random sequence is clouded by difficult philoshphical issues.
>:-(
>VPN
>

Rodger Rosenbaum

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Nov 26, 2002, 1:41:58 AM11/26/02
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On Mon, 25 Nov 2002 20:49:23 GMT, "Joseph K. Horn" <Joe...@jps.net> wrote:

>Gene Wright emailed me to say:
>
>> If you're going to call 1/3 a zero probability event,
>> then ALL random numbers of infinite length
>> are zero probability events.

We seem to have a paradox here. Not only are all RANDOM numbers of infinite length
zero probability events, all numbers of any kind would also be zero probability events.
Does this mean that we can't write down any number? Knuth discusses what are called
infinity-distributed sequences, and mentions a proof that infinite sequences can only be
infinity distributed (and only then are they "truly" random) if they are the decimal
representation of a transcendantal number. Since the cardinality of the transcendentals
is higher than the merely algebraics, if we can find some way to write down
(metaphorically, you understand) an infinitely long decimal number, it's more (infinitely
more, in fact) likely to be random than not (in the sense that its digits form a random
sequence).

Thomas Becker

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Nov 26, 2002, 3:22:55 AM11/26/02
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According to my layman definition a random number is one
that isn't predictable. To get one in the range from 1 to
6 you could throw a dice. I don't see why the probability
of a random number, a random digit, or the Nth element in
a random sequence to be X is zero.

I agree, any chosen number isn't random - only *before* it
appears. (Isn't that related to quantum mechanics?)

Cheers,
Thomas

Peter Geelhoed

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Nov 26, 2002, 4:09:15 AM11/26/02
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"Joseph K. Horn" wrote:

> How to Generate Random Numbers

I once build a TRUE random number genarator from a piece of Strontium-90.
Here is how it worked:
The time between the decay of two atoms is a random variable.
The chance that the time between event_0 and event_1 is larger than
the time between event_1 and event_2 is (slightly larger *) than 0.5.
We used this to determine wheter these three events were 0 or 1.
This is a TRUE RNG.


* Since an event decreases the number of atoms and therefore the average time
until the next event increases. The halflife of Sr-90 is 28.8 years. The
activity of
our sample was 1mCi and with some nuclear statistics you can calculate that the

decrease in activity between two events EXP(-2E-17)=1-2E-17.
Which was slightly beyond our measuring capability.


--
ir. P.F.Geelhoed
Delft University of Technology
Laboratory for Aero & Hydrodynamics
Leeghwaterstraat 21, 2628 CA Delft, The Netherlands
+31-15-2786656 / +31-15-2782947 (fax)

Erwann ABALEA

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Nov 26, 2002, 4:18:38 AM11/26/02
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On Tue, 26 Nov 2002, Veli-Pekka Nousiainen wrote:

> "Rodger Rosenbaum" <rodg...@siteconnect.com> wrote in message
> news:7dv5uu0d2e063vhav...@4ax.com...
> > How many elements must the sequence have before it is a sequence of
> "random" numbers?
> > 20? 10? 3? 2? Why not one?
> How about two? Then at least I can calculate the standard deviation.
> :-D

You're mixing probability and statistics. You're wrong. Once you're
measuring statistics, nothing's random anymore, so calculating
probabilities is a non-sense.

Randomness is defined as the probability to predict the next digit.

--
Erwann ABALEA <erw...@abalea.com> - RSA PGP Key ID: 0x2D0EABD5
-----
Ceci était mon dernier message, quels que soient les messages suivants,
je n'y répondrai pas car ce ne sont que des opinions.
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Veli-Pekka Nousiainen

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Nov 26, 2002, 5:45:09 AM11/26/02
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Now I've lost it totally
??? =(8-/)
VPN

"Rodger Rosenbaum" <rodg...@siteconnect.com> wrote in message
news:of46uugqqmanorj19...@4ax.com...

> On Tue, 26 Nov 2002 07:28:32 +0200, "Veli-Pekka Nousiainen"
<DROP...@welho.com> wrote:
X

Veli-Pekka Nousiainen

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Nov 26, 2002, 5:47:36 AM11/26/02
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"Peter Geelhoed" <pe...@dutw1479.wbmt.tudelft.nl> wrote in message
news:3DE33A3A...@dutw1479.wbmt.tudelft.nl...
X

> The time between the decay of two atoms is a random variable.
Actually it is NOT, but the current science cannot express this yet.
;-)
VPN


Veli-Pekka Nousiainen

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Nov 26, 2002, 5:49:14 AM11/26/02
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"Erwann ABALEA" <erw...@abalea.com> wrote in message
news:Pine.LNX.4.33.021126...@patchwork.seclogd.org...

> On Tue, 26 Nov 2002, Veli-Pekka Nousiainen wrote:
X

> Randomness is defined as the probability to predict the next digit.

OK - then - how can I predict the next digit digit
without a sequence of at least TWO digits?

Opinions?
VPN


Erwann ABALEA

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Nov 26, 2002, 8:13:25 AM11/26/02
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On Tue, 26 Nov 2002, Veli-Pekka Nousiainen wrote:

> "Erwann ABALEA" <erw...@abalea.com> wrote in message
> news:Pine.LNX.4.33.021126...@patchwork.seclogd.org...
> > On Tue, 26 Nov 2002, Veli-Pekka Nousiainen wrote:
> X
> > Randomness is defined as the probability to predict the next digit.
>
> OK - then - how can I predict the next digit digit
> without a sequence of at least TWO digits?

By knowing how the generator works internally, and its current state.

If your generator is biased, the prediction can be better than 1/n (where
n is the total number of different digits your generator can produce).

If the generator is a really good one (for example, a Blum-Blum-Shub one),
you can't predict anything even knowing an infinite number of produced
digits: not only the next digit, but also the *previous* ones.

If the generator is a really bad one (for example, the one used in
Microsoft products (Visual XX, Excel, ...)), the value returned by the
call to rand() *is* the internal state. In this case, you only have to get
*one* number to be able to predict the rest of the suite. (of course, if
you don't call srand(), then you don't have to know anything).

--
Erwann ABALEA <erw...@abalea.com> - RSA PGP Key ID: 0x2D0EABD5
-----

Quel est le MEILLEUR SITE avec une tonne de Javascript's que vous
connaissez ??
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Erwann ABALEA

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Nov 26, 2002, 8:19:33 AM11/26/02
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On Tue, 26 Nov 2002, Veli-Pekka Nousiainen wrote:

It is actually. You can't discuss about the quality of the randomness of a
source if you're able to examine it carefully while working. You *must*
consider the random generator as a black box. You only know what it is
composed of, how it works, but not its internal state.

That sort of generator is a true random number generator, as opposed to a
software one (i.e. produced by a deterministic finite state machine such
as a computer). These later ones are called pseudo-random number
generators, even if some of them have been proved to be perfect.

--
Erwann ABALEA <erw...@abalea.com> - RSA PGP Key ID: 0x2D0EABD5
-----

Je pense qu'un lecteur assidu se reconnaitra. James, si tu veux que
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Nick Karagiaouroglou

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Nov 26, 2002, 10:41:22 AM11/26/02
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Hi random folks! ;-)

Jumping in, if you don't mind, may I ask:



> > The Rand Corporation thought they had found some random digits a while
> ago.
> > http://www.rand.org/publications/classics/randomdigits/
>
> They found some very large random *sequences* of digits. But if you take
> any one of those digits out of its table, and write it down by itself on a
> scrap of paper, is it still a random digit? No.

Don't you think that this is kind of..., well, question that forms its
own answer? If you write the whole random sequence on paper and you
have it in front of your eyes, does it suddenly become un-random?
Since we can only have finite sequences of random numbers, it looks to
me like we will never have a sequence with all random numbers included
with the same probability. Any finite sequence of real random numbers
ommits infinite many possible numbers, exactly because it is finite
(?).

> Further proof: Is 0 a random digit? No. Is 1? No. Etc, up to 9. And
> those are the only digits there are. Therefore, by process of brute
> elimination, there is no such thing as a random digit.

Why not?

> Ah, but is { 0 8 6 1 1 1 2 9 1 8 8 9 3 0 2 4 6 } a random sequence of
> digits? It is, according to the HP32SII, which generates those after
> clearing the random seed and running RANDOM 10 * IP repeatedly. Now, I see
> a 4 in that list of "random digits". Does that mean that 4 is a random
> digit? No. There is one 4 in that list, but there are four 1's. Does that
> mean that 1 is "more random" than 4? There is not a single 5 or 7 in the
> list; does that mean that they are not random digits?

Can we ask at all the question about randomness for a single digit?
Does it then make sense?

> The sequence is random. The digits are not.
>

And what if someone comes out and says: "Hurrah! I have found an
analytic closed form of the random sequence that proves that it is not
random because it will never produce some numbers of the range?"

Greetings,
Nick.

Nick Karagiaouroglou

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Nov 26, 2002, 11:01:39 AM11/26/02
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Rodger Rosenbaum <rodg...@siteconnect.com> wrote in message news:<7dv5uu0d2e063vhav...@4ax.com>...

> On Tue, 26 Nov 2002 03:55:48 GMT, "Joseph K. Horn" <Joe...@jps.net> wrote:
>
> >Rodger Rosenbaum wrote:
> >
> >>> There are no random numbers.
> >>> There are only random sequences.
> >>
> >> If there are no random numbers,
> >> then how much less are there random digits?
> >
> >There is no such thing as a random digit.
> >There are only random *sequences* of digits.
> >
> >Ah, but your point is well taken; it refines my statement to this:
> >
> >There is no such thing as a random number.
> >There are only random *sequences* of numbers.
> >
> >Can we agree on that?
> >
> >> The Rand Corporation thought they had found some random digits a while
> ago.
> >> http://www.rand.org/publications/classics/randomdigits/
> >
> >They found some very large random *sequences* of digits. But if you take
> >any one of those digits out of its table, and write it down by itself on a
> >scrap of paper, is it still a random digit? No.
>
> How many elements must the sequence have before it is a sequence of "random" numbers?
> 20? 10? 3? 2? Why not one?

The more digits you have the better, I think. Though there is no sharp
limit, a sequence with as many digits as possible is better.
(Statistically).

I thought that randomness was defined through the totally uniform
distribution of the produced numbers over some number range. That the
probability to pick a distinct number out of the sequence is exactly
equal to the probability of picking some other.

Taken from http://mathworld.wolfram.com/RandomNumber.html
"A random number is a number chosen as if by chance from some
specified distribution such that selection of a large set of these
numbers reproduces the underlying distribution."
Here you see why 1 single digit may be a sequence but can hardly be
used for saying that it has something to do with randomness,

From the same URL:
"Almost always, such numbers are also required to be independent, so
that there are no correlations between successive numbers.
Computer-generated random numbers are sometimes called pseudorandom
numbers, while the term "random" is reserved for the output of
unpredictable physical processes. When used without qualification, the
word "random" usually means "random with a uniform distribution."
Other distributions are of course possible. For example, the
Box-Muller transformation allows pairs of uniform random numbers to be
transformed to corresponding random numbers having a two-dimensional
normal distribution."

So random (what I think Joe means) and unpredictable (what I think
that you mean) is not quite the same.

From the same URL:
"It is impossible to produce an arbitrarily long string of random
digits and prove it is random. Strangely, it is also very difficult
for humans to produce a string of random digits, and computer programs
can be written which, on average, actually predict some of the digits
humans will write down based on previous ones."

That's interesting isn't it?


>> the digits of each number
>>form a sequence that is also random.
>
> Well then, since the exact number 1/3 could be returned by a random number >generator,
> and has a decimal representation of .33333333333333333333..., we seem to get >the result
> that the sequence {3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3...} is a sequence of >random
> numbers, eh?

Hard to say. If we allow that the random number has to be produced in
the "number range" [3-3], then the above sequence is the (only
possible) perfect random sequence, not because you can't predict what
comes next but because each possible number (here only 3) is there
with the same probability. The question is more if the above "number
range" makes sense or not.

Greetings,
Nick.

Bruce Horrocks

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Nov 26, 2002, 4:50:01 PM11/26/02
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In message <EjIE9.1697$vY1.5...@reader1.news.jippii.net>, Veli-Pekka
Nousiainen <DROP...@welho.com> writes

Isn't this one of those experiments that can only be performed by a
specially trained black cat in a hat or a box or something, singing
"Memory" but whenever you look at the cat it drops dead from
embarrassment?
--
Bruce Horrocks
Hampshire
England
b...@granby.demon.co.uk

Bruce Horrocks

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Nov 26, 2002, 4:39:17 PM11/26/02
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In message <7dv5uu0d2e063vhav...@4ax.com>, Rodger
Rosenbaum <rodg...@siteconnect.com> writes

[argument snipped]
So far I'm with you rather than Joe...

>digits selected by the same method? The problem here seems to be that
>the things we
>associate with randomness such as lack of obvious patterns,
>non-predictability of some

Bear this in mind.

> Well then, since the exact number 1/3 could be returned by a random
>number generator,
>and has a decimal representation of .33333333333333333333..., we seem
>to get the result
>that the sequence {3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3...} is a
>sequence of random
>numbers, eh?

I disagree here. 1/3 is generated randomly - without obvious pattern,
non-predictably etc. as you say above. However thinking about 1/3 as
0.3333etc is a red herring.

Converting 1/3 into decimal, stripping the point, splitting each
successive digit and inserting into a list is a purely mechanical
process - nothing random (or pseudo-random) about that.

So forget the representation.

Joe:
Are you going to go to your local state lottery organiser and tell them
that the numbers painted on their balls are not random and therefore the
winning number each week has not been chosen randomly? It's not the
balls, it's the mixing up and unbiased selection that makes *each
number* random.

Does the fact that as the lottery balls are drawn they form a sequence
matter? One could argue that this week's lottery draw is simply the
continuation of last week's sequence, which was a continuation of the
week before's. Or is it only a continuation when the same machine and
the same set of balls are used? Answer: it's irrelevant.

Regards,

Rodger Rosenbaum

unread,
Nov 26, 2002, 7:31:29 PM11/26/02
to
On Tue, 26 Nov 2002 21:39:17 +0000, Bruce Horrocks <b...@granby.demon.co.uk> wrote:

>In message <7dv5uu0d2e063vhav...@4ax.com>, Rodger
>Rosenbaum <rodg...@siteconnect.com> writes
>
>[argument snipped]
>So far I'm with you rather than Joe...
>
>>digits selected by the same method? The problem here seems to be that
>>the things we
>>associate with randomness such as lack of obvious patterns,
>>non-predictability of some
>
>Bear this in mind.
>
>> Well then, since the exact number 1/3 could be returned by a random
>>number generator,
>>and has a decimal representation of .33333333333333333333..., we seem
>>to get the result
>>that the sequence {3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3...} is a
>>sequence of random
>>numbers, eh?
>
>I disagree here. 1/3 is generated randomly - without obvious pattern,
>non-predictably etc. as you say above. However thinking about 1/3 as
>0.3333etc is a red herring.

My point here was that something Joe said earlier, and I quote
"the digits of each number form a sequence that is also random", leads to my
conclusion.
I pointed out that 1/3 could certainly be returned by a random number generator and
therefore its digits are random according to what Joe said.

John H Meyers

unread,
Nov 27, 2002, 2:50:29 AM11/27/02
to
Seen:

> I thought that randomness was defined through the totally uniform
> distribution of the produced numbers over some number range.

Oh, do you mean like the sequence
0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,... ?

Isn't that a "totally uniform" distribution?

Or, do you mean that if, out of a 100,000 digit sequence,
there were *exactly* 10,000 of each digit? That would be
"totally uniform," but highly improbable, would it not?

It can be a tricky subject unless one is really clear
about what one really means, and also not continually
changing the meaning, perhaps without even realizing it,
producing apparent contradiction.

One framework in which to put it is in terms of knowledge.

You know the past, but not the future; actually, this can
be turned around, to say that what you know and what you
don't know are what define the realms of "past" and "future,"
and that "time" is the successive unfoldment of knowledge.

The degree of randomness is then the degree of your
not being able to know. If you have seen the next page
in the Rand Corp's list of digits (was that Herman Kahn
mentioned in the list of references?), then you know
some information, while someone else who has not seen
may not know, and if there exists no significantly simpler
pattern for predicting the unknown digits than the digits
themselves, then that defines their condition of being
a random sequence, from the perspective of one who
might yet not know the sequence.

Inability to find a pattern does not demonstrate
the non-existence of a pattern; a battery of common tests
does, however, often detect simpler patterns, and thus
can demonstrate lack of randomness.

A good pseudo-random number generator or good cryptography
should minimize the detectability of such patterns;
it is a challenge to generate longer streams of output
than the saved "system state" without making such patterns
detectable, and affording knowledge of the rest of the unknown
from a smaller sample of the known.

What we call "science" generally has an exactly related aim,
to know much more (and to be able to predict or cause effects),
from as little an original collection of the known
(and also, importantly, with least effort) as is possible.

So it all comes down to our state of knowledge vs. ignorance;
should it ever turn out, in our own personal experience,
that we can have a state of consciousness that is simultaneously
one of absolute simplicity and yet all-knowing, one in which
thought organizes its own accomplishment, with least effort,
and which every conscious person can have, then we will have
a supreme science, a Science of Creative Intelligence.


Some theory (Physics):
http://sankey.ws/qm.html
http://sankey.ws/universe.html

Practice (and music):
http://sankey.ws/energy.html
http://sankey.ws/string.html
http://sankey.ws/harpsichord.html
http://sankey.ws/scarlattirec.html
http://sankey.ws/my.html

"He who receives an idea from me, receives instruction himself
without lessening mine; as he who lights his taper at mine,
receives light without darkening me." (Thomas Jefferson)
http://sankey.ws/copyright.html


"Everyone on board our Spaceship Earth can live abundantly
and successfully... Humanity has the option to make it.
We must choose it before it expires..." [Buckminster Fuller]

Some related thoughts from Buckminster Fuller were quoted in:
http://groups.google.com/groups?selm=3D788568.92FD49E4%40miu.edu


With best wishes from http://www.mum.edu

.


-----= Posted via Newsfeeds.Com, Uncensored Usenet News =-----
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Nick Karagiaouroglou

unread,
Nov 27, 2002, 9:20:01 AM11/27/02
to
John H Meyers <jhme...@miu.edu> wrote in message news:<3DE47945...@miu.edu>...

> Seen:
>
> > I thought that randomness was defined through the totally uniform
> > distribution of the produced numbers over some number range.
>
> Oh, do you mean like the sequence
> 0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,... ?
>
> Isn't that a "totally uniform" distribution?

Exactly such an example I wanted to use today for telling randomness
from predictability. Thanks for saving me from typing it ;-)

But seriously, according to the strict mathematical definition isn't
that a random sequence? You can of course predict "what comes next",
but only because you *already have seen* the sequence itself, and have
observed the underlying pattern. This confuses me a little bit. What
is this (or any other) sequence?

1) Is it the result of already having randomly picked numbers out of a
pool 0-9 that contains all numbers from 0 to 9 "equally often"? If so,
could you predict that this very sequence will be formed if you pick
30 (or what so ever) numbers out off the pool? I believe not. If the
sequence represents the outcoming of some experiment that we didn't do
yet, then you and me and anybody else couldn't predict what number
would appear next, or am I wrong here? The sequence is then one of the
possible outcomes of picking randomly numbers out of a pool which is
made in such a way, that all numbers are equally likely to be picked
out. The special case 0,1,2,3... isn't special at all, it appears
appealing to us because it is something that we recognize easily. But
any other sequence would be exactly as probable before we make the
experiment.

2) Or is the sequence itself the pool which I should pick numbers
from? Though this is not the strict mathematical definition, if we
accept it, then "what comes next" becomes predictable, *if* we know
which position of the sequence we are at.

The mathematical definition of a random number is just that, if you
pick a number out of some pool (i.e. distribiution) then all numbers
are equally likely to be picked out. So a sufficient large number of
picked numbers reproduces the distribution from which the numbers were
picked. If you have a pool of integers from 0 to 9 and each of these
integers is available the same "number of times" in the pool, then
picking randomly 10 numbers from the pool can make the sequence
0,1,2,3,4,5,6,7,8,9 of *picked up* numbers, exactly like it could
create 3,4,9,0,1,6,2,5,7,8. And here is where I have my difficulties.
The so created sequence is unpredictable *until I see it*. After I
have seen it, of course I can find always some
formula/algorithm/whatsoever that reproduces this sequence. But for me
this doesn't make the randomly created sequence less random, because
the next time, when I create another random sequence using the same
pool and the same procedure of picking, I still can't say what comes
next.


> Or, do you mean that if, out of a 100,000 digit sequence,
> there were *exactly* 10,000 of each digit? That would be
> "totally uniform," but highly improbable, would it not?

I can't understand what you mean here, John. Can you elaborate?

> You know the past, but not the future; actually, this can
> be turned around, to say that what you know and what you
> don't know are what define the realms of "past" and "future,"
> and that "time" is the successive unfoldment of knowledge.

Exactly. But here I have two possibilities. Is the sequence the past,
i.e. is it already created when I justify if it is random or not? Or
does it yet have to be created by picking numbers? If I forget for a
moment the math definition of "random", then what sequence can be
random at all, when there is always a way to reproduce an already
created sequence by some formula? I think that this is why the
"randomness" is defined as it is defined mathematically. If we couple
randomness directly to unpredictability, then we'll face difficulties.

> Inability to find a pattern does not demonstrate
> the non-existence of a pattern; a battery of common tests
> does, however, often detect simpler patterns, and thus
> can demonstrate lack of randomness.

Pattern recognition is always possible on a *known* sequence of
numbers, events or anything else. The problem is to predict what
pattern the next test (of the same kind) will produce. But speaking
about patterns, perhaps we humans have the ability to regognize too
much patterns, or even to recognize patterns there are none. (Example:
Husband recognizes pattern: "When wife comes smiling to me, she wants
me to do something for her" ;-))

> A good pseudo-random number generator or good cryptography
> should minimize the detectability of such patterns;
> it is a challenge to generate longer streams of output
> than the saved "system state" without making such patterns
> detectable, and affording knowledge of the rest of the unknown
> from a smaller sample of the known.

This introduces an additional concept, mapping. A known cyphered
message, received as a sequence of bytes for example, can be
reproduced, in the sence that you can create exactly the same sequence
of bytes, by using some algorithm. But you can't map this sequence
onto another, so that the "human readable" message is created. And the
fact that some given algorithm creates this cyphred sequence of bytes
of course doesn't help us for the next message.

But let's stay on simpler things for the start, just random sequences
of numbers.

> What we call "science" generally has an exactly related aim,
> to know much more (and to be able to predict or cause effects),
> from as little an original collection of the known
> (and also, importantly, with least effort) as is possible.

Predictability is not the same like randomness. When we examine some
nature phenomenon, then we could say that we just "pick" some values
out of a pool of such values. But the pool doesn't have an uniform
distribution of these values over the range of possible values. Some
of them are strongly prefered, while most of the others are almost
neglected. The "pattern" for this is just what we call "law of
nature", isn't it? We make a finite number of experiments, start to
"smell" that there is a pattern, and the we find something like
F=dp/dt which explains why some of the possible values for, say F, are
so strongly prefered. But if we have a totally uniform distribution,
then *before* we pick the numbers, we can't say what will be picked
next. Of course after picking them, we can always construct some
formula that constructs exactly the same sequence, like that which we
just created. But the next time we pick numbers again out of the
uniform distribution, to make a new sequence, we will find something
completely different, because the distribution is uniform. So having
an already constructed and known sequence, doesn't make it un-random
just because we can reproduce it.


> So it all comes down to our state of knowledge vs. ignorance;
> should it ever turn out, in our own personal experience,
> that we can have a state of consciousness that is simultaneously
> one of absolute simplicity and yet all-knowing, one in which
> thought organizes its own accomplishment, with least effort,
> and which every conscious person can have, then we will have
> a supreme science, a Science of Creative Intelligence.

It appears to me that you must be way ahead of me, since I can't
understand a single word of what you say here. But please, even if I
am too stupid to realize what high meanings these words could have,
allow me to enjoy my journey to that state of consiousness (if it
exists), let me be slow and don't put me in a hurry, OK? Let me
examine each and every single stone on the way, wherever this way
might take me. And even if I miss the high target that you already
reached and land at the cottage of Trabakoulas, it doesn't matter. The
journey is enough for me.


> "Everyone on board our Spaceship Earth can live abundantly
> and successfully... Humanity has the option to make it.
> We must choose it before it expires..." [Buckminster Fuller]

And I have my own options, maybe wrong options, but even if I am
mistaken in these options, hey! they are *my* mistakes. This is me.

> With best wishes from http://www.mum.edu

Even more best wishes from just the mistaken me.
Nick.

Nick Karagiaouroglou

unread,
Nov 27, 2002, 9:36:21 AM11/27/02
to
"Veli-Pekka Nousiainen" <DROP...@welho.com> wrote in message news:<clIE9.1698$vY1.5...@reader1.news.jippii.net>...

Hi Veli-Pekka!

Considering randomness you should predict not what digit comes in an
*already* constructed sequence of digits, but what the next digit will
be if the sequence is just being made. It is what digit you will pick
next (with closed eyes) out of a bucket of digits, in order to make
the sequence. If you can't say what digit you will pick next, then the
so constructed sequence is a random sequence, no matter if you can
somehow reproduce it after you finished picking numbers. It is the way
you constructed the sequence that makes it random, not the fact that
you can predict its digits when it is ready.

Predictable greetings,
Nick.

Joseph K. Horn

unread,
Nov 27, 2002, 1:49:27 PM11/27/02
to
Rodger Rosenbaum wrote:

> The issue I am raising is whether Joe's point is valid. The question of
what
> constitutes a random sequence is clouded by difficult philoshphical
issues.

More troublesome than difficult philosophical issues are difficult
philosophers, of whom we constitute two. ;-)

PHILOSOPHICALLY, an event is truly random if and only if it is unbiased
*AND* unpredictable *AND* unreproducible. However, every *specific* number
lacks all three of those qualities, and hence is thrice non-random.

Non-randomly picked example: 17. 17 is biased; it *insists* on being 17.
It's predictable; it's *always* equal to 17. And it's reproduceable: 17 17
17 17 71 (oops) 17 17 17 ... as many as you want.

Replace 17 above with any other number and the paragraph remains true.
Therefore all numbers are intrinsically non-random when taken alone, Q.E.D,
+/- 3dB.

NUMERICALLY, a numeric sequence is "sufficiently random" if it passes Test
Suite X chosen by Person Y for purpose Z, where X and Y and Z are
unpredictable but highly biased and reproducible. That's so subjective that
it boils down to, "It's random if you think it's random," leading me to
suspect that numeric randomness is an art, not a science.

-Joe-

Free online prime testing and factoring calculators from RAND Corporation:
http://www-3.engineering.com/cnc/Mathematics/NumberTheory?discipline=Mathema
tics


Rodger Rosenbaum

unread,
Nov 27, 2002, 4:20:49 PM11/27/02
to
On Wed, 27 Nov 2002 18:49:27 GMT, "Joseph K. Horn" <Joe...@jps.net> wrote:

>Rodger Rosenbaum wrote:
>
>> The issue I am raising is whether Joe's point is valid. The question of
>what
>> constitutes a random sequence is clouded by difficult philoshphical
>issues.
>
>More troublesome than difficult philosophical issues are difficult
>philosophers, of whom we constitute two. ;-)
>
>PHILOSOPHICALLY, an event is truly random if and only if it is unbiased
>*AND* unpredictable *AND* unreproducible. However, every *specific* number
>lacks all three of those qualities, and hence is thrice non-random.
>
>Non-randomly picked example: 17. 17 is biased; it *insists* on being 17.
>It's predictable; it's *always* equal to 17. And it's reproduceable: 17 17
>17 17 71 (oops) 17 17 17 ... as many as you want.

Isn't this called "begging the question"? You select an admittedly non-random number
and then proceed to show that it doesn't have randomness properties.

>
>Replace 17 above with any other number and the paragraph remains true.
>Therefore all numbers are intrinsically non-random when taken alone, Q.E.D,
>+/- 3dB.

As I said in an earlier post, what makes a number random is not some intrinsic
property, but rather the process by which it is selected. Kosher meat's "kosherness"
isn't intrinsic; it's a certain process that makes it kosher. The numbers in a sequence
are only random if they are selected by a random process, and the sequence will likely
have certain properties and lack certain others. If the sequence is large, we can apply
tests to find out if it is likely to have been generated by a random process. As the
sequence becomes smaller, these tests don't work well. But this doesn't change the fact
that the sequence is random (meaning that it was generated by a random process). It only
means that we cannot show a posteriori that it's random. But we can know it by knowing
how it was generated.

If we have a large sequence that all agree is random, and select a very small subset,
does that subset lose its randomness? I think not; it's just that taken in isolation and
without knowledge about how it was created, the usual tests cannot tell us whether it is
likely to have been created by a random process, because it is very small. But if I know
that it was created by a random process because I did the creating, then I can know that
it is a random sequence.

A thing may have a quality which is not discernible from the thing itself. The power
companies are starting to sell electricity which has been generated by "green"
technologies such as windmills. We can't tell from the electricity itself if it has been
generated by a "green" technology, but it it has, then it IS "green" and we call it
"green" (we humans DO like to name things, and then argue about what the names mean). We
cannot know of this quality by examining the electricity; we have to know that it's
"green" by knowing how it was generated. So with small random sequences.

Therefore I assert that all numbers are not necessarily non-random when taken alone. I
take an agnostic position; given an isolated number and with no other information, the
number might be random (selected by a random process) and it might not be. I cannot know.
However, it is only I who cannot know. The (mathematically literate) person who selected
the number knows and they could tell me; then I would know. But given a large sequence of
numbers, there are tests I can apply which can indicate whether the sequence was generated
by a random process.

Sequences which are generated by pseudo-random generators are completely deterministic,
non-random, and yet they can pass all the tests designed to show randomness and are useful
for the purposes to which random numbers are put, even though they are not truly random.
Perhaps to be precise, we should say that the tests don't so much tell me if the sequence
was generated by a random process, as they tell me if it was generated by a non-random
process. They can falsify the hypothesis that the sequence is random.

Q.E.D.

Joseph K. Horn

unread,
Nov 27, 2002, 4:38:20 PM11/27/02
to
Bruce Horrocks wrote:

> Joe:
> Are you going to go to your local state lottery organiser and tell them
> that the numbers painted on their balls are not random and therefore the
> winning number each week has not been chosen randomly?

No! *I* haven't got the balls! ... Um, what's that? ... Oh, *so* sorry.
Unfortunate choice of words. Ahem. What I mean is, the winning numbers are
not random in and of themselves, because if they were, they could just use
the same numbers every time, because they're random, and that's all we need,
right? No, of course not. The *sequence* of winning numbers is generated
by a *random process*, and as long as that process stays random, it produces
a "random sequence" from which winning numbers are *sequentially* chosen.
It's not the numbers that have the property of randomness; it's the
sequence, because of the process that generated it.

Causality. Essential concept. Without it, we can only have "sufficiently
pseudo-random", but not "random".

> It's not the balls, it's the mixing up and unbiased
> selection that makes *each number* random.

If that were correct, that "each number" in the current winner's list is a
"random number", we could therefore reuse those exact same numbers
(pre-determined to be truly "random numbers") for each lottery! No need for
that machine any more! Of course, after a few lotteries, everybody would
notice how the same "random numbers" keep appearing, and everybody would bet
on those, and everybody would win!

No. It's the *sequence* of winning numbers that's random, not any of the
numbers themselves.

Is sequence S random? Maybe. Depends on how it was generated.

Is number X random? No. Is 6 a random number? Is Giovanni H. Jonesmythe a
random person? If you say, "That depends on the sequence from which you
extracted that example," then I rest my case. It's the *sequence* that's
random, not the elements therein, just as the set of primes is an infinite
set, but none of the primes themselves are infinite. Randomness is a
property of *sequences*, not of numbers, and sequences *inherit* that
property from the process that creates them. No, the elements do not
inherit the properties of the sequence, as shown already with the case of
prime numbers, and just about every other integer sequence in Sloane's
delightful almost-infinite online set at
http://www.research.att.com/~njas/sequences/index.html

> One could argue that this week's lottery draw is simply the
> continuation of last week's sequence, which was a continuation of the
> week before's.

One could argue? It's not just arguable; it's an historical fact. Print
all the winning numbers ever picked, and what do you have? Yes, quite. A
sequence. A *random* sequence, if the machine uses a physically random
process to create that sequence.

> Or is it only a continuation when the same machine and
> the same set of balls are used? Answer: it's irrelevant.

Au contraire! It's not only relevant, it's essential to the discussion!
The randomness of a sequence *depends* on the process that created the
sequence, doesn't it? Where else would this quality of "randomness" come
from? Do we *impose* it upon unsuspecting numbers regardless of their
origin and without their consent? Is it not rather *caused* by the process
that generated the sequence?

<soapbox>

It all depends on this: Do I believe that mathematics is a useful but
imperfect abstraction from external physical reality, or do I believe that
so-called "external reality" is merely an imperfect shadow (cast on the cave
wall of my consciousness) of imponderable, ineffable, mathematically pure
realities? In brief, am I a Realist (aka "scientist") or an Idealist (aka
"Platonist")? And am I such by my own free will, or was it determined by
the Fates? And if there really is a Universal Consciousness, could it be
wrong?

Reporter: "And who have we here?"
Joe Q Random: "Here's my wife, Betty Random, and here are all the Random
children."
Reporter: "Um, that boy over there doesn't look like a Random child to me."
Betty: "Yes, but I can assure you that his resemblance to John Q Postman is
a pure coincidence."
Reporter: "Ah, the unpredictable randomness of genetics!"
Joe: "Exactly."

</soapbox>

-Joe- -Realist-
A mind is a terrible thing to boggle.


Joseph K. Horn

unread,
Nov 27, 2002, 4:44:00 PM11/27/02
to
Nick Karagiaouroglou wrote:

> > Further proof: Is 0 a random digit? No. Is 1? No. Etc, up to 9. And
> > those are the only digits there are. Therefore, by process of brute
> > elimination, there is no such thing as a random digit.
>
> Why not?

Ok, let's start from the top.

Is 3 a random digit?

-Joe-


Joseph K. Horn

unread,
Nov 27, 2002, 5:22:55 PM11/27/02
to
Rodger Rosenbaum wrote:

> We seem to have a paradox here. Not only are all
> RANDOM numbers of infinite length zero probability
> events, all numbers of any kind would also be zero
> probability events.

That is correct, since all numbers are actually of infinite length (in "show
all leading and trailing zeros" mode). No paradox, just reality.

> Does this mean that we can't write down any number?

No, it doesn't mean that, and here's why. I can write down the number 0.153
exactly. But that's because 0.153 is not a random number! I picked it on
purpose, not randomly! If I were to create a TRUE random decimal-number
generator, it would NEVER pick 0.153! That's what probability theory says,
anyway: the probability of an event is equal to the number of successful
outcomes divided by the total number of possible outcomes. There is only
one decimal number equal to 0.153, and there are infinitely many possible
decimal numbers (or MORE than infinitely many, if Hilbert was right). So
P(0.153) = 1/infinity = zero.

The only alternative I can think of is to reject the notion that 1/infinity
= zero, and go instead with the much more palatable notion that 1/infinity =
infinitesimal. Would you be comfortable with the statement, "The
probability of a random decimal being 0.153 is infinitesimal"? I hope not,
because "1/infinitesimal = infinity" implies that 1/0 must be *more* than
infinity, which makes no sense, so division by zero must remain undefined,
which throws most of the calculus into the ash can, because it *depends* on
the concept of defining division by zero for patching the holes in
discontinuous functions. Hence, by reductio ad absurdum, I reject the
alternative, and stick with the common notion that 0 and infinity are
reciprocals.

This whole discussion reminds me of my students' refusal to accept that
0.99999... is *exactly equal* to 1. Not merely approaching it as a limit;
*equal* to it. Counterintuitive to the max, but mathematically certain.

> ... if we can find some way to write down (metaphorically,


> you understand) an infinitely long decimal number, it's more
> (infinitely more, in fact) likely to be random than not
> (in the sense that its digits form a random sequence).

Exactly. The probability of a rational number being generated by a truly
random number generator is zero, unlike all HP pseudo-random number
generators which ONLY produce rational numbers. I wanted to mention that in
the original posting, but I thought to myself, "Nah, leave it out; somebody
will probably disagree with it, even though it's mathematically certain."

-Joe-


Steen Schmidt

unread,
Nov 27, 2002, 5:45:42 PM11/27/02
to
> This whole discussion reminds me of my students' refusal to accept that
> 0.99999... is *exactly equal* to 1. Not merely approaching it as a limit;
> *equal* to it. Counterintuitive to the max, but mathematically certain.

It's related to the concept of infinity, which I also found really hard to
grasp at first. I hope you wasn't hard on him or her? ;-)

Only time and thought - not lessons - will let the student see the truth...

Regards
Steen


Rodger Rosenbaum

unread,
Nov 27, 2002, 6:05:47 PM11/27/02
to
On Wed, 27 Nov 2002 22:22:55 GMT, "Joseph K. Horn" <Joe...@jps.net> wrote:

>Rodger Rosenbaum wrote:
>
>> We seem to have a paradox here. Not only are all
>> RANDOM numbers of infinite length zero probability
>> events, all numbers of any kind would also be zero
>> probability events.
>
>That is correct, since all numbers are actually of infinite length (in "show
>all leading and trailing zeros" mode). No paradox, just reality.
>
>> Does this mean that we can't write down any number?
>
>No, it doesn't mean that, and here's why. I can write down the number 0.153
>exactly. But that's because 0.153 is not a random number!

So we can write down some numbers.

I picked it on
>purpose, not randomly! If I were to create a TRUE random decimal-number
>generator, it would NEVER pick 0.153! That's what probability theory says,
>anyway: the probability of an event is equal to the number of successful
>outcomes divided by the total number of possible outcomes. There is only
>one decimal number equal to 0.153, and there are infinitely many possible
>decimal numbers (or MORE than infinitely many, if Hilbert

^Cantor^


was right). So
>P(0.153) = 1/infinity = zero.

But by this argument, we can never write down a random number since every other number
besides 0.153 has the same probability of being picked, namely P(random n)=1/infinity.

Joseph K. Horn

unread,
Nov 27, 2002, 6:53:42 PM11/27/02
to
Rodger Rosenbaum wrote:

>> Non-randomly picked example: 17. 17 is biased; it
>> *insists* on being 17. It's predictable; it's *always*
>> equal to 17. And it's reproduceable: 17 17 17 17 71
>> (oops) 17 17 17 ... as many as you want.
>
> Isn't this called "begging the question"? You select
> an admittedly non-random number and then proceed
> to show that it doesn't have randomness properties.

Ooh, you caught me red handed! You're right; 17 is in fact one of the
well-known non-random numbers, and I knew that going in. How very uncouth
of me. Please let me know which random number you'd like me to use next
time. ;-)

> As I said in an earlier post, what makes a number random is not some
intrinsic
> property, but rather the process by which it is selected.

Is that another way of saying, "Numbers aren't random, but processes of
selecting them can be"? If so, then we agree.

> Kosher meat's "kosherness" isn't intrinsic; it's a certain
> process that makes it kosher.

Aha! As I understand it, a rabbi's blessing creates a *relationship*
between the food and God, and it's that *relationship* which makes it
kosher. Is that correct? If so, I see what you mean: X (some person or
place or thing or number or other concept) can really have a relation of
some kind with Y, and that relationship is a real, actual quality of X, but
it cannot be *detected* by any kind of examination of X alone. [For
example, I own this HP32SII. There is a real relationship of ownership
there; it *really is* mine. But there is no way that anybody could discern
that fact by examining it, since that relationship has not caused any unique
physical changes to it.] Only by one's *knowledge* (thanks, JHM!) of X and
Y can their relationship be known.

Similarly, if a single number or finite sequence of numbers is extracted
from an infinite random sequence, the "randomness" of the sequence might not
be detectable in the finite sequence, and is certainly not detectable in the
single number, BUT the fact remains that there is a real relationship
between the random sequence and the finite sequence... and even the single
number!

Methinks there is a convergence of ideas happening here. I *love* it when
that happens!

-Joe-


Joseph K. Horn

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Nov 27, 2002, 7:06:58 PM11/27/02
to
Rodger Rosenbaum wrote:

>>decimal numbers (or MORE than infinitely many, if

>> Hilbert was right.
> ^Cantor^

Rats. I picked a random mathematician and it was the wrong one. Phooey.

Hilbert *is* a random mathematician, right? Or *was* until 1943, when he
became terribly ill-conditioned for any kind of mathematics other than
decomposition.

> But by this argument, we can never write down a
> random number since every other number besides
> 0.153 has the same probability of being picked,
> namely P(random n)=1/infinity.

Correct. Give me a counterexample, and I'll recant.

-Joe-


John H Meyers

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Nov 27, 2002, 8:50:05 PM11/27/02
to
JKH wrote:

> Free online prime testing and factoring calculators
> from RAND Corporation:

> http://www-3.engineering.com/cnc/Mathematics/NumberTheory?discipline=Mathematics


I gave "Integer factorization" a try, banging out this input, at random:

9753195795197859751975951795979579259725975959597

A "results" page shortly appeared, seemingly with no result
(no factors, nor even a copy of the original value,
but no statement that it was prime, either).

I clicked "Download Results" and got the same page image;
I repeated the calculation, requesting results by email,
and got the same again.

The HP49G, meanwhile, got
'4201*2321636704403203940008557913825179542900732197'

Maybe we don't know for sure whether that second factor is really prime,
but it seems to have proceeded further than the web calculator
(provided by RAND corporation? The U.S. government's "think tank"?)

[r->] [OFF]

Veli-Pekka Nousiainen

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Nov 27, 2002, 9:25:31 PM11/27/02
to
"Joseph K. Horn" <Joe...@jps.net> wrote in message
news:gXaF9.4338$ta5.5...@newsread2.prod.itd.earthlink.net...

> No. It's the *sequence* of winning numbers that's random, not any of the
> numbers themselves.

I've got it now!
You have pretty understandable lessons you preach!
(pun intended)
VPN


Veli-Pekka Nousiainen

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Nov 27, 2002, 9:28:59 PM11/27/02
to
No
The cat is at quantum stage
where it is dead and alive on the same time
and when you measure it by looking at the box
you then kill the cat (or something like that :)
VPN
"Bruce Horrocks" <b...@granby.demon.co.uk> wrote in message
news:hhU01XGJ...@granby.demon.co.uk...
X

Veli-Pekka Nousiainen

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Nov 27, 2002, 9:37:15 PM11/27/02
to
"Nick Karagiaouroglou" <n...@imos-consulting.com> wrote in message
news:cd9ca36b.02112...@posting.google.com...
X

> It is the way
> you constructed the sequence that makes it random, not the fact that
> you can predict its digits when it is ready.

Thanks!


Veli-Pekka Nousiainen

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Nov 27, 2002, 9:46:57 PM11/27/02
to
"John H Meyers" <jhme...@miu.edu> wrote in message
news:3DE5764D...@miu.edu...

> JKH wrote:
>
> > Free online prime testing and factoring calculators
> > from RAND Corporation:
>
> >
http://www-3.engineering.com/cnc/Mathematics/NumberTheory?discipline=Mathema
tics
>
>
> I gave "Integer factorization" a try, banging out this input, at random:
>
> 9753195795197859751975951795979579259725975959597
>
> A "results" page shortly appeared, seemingly with no result
> (no factors, nor even a copy of the original value,
> but no statement that it was prime, either).
>
> I clicked "Download Results" and got the same page image;
> I repeated the calculation, requesting results by email,
> and got the same again.
>
> The HP49G, meanwhile, got
> '4201*2321636704403203940008557913825179542900732197'
>
> Maybe we don't know for sure whether that second factor is really prime,
> but it seems to have proceeded further than the web calculator
> (provided by RAND corporation? The U.S. government's "think tank"?)
BAH!
I got the right answer from that URL after the Java applet loaded.
The same one that your HP 49G gave.
VPN


Veli-Pekka Nousiainen

unread,
Nov 27, 2002, 9:59:23 PM11/27/02
to
My stomach hurts!
If you don't soon drop this subject
I will burst and it's all your fault - Horn & Rosenbaum
(A Catholic & a Jew - am I right? If I can predict that, it's not random)
Hey!
There are no randomness at all because God knows ALL the numbers
even the infinite numbers. God > Cantor's infinity > infinity > 1 > 0
;-)
VPN
PS: I still think that atoms breaking down are not doing it randomly.
This phenomenon just passes the Knuth's Spectral test
(with flying colors - can colors fly? :)

"Joseph K. Horn" <Joe...@jps.net> wrote in message
news:C6dF9.4592$ta5.5...@newsread2.prod.itd.earthlink.net...

Nick Karagiaouroglou

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Nov 28, 2002, 7:00:45 AM11/28/02
to
"Joseph K. Horn" <Joe...@jps.net> wrote in message news:<A0bF9.4345$ta5.5...@newsread2.prod.itd.earthlink.net>...

Hi Joe!

Well, if you give me a 3 and you tell me "that's the number, boy!",
then of course it is not random. But this is not the question. The
question is, how was this 3, as a sequence iof only one digit, was
constructed. If you tell me to close my eyes, and just pick a number
out of a bucket full of 1's, 2's ... and so on, and if it just happens
that I pick this 3, then this 3 *is* a random sequence of only 1
number, not because I don't know now what number I have in my hands,
but because I couldn't tell that before I picked it up. The way the
sequence was constructed is imposrtant, not if we can somehow
re-construct it after we already have it.

I'll answer to you with another question. As John asked, is
0,1,2,3,4,5,6,7,8,9 a random sequence? Be careful here, because if you
say no, then I have to ask you if you think that this sequence is
impossible to create with the random number generator of the HP49G.
And if you think that it is really impossible, then consider the
sequence 7,2,5,9,1,6,3,8,4,0. Is this more or less "possible"?

In fact they are both exactly as "possible", the only difference being
that on the first we all readily recognize a pattern, while on the
second.., well it is harder.

BTW, there is no single finite sequence on this world, that can't be
"patternized" *after* it has been constructed. To use the same example
as above, enter [[1 2 3 4 5 6 7 8 9][7 2 5 9 1 6 3 8 4 0 ]] and press
[LAGRANGE] on the HP49G. You get a polynomial that "predicts" all
digits, that is if X=1, then it spits out the digit 7, if X=2 then it
spits out the digit 2, and so on, X being the ordinal number of the
the digit in the sequence. You can do that with each an every
sequence, after you have randomly constructed it. Does this mean that
your hands obeyed the "law of the Lagrange polynomial" when they
picked the numbers out of the bucket? Of course no. The important
thing is how the sequence was created, not if you can re-construct it
later on using some rule. And that because the next time you do the
same experiment, picking numbers out of the bucket, the rule that you
used to re-construct the previous sequence will be useless. If it were
that way, that "random" meant unpredictability of the next digit of an
already constructed sequence, then there would be not a single random
sequence on this planet.

In exactly the same way, if there is no way of saying what number I'll
pick out of the bucket of numbers, then the resulting sequence of
picked numbers is random, even if it only consists of a single number.
It get's the lable "random" out of the way it was constructed, not
because I know that it is 3 after having picked the number.

I hope that my "random chosen" vocabulary was not too random for
understanding.
Predictabel greetings,
Nick.

Nick Karagiaouroglou

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Nov 28, 2002, 7:10:45 AM11/28/02
to
Thomas Becker <thbe...@arcor.de> wrote in message news:<arvb1f$n0q$1...@newsread1.arcor-online.net>...
> According to my layman definition a random number is one
> that isn't predictable. To get one in the range from 1 to
> 6 you could throw a dice. I don't see why the probability
> of a random number, a random digit, or the Nth element in
> a random sequence to be X is zero.
>
> I agree, any chosen number isn't random - only *before* it
> appears. (Isn't that related to quantum mechanics?)
>

Hi Thomas!

You grasped the meaning right away. I keep telling about this fine
distinction of before and after the generation of the number, but
nobody seems to be listening. "Random" stands for how it was made, and
if we could predict what comes out, *before* it came out, not after
it.

The relationship to QM is only of very general nature, only an
analogy, at least for our current knowledge. With QM you can of couse
make very good random number generators but not every random number
generator works a' la QM.

Greetings,
Nick.

Joseph K. Horn

unread,
Nov 28, 2002, 11:18:30 AM11/28/02
to
John H Meyers wrote:

> JKH wrote:
>
> > Free online prime testing and factoring calculators
> > from RAND Corporation:
>
> >
http://www-3.engineering.com/cnc/Mathematics/NumberTheory?discipline=Mathema
tics
>
>
> I gave "Integer factorization" a try, banging out this input, at random:
>
> 9753195795197859751975951795979579259725975959597
>
> A "results" page shortly appeared, seemingly with no result
> (no factors, nor even a copy of the original value,
> but no statement that it was prime, either).

Please try again, and be sure to read everything shown. The screen is
supposed to refresh itself every 6 seconds, and will show the result as soon
as it is found. I just tried your example, and it factored by the first
screen refresh (6 seconds). The delay you encountered may have been due to
Internet delays of one kind or another.

-Joe-


Joseph K. Horn

unread,
Nov 28, 2002, 12:24:00 PM11/28/02
to
[Warning! This thread is getting more odd and more random as it goes on. It
seems to be approaching maximum entropy. In any case, it went off-topic long
ago. You have been warned! -jkh-]

Veli-Pekka Nousiainen writes:

> My stomach hurts!

Try the Mathematical Digest. ;-)

> There are no randomness at all because God knows ALL
> the numbers even the infinite numbers.

That nothing is random from God's point of view was proven handily by Thomas
Aquinas some 750 years ago. But that lends no support to the hypothesis
that nothing is random from *our* point of view. After all, God's justice
and God's mercy are both infinite and identical, but ours are neither by a
long shot!

> God > Cantor's infinity > infinity > 1 > 0

My way of looking at it is: if 1/infinitesimal = infinity, then 1/0 = God.
Perhaps the Big Bang really *did* happen when God divided by zero.

What are the reciprocals of transfinite numbers, anyhow? Subfinite numbers?
Hypofinite numbers? Numericules? Diminutives? Speckulations? Weenies?

> PS: I still think that atoms breaking down are not doing it randomly.
> This phenomenon just passes the Knuth's Spectral test

I agree. Furthermore, due to the quantum lumpiness of physical reality, we
shouldn't call it a space-time continuum at all; it's a space-time
contiguity (we ought to call it a contiguum, but there's no such word in
English). In a way, Pythagoras was right; there are no irrational numbers
in reality, only integers (quanta) and their ratios. So even so-called
"random physical processes" only have a finite number of possible states,
and hence are not "truly" random, but (like computer RNG's) can be
sufficiently *chaotic* to survive the various tests of randomness.

Whatever it is, it's all good.

-Joe-


Joseph K. Horn

unread,
Nov 28, 2002, 12:56:57 PM11/28/02
to
Nick Karagiaouroglou wrote:

> I keep telling about this fine distinction of before and after
> the generation of the number, but nobody seems to be listening.

Don't worry; we're listening.

> "Random" stands for how it was made...

I agree! Numbers contain within themselves no information regarding where
they came from... but it makes perfect sense to ask "Is the PROCESS that
PICKED this number a random PROCESS?"

Is celery a healthy food? Most certainly not! It's not healthy at all:
it's DEAD! But we *call* it "healthy" due to its relationship to *our*
health when we eat it. [Thank you, Thomas Aquinas, for that example.]
Similarly, we call the results of random number generators "random numbers"
not because the numbers themselves are actually random (they're not!) but
because of their relationship to the randomness of the process that
generated them. In other words, we think of RNG's as random-number
generators (that is, generators of random numbers) but they are really
random number-generators (that is, random generators of numbers). I can
randomly generate a specific number, but I can't generate a random number,
unless we agree to understand the term "random number" similarly to the
illogical but understood term "healthy food." Don't carry this analogy too
far, or I'll make you eat some celery. >:-b

-Joe- -stalker-


Wolfgang Rautenberg

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Nov 28, 2002, 1:57:29 PM11/28/02
to
Joseph K. Horn wrote:
> What are the reciprocals of transfinite numbers, anyhow? Subfinite numbers?
> Hypofinite numbers? Numericules? Diminutives? Speckulations? Weenies?

Cantor proved for cardinal numbers a,b that a x b = max{a,b}
if at least one of them is transfinite. Thus, no transfinite
cardinal can have a reciprocal. Slighly more complicated
are the arithmetical formulas for transfinite ordinals,
but also such an ordinal has no reciprocal.

The situation changes drastically when looking at
non-standard-analysis, a new disciplie which has to do
with real closed extensions of the real number field
in which exist infinite numbers, i.e., those greater
that all reals. Such a number has a positive reciprocal
smaller than all positive reals, hence, is an infinitesimal
in the sence of Newton, Leibniz and Euler. With the
aid of his infinitesimals, Euler developed the exponential
series for the function e^x already on page 1 of this book!
Then he takes just a little rest in the theory to compute
e from e = 1 + 1/1! +1/2! + ... with a precision of 25
decimals. Eulers book shows how fast one can attend results
for which at present (without infinitesimals) at least
100 pages of preparation are needed :-)

Infinitesimals were eliminated from analysis by Cauchy
and Weierstraß because of lack of a logical foundation.
But to the great surprise of the classical analysis,
where reintroduced in about 1960 by Abraham Robinson
who explored modeltheoretic methods of Alfred Tarski.


Both Alfred Tarski and Abraham Robinson emigrated just
before world war to from Europe to the USA. At many
universities in the world, non-standard analysis is
already taught but it will be still about a decade
to learn on these new numbers in colleges and
high-scools.

- Wolfgang

PS. The USA receveid so much intellectual potential
from Europe before and after world-war II for *nothing*
that the should perhaps feel obliged to help the rest
of the world at least in science and engineering.

E

Wolfgang Rautenberg

unread,
Nov 28, 2002, 2:20:24 PM11/28/02
to
I forgot to mention the title of Eulers book
recommended occasionally to my math students:

L. Euler, Introductio in Analysin infinitorum,
Collected works, vol 8, Leipzig 1922.

- Wolfgang

Veli-Pekka Nousiainen

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Nov 28, 2002, 7:13:09 PM11/28/02
to
Any ISBN numbers?
What translations are available?
Finnish would be good, English acceptable.
My Svenska & Deutch are not good enough for math stuff.
Remember, my IQ is almost certainly below 150.
VPN
"Wolfgang Rautenberg" <ra...@math.fu-berlin.de> wrote in message
news:3DE66C78...@math.fu-berlin.de...

Thomas Becker

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Nov 29, 2002, 2:39:47 AM11/29/02
to
Well, I think it is still healthier to eat healthy food, also in
the sense you are refering to. Celery as a plant isn't dead from
the beginning. Most food dies during the expensive processing to
make it suitable for the food industry.

Cheers,
Thomas

Joseph K. Horn wrote:
[...]


> Is celery a healthy food? Most certainly not! It's not healthy at all:
> it's DEAD! But we *call* it "healthy" due to its relationship to *our*
> health when we eat it. [Thank you, Thomas Aquinas, for that example.]

[...]

Nick Karagiaouroglou

unread,
Nov 29, 2002, 8:36:23 AM11/29/02
to
"Joseph K. Horn" <Joe...@jps.net> wrote in message news:<JNsF9.5183$yy.6...@newsread1.prod.itd.earthlink.net>...

> Nick Karagiaouroglou wrote:
>
> > I keep telling about this fine distinction of before and after
> > the generation of the number, but nobody seems to be listening.
>
> Don't worry; we're listening.

Hmm...

> > "Random" stands for how it was made...
>
> I agree! Numbers contain within themselves no information regarding where
> they came from... but it makes perfect sense to ask "Is the PROCESS that
> PICKED this number a random PROCESS?"

OK. Then the process itself must be examined thoroughly. If after
examination we still have no clue, then it could be that it is a
random process, or it could be that we just can't find that it is not
a random process.

In the case of Nick closes his eyes and picks a celery out of Joe's
bucket, I think, nobody can find out what makes Nick pick the one and
not the other celery. So we call it random process. Perhaps it isn't,
perhaps there is a fantastic equation with three trillions parameters
that predicts that I'm going to pick the smallest one.

> Is celery a healthy food? Most certainly not! It's not healthy at all:
> it's DEAD! But we *call* it "healthy" due to its relationship to *our*
> health when we eat it. [Thank you, Thomas Aquinas, for that example.]
> Similarly, we call the results of random number generators "random numbers"
> not because the numbers themselves are actually random (they're not!) but
> because of their relationship to the randomness of the process that
> generated them. In other words, we think of RNG's as random-number
> generators (that is, generators of random numbers) but they are really
> random number-generators (that is, random generators of numbers). I can
> randomly generate a specific number, but I can't generate a random number,
> unless we agree to understand the term "random number" similarly to the
> illogical but understood term "healthy food." Don't carry this analogy too
> far, or I'll make you eat some celery. >:-b

No please, anything else but not that! ;-)

The celery can be considered as the case of a one celeried sequence,
so it is the sequence which can be called "random" sequence of
celeries. You totally confused me, Joe! *8-S

Celereatings,
Nick.

John H Meyers

unread,
Nov 29, 2002, 1:21:51 PM11/29/02
to
JKH wrote:

> the probability of an event is equal to the number of successful
> outcomes divided by the total number of possible outcomes.

Some words seem to be missing here, like "finite set of outcomes,"
and "equally likely."


> There is only one decimal number equal to 0.153

But there are two infinite digit sequences equal to that value,
as there are for every value expressed by only a finite
number of digits, so actually that kind of value
must be twice as likely as any other :)

> and there are infinitely many possible decimal numbers
> (or MORE than infinitely many, if Hilbert was right).

How do you know that numbers, as far as they might represent
quantities in nature, aren't actually all quantized,
and even all selected from a finite set in every case?

Even spatial distances might appear to be quantized,
in some theories, which if true would mean that
Hilbert's finding more points on a line than
can be matched with the set of all integers
might only be possible in the imagination,
and not in the physical world.

> "1/infinitesimal = infinity" implies that 1/0 must be *more* than
> infinity, which makes no sense

Maybe it's equal to the number of points on a line,
which you already said was more than infinity :)

> so division by zero must remain undefined,
> which throws most of the calculus into the ash can, because it *depends*
> on the concept of defining division by zero for patching the holes in
> discontinuous functions.

I thought it depended only on defining limits of sequences.

> Hence, by reductio ad absurdum, I reject the alternative,
> and stick with the common notion that 0 and infinity are reciprocals.

It's already been pointed out that "infinities" are of various kinds,
some being "greater than" others, and so must there be
various kinds of zero. But why should engineers care?
They only know how to build finite structures,
to approximate accuracies, and leave perfection (whatever that is) to God :)

Perhaps the notion that we can conceive, in our consciousness,
of that which never exists in physical form, is some evidence,
or at least a hint, that consciousness lies beyond the material realm.

Structural and societal engineering (following the HP49G stuff):
http://groups.google.com/groups?selm=955lok%24bf7%241%40nnrp1.deja.com

John H Meyers

unread,
Nov 29, 2002, 1:56:43 PM11/29/02
to
VPN wrote:

> I got the right answer from that URL after the Java applet loaded.

I had Java disabled; they didn't seem to say anything
that would have given notice that Java was required
to display results.

This reminds me of an electrical outlet in a European hotel
which looked like a 220V outlet, into which we plugged
some very expensive EEG gear. It turned out to be
a 380V circuit, however :-(((((

http://www.me.gatech.edu/me/publicat/AugTranscript.htm

-[]-

Mizrandir

unread,
Nov 30, 2002, 8:00:39 AM11/30/02
to
Thomas Becker <thbe...@arcor.de> wrote in message news:<arvb1f$n0q$1...@newsread1.arcor-online.net>...
> According to my layman definition a random number is one
> that isn't predictable. To get one in the range from 1 to
> 6 you could throw a dice. I don't see why the probability
> of a random number, a random digit, or the Nth element in
> a random sequence to be X is zero.

People are getting confused here, the probability of a random number
is zero only if it is comes from an infinite set. If it comes from a
finite set it indeed has a probability of appearing (like in the dice
example). Talking about a sequence, it will have a non zero
probability of appearing only if it is a finite sequence comming from
a finite set of possible numbers (for example the sequence { 1 1 4 6 2
4 } when throwing a dice.

PS: Excuse me if I'm not using the exact mathematical terms: english
is not my mother language and i'm an engineering student, not a
mathematician.

miz.

John H Meyers

unread,
Nov 30, 2002, 5:32:44 PM11/30/02
to
The Rand Corp publication of a random digit sequence
http://www.rand.org/publications/classics/randomdigits
mentions von Neumann and cites two papers by "H. Kahn"

So here's some more historical information about Herman Kahn:

http://www.alteich.com/links/kahn.htm

"In 1952 Herman Kahn became involved with von Neumann
in the design of the hydrogen bomb"
http://trace.ntu.ac.uk/frame2/articles/borg/kahn.html
http://trace.ntu.ac.uk/frame2/articles/borg/concl.html

"Perhaps the most damaging effect of game theory in the hands of RAND
was the paranoid bias it introduced into the modeling of an enemy's psyche."

"While working at RAND, Kahn settled in with a group working on
nuclear strategy known as the Strategic Objectives Committee.
Its members recognized that an all out nuclear war with an initial
strategy to attack cities was not feasible. In response to such a
strategy, Kahn (only half jokingly) proposed his 'Doomsday Machine,'
a massive computer connected to a stockpile of hydrogen bombs.
When the computer sensed imminent and intolerable danger from a
Soviet attack, it would detonate the bombs and cover the planet
with radiation fallout and billions of dead.

No one laughed (except for Stanley Kubrick,
whose 1964 dark comedy, 'Dr. Strangelove,
or: How I learned to stop worrying and love the Bomb,'
parodied Kahn's Doomsday Device)."

Dr. Strangelove:
http://www.filmsite.org/drst.html

Joseph K. Horn

unread,
Nov 30, 2002, 9:36:22 PM11/30/02
to
Yikes! Thanks, John, for that sobering posting. It gives "The Wrath of
Kahn" a whole new meaning. >:-O (Or was that Khan? I can't remember.)

Also, I'm very glad to know that *that* Kahn is not the same as the Kahan
who helped HP optimize its calculators' accuracy back in the 70's. Just one
letter away. So close, and yet... so far. Whew!

-Joe- (one letter away from Joke... Jove... Joel... Joey...)
-Horn- (one letter away from Shorn... Thorn... Horny...)


Lance Reichert -- Forensic Lepidopterist

unread,
Dec 1, 2002, 1:34:23 AM12/1/02
to

"Joseph K. Horn" <Joe...@jps.net> wrote in message
news:Xs8F9.4121$ta5.5...@newsread2.prod.itd.earthlink.net...

> NUMERICALLY, a numeric sequence is "sufficiently random" if it passes Test
> Suite X chosen by Person Y for purpose Z, where X and Y and Z are
> unpredictable but highly biased and reproducible. That's so subjective
that
> it boils down to, "It's random if you think it's random," leading me to
> suspect that numeric randomness is an art, not a science.

If I can meld concepts from several CS classes correctly, an infinite
sequence of symbols is truly random if and only if there is no finite way to
completely describe it. (If I'd actually taken an information theory class
rather than just discussed it with the students, I'd be able to state that
more rigorously.)

No statistical analysis can completely describe such a sequence. Mean,
mode, median, standard deviation, and kurtosis together are insufficient to
completely describe a random sequence. (For proof, I offer two numeric
sequences, identical except that the second has exchanged the positions of
the first's seventh and 17th elements. Both sequences would have identical
statistics, yet they are different, so their complete descriptions must
differ.)

John's sequence
"0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,... "
appears to be completely described by "the repetition of the subsequence
'0,1,2,3,4,5,6,7,8,9'" The infinite sequence produced by a linear
congruential pseudo-random number generator (which is what most calculators
and programming language libraries offer) is completely described by the
mathematical transform used to generate the next number and the seed number.
Since it is cyclic, it could also be described similarly to John's sequence.

The sequence that begins "1,3,5,7,9,11,13,15,17,19,21,..." is both infinite
and non-cyclic, but it can be completely described as "the odd natural
numbers, in ascending order," so being infinite, acyclic, and having an
infinite alphabet are not sufficient.

No examination of a finite subsequence can prove that the sequence from
which it was taken is random because there exists a cyclic, non-random
sequence from which that subsequence could have been drawn.

Since none of our experiments will ever exhaust an infinite sequence of
input symbols, we really don't care about truly random input, and selection
of a possibly non-random, but apparently-random subsequence is where the art
arises.

Lance ==)-----------
lrre...@inav.your.pants.net
Drop your.pants. to reply to me directly.


Lance Reichert -- Forensic Lepidopterist

unread,
Dec 1, 2002, 1:39:41 AM12/1/02
to

"Rodger Rosenbaum" <rodg...@siteconnect.com> wrote in message
news:qn9auucephc3f833b...@4ax.com...
> On Wed, 27 Nov 2002 18:49:27 GMT, "Joseph K. Horn" <Joe...@jps.net>

wrote:
> >Non-randomly picked example: 17. 17 is biased; it *insists* on being 17.
> >It's predictable; it's *always* equal to 17. And it's reproduceable: 17
17
> >17 17 71 (oops) 17 17 17 ... as many as you want.
>
> Isn't this called "begging the question"? You select an admittedly
non-random number
> and then proceed to show that it doesn't have randomness properties.

Actually, to quote my Modelling and Simulation professor (lo these many
decades ago), "17 is the MOST random number." When questioned about why
this is so, he'd mumble something about its lack of unique properties and
duck out the door.

Lance ==)----------------
Aging and unreformed college student
lrre...@inav.your.pants.net
Drop your.pants. to reply directly.


Lance Reichert -- Forensic Lepidopterist

unread,
Dec 1, 2002, 1:59:55 AM12/1/02
to

"Rodger Rosenbaum" <rodg...@siteconnect.com> wrote in message
news:qn9auucephc3f833b...@4ax.com...
> If we have a large sequence that all agree is random, and select a very
small subset,
> does that subset lose its randomness? I think not; it's just that taken
in isolation and
> without knowledge about how it was created, the usual tests cannot tell us
whether it is
> likely to have been created by a random process, because it is very small.
But if I know
> that it was created by a random process because I did the creating, then I
can know that
> it is a random sequence.

Actually no. By selecting the subsequence, you have tainted its randomness,
unless the selection itself was random.

I was just discussing Dungeons & Dragons(tm) with my boys, so this example
comes to mind: A character's attributes (of which there are six) are
generated in a supposedly random manner, by throwing three six-sided dice
and summing their pips. However, the rules book discusses several ways of
improving a characters' survivability by tampering with the results. One
method was to throw four dice, and discard the low one before summing the
others; another was to roll up all six attribute numbers, but reserve
assigning those numbers to any particular attribute until you've seen all
six (for this reason, most low-level characters are ugly rather than
sickly).

In both cases, the individual die rolls are part of a truly random sequence,
but selection of which rolls to use for what removes (or reduces, if that
has meaning) their original randomness.

> Perhaps to be precise, we should say that the tests don't so much tell me
if the sequence
> was generated by a random process, as they tell me if it was generated by
a non-random
> process. They can falsify the hypothesis that the sequence is random.

Not even that -- our D&D character could have all six attribute scores
legitimately rolled up as 18s (one chance in 101,559,956,668,416). The
process was still random, but no string of 18 consecutive rolls of six could
pass any "test" of randomness, nor would the Dungeon Master allow such a
character.

Lance ==)--------------
Graduate of the Las Vegas school of probability analysis.
lrre...@inav.your.pants.net
Drop your.pants. to respond directly.