On Sat, 2024-03-30 at 15:49 +0100, David Brown wrote:
> On 29/03/2024 19:35, Keith Thompson wrote:
> > David Brown <david.brown@hesbynett.no> writes:
> > [...]
> > > I think mathematics is a great hobby.  It's a shame to see someone
> > > spend their time and effort on doing it so badly.
> >
> > It's also a shame to see someone engaging here in a discussion that has
> > nothing to do with C++.  David, if you must feed this particular troll,
> > I suggest doing so in comp.theory.
> >
> > *You don't have to reply to everything.*
> >
>
> It is Easter, and Usenet traffic is low.  No, I don't have to reply to
> everything (and I don't - I have replied to very few of wij's broken
> maths threads), and this thread will soon die away.  I am trying to get
> some idea of why wij thinks the way he does, and perhaps even help him
> think differently (though that's quite optimistic).
>

Persuade me and readers with proof, otherwise you lie or spread lies (from the moment)

On 30/03/2024 16:14, wij wrote:
> On Sat, 2024-03-30 at 15:49 +0100, David Brown wrote:
>> On 29/03/2024 19:35, Keith Thompson wrote:
>>> David Brown <david.brown@hesbynett.no> writes:
>>> [...]
>>>> I think mathematics is a great hobby.  It's a shame to see someone
>>>> spend their time and effort on doing it so badly.
>>>
>>> It's also a shame to see someone engaging here in a discussion that has
>>> nothing to do with C++.  David, if you must feed this particular troll,
>>> I suggest doing so in comp.theory.
>>>
>>> *You don't have to reply to everything.*
>>>
>>
>> It is Easter, and Usenet traffic is low.  No, I don't have to reply to
>> everything (and I don't - I have replied to very few of wij's broken
>> maths threads), and this thread will soon die away.  I am trying to get
>> some idea of why wij thinks the way he does, and perhaps even help him
>> think differently (though that's quite optimistic).
>>
>
> Persuade me and readers with proof, otherwise you lie or spread lies (from the moment)
>

You were given a proof, but rejected it for no reason other than it
showed that your jumble of claims was incorrect. Thus I don't think
there is any point in trying to give more detailed proofs. But if you
like, I can give some links to other people's proofs - starting with
proving that 0.999... equals 1. If you agree with these, maybe we can
move on to proving that 1/3 equals 0.33... repeating, and then further
onto showing that repeating decimals are rational. So let me know which
of these links you agree with, or disagree with (preferably with reasons
or justification for disagreeing with them).

<https://en.wikipedia.org/wiki/0.999...>
<https://www.purplemath.com/modules/howcan1.htm>
<https://brilliant.org/wiki/is-0999-equal-1/>

(I don't need to persuade any other readers - they already know this stuff.)

And if you think I am lying, you can add lying to your list of concepts
that you don't understand.

On Sat, 2024-03-30 at 19:26 +0100, David Brown wrote:
> On 30/03/2024 16:14, wij wrote:
> > On Sat, 2024-03-30 at 15:49 +0100, David Brown wrote:
> > > On 29/03/2024 19:35, Keith Thompson wrote:
> > > > David Brown <david.brown@hesbynett.no> writes:
> > > > [...]
> > > > > I think mathematics is a great hobby.  It's a shame to see someone
> > > > > spend their time and effort on doing it so badly.
> > > >
> > > > It's also a shame to see someone engaging here in a discussion that has
> > > > nothing to do with C++.  David, if you must feed this particular troll,
> > > > I suggest doing so in comp.theory.
> > > >
> > > > *You don't have to reply to everything.*
> > > >
> > >
> > > It is Easter, and Usenet traffic is low.  No, I don't have to reply to
> > > everything (and I don't - I have replied to very few of wij's broken
> > > maths threads), and this thread will soon die away.  I am trying to get
> > > some idea of why wij thinks the way he does, and perhaps even help him
> > > think differently (though that's quite optimistic).
> > >
> >
> > Persuade me and readers with proof, otherwise you lie or spread lies (from the moment)
> >
>
> You were given a proof, but rejected it for no reason other than it
> showed that your jumble of claims was incorrect.  Thus I don't think
> there is any point in trying to give more detailed proofs.  But if you
> like, I can give some links to other people's proofs - starting with
> proving that 0.999... equals 1.  If you agree with these, maybe we can
> move on to proving that 1/3 equals 0.33... repeating, and then further
> onto showing that repeating decimals are rational.  So let me know which
> of these links you agree with, or disagree with (preferably with reasons
> or justification for disagreeing with them).
>
> <https://en.wikipedia.org/wiki/0.999...>
> <https://www.purplemath.com/modules/howcan1.htm>
> <https://brilliant.org/wiki/is-0999-equal-1/>
>
>
> (I don't need to persuade any other readers - they already know this stuff.)
>
>
> And if you think I am lying, you can add lying to your list of concepts
> that you don't understand.
>
>
I cannot read English fast. I will pick the one proof not in my proof.
Archimedean property just states that infinitesmal does not exit, IIUC. It is
an assertion, not a proof. But I think, if infinitesimal does not exit, what did
those calculus pioneers baffled at?
(I just have a thought, with Archimedean property, you cannot say "infinite repeating"
because there is no 1/∞, this applies to the decimal representation of √2)

The second one's proof depends on magic trick to make people believe, but
neither a valid proof. I have shown how the magic works in my post.

The third one's link, ... the same.

You saw my Simple Enough Proof For Kids (kids know what the Emperor's Cloth is)
and can't disprove it or prove your belief. What should I interpret if you attack too hard?

Paavo Helde <eesnimi@osa.pri.ee> writes:

> 27.03.2024 04:42 Chris M. Thomasson kirjutas:
>
>> On 3/26/2024 2:51 PM, Paavo Helde wrote:
>>
>>> 26.03.2024 22:13 Chris M. Thomasson kirjutas:
>>>
>>>> On 3/26/2024 7:51 AM, wij wrote:
>>>> [...]
>>>>
>>>> Repeating decimals are rational, say
>>>>
>>>> 0.142857 142857 142857
>>>>
>>>> That is just 1 / 7 represented in base 10.
>>>>
>>>> Now, think of using a TRNG to create each digit...
>>>>
>>>> That would be, irrational... ;^)
>>>
>>> Any number represented by stored digits on Earth has finite number
>>> of digits (because Earth is finite) and therefore is rational,
>>> regardless of how the digits are generated.
>>
>> Even with a TRNG?
>
> TRNG is defined as a device having access to a "physical entropy
> source". For producing an irrational number it ought to produce an
> infinite never-repeating sequence of digits.
>
> It's not clear if any physical mechanism would in principle produce a
> non-repeating infinite sequence of digits, the fact that the world is
> quantized might rather speak against it.

The world being quantized is irrelevant. Flipping a coin has two
"quantized" outcomes - either heads or tails. However, if the two
outcomes are completely unpredictable then the coin can serve as a
True random number generator. All that is known about quantum
mechanics has found (as I understand it) that it is statistical
only and completely unpredictable. And that property suffices
to be a TRNG.

> Even if it did, the lifetime
> of a physical device is finite, so it can only produce a finite
> sequence of digits. Even if the lifetime of the hypothetical device
> would be infinite, we do not possess the infinite space and time to
> store the result. Such infinities are strictly the territory of
> mathematics, in real world we can have only finite sequence of digits.

None of that matters. All that does matter is that if we know any
finite number of outputs we still don't know anything about what the
next output will be. There is no way to prove, in a mathematical
sense, that a proposed TRNG is in fact a mathematically perfect
TRNG. But we can make TRNGs in the sense that they have resisted
all efforts to successfully predict their behavior (in a way that
is statistically significant).

>>> If you want to represent irrational numbers you need to use some
>>> other encoding schema, e.g. "sqrt(2)" (8 bytes, voila!).
>>
>> pi? ;^D
>
> In mathematics, a notion like pi actually defines an irrational number
> by fixing its properties. For some such irrational numbers it is
> possible to give an infinite algorithm which produces the sequence of
> its digits. The funny thing is that after fixing the number there is
> no randomness any more, so e.g. a machine computing subsequent digits
> of pi would make a pretty poor RNG ;-)

What you're saying about pi is true of any deterministic random
number generator. Yet there are lots of deterministic RNG's that do
a fine job of generating random numbers. I know of no mathematical
result that says consecutive digits of pi would fail any of the many
statistical tests for RNG's that have been devised. As long as the
starting point is not known, AFAIAA there is no reason to think
consecutive digits of pi would be any worse as a random number
generator than any other deterministic random number generator.

(Note: what I am calling deterministic random number generators
are sometimes referred to as PRNGs, for "pseudo random number
generators".)

Incidentally, pi is not just irrational but transcendental. The
property of being irrational is not by itself guarantee that a
number would be a good source of random digits. Probably that
statement is true for (arbitrary) transcendental numbers also.
But pi is not just any transcendental number. :)

On 26/04/2024 01:33, Tim Rentsch wrote:
> Paavo Helde <eesnimi@osa.pri.ee> writes:
>
>> 27.03.2024 04:42 Chris M. Thomasson kirjutas:
>>
>>> On 3/26/2024 2:51 PM, Paavo Helde wrote:
>>>
>>>> 26.03.2024 22:13 Chris M. Thomasson kirjutas:
>>>>
>>>>> On 3/26/2024 7:51 AM, wij wrote:
>>>>> [...]
>>>>>
>>>>> Repeating decimals are rational, say
>>>>>
>>>>> 0.142857 142857 142857
>>>>>
>>>>> That is just 1 / 7 represented in base 10.
>>>>>
>>>>> Now, think of using a TRNG to create each digit...
>>>>>
>>>>> That would be, irrational... ;^)
>>>>
>>>> Any number represented by stored digits on Earth has finite number
>>>> of digits (because Earth is finite) and therefore is rational,
>>>> regardless of how the digits are generated.
>>>
>>> Even with a TRNG?
>>
>> TRNG is defined as a device having access to a "physical entropy
>> source". For producing an irrational number it ought to produce an
>> infinite never-repeating sequence of digits.
>>
>> It's not clear if any physical mechanism would in principle produce a
>> non-repeating infinite sequence of digits, the fact that the world is
>> quantized might rather speak against it.
>
> The world being quantized is irrelevant. Flipping a coin has two
> "quantized" outcomes - either heads or tails. However, if the two
> outcomes are completely unpredictable then the coin can serve as a
> True random number generator. All that is known about quantum
> mechanics has found (as I understand it) that it is statistical
> only and completely unpredictable. And that property suffices
> to be a TRNG.

Yes, quantum effects are great for TRNG's. To have a RNG, you want two
things: a known and consistent output distribution (it doesn't have to
an even distribution, but you need to know it to compensate for it), and
it needs to be non-deterministic - that is, you can't predict the next
output from previous outputs or other information. If it would be
theoretically possible to predict the output, it's a PRNG. If it is not
possible even theoretically, it is a TRNG.

>
>> Even if it did, the lifetime
>> of a physical device is finite, so it can only produce a finite
>> sequence of digits. Even if the lifetime of the hypothetical device
>> would be infinite, we do not possess the infinite space and time to
>> store the result. Such infinities are strictly the territory of
>> mathematics, in real world we can have only finite sequence of digits.
>
> None of that matters. All that does matter is that if we know any
> finite number of outputs we still don't know anything about what the
> next output will be.

Yes.

> There is no way to prove, in a mathematical
> sense, that a proposed TRNG is in fact a mathematically perfect
> TRNG.

We can be sure that many types of TRNG are "true" random number
generators in the "physics sense". That is not as strong as a
"mathematical sense", in that the laws of physics are always open to
improvement as we learn more about the universe. But things like
thermal noise in electronics components is known - by current physics
theory - to be truly random. (Measuring it disturbs things, unfortunately.)

> But we can make TRNGs in the sense that they have resisted
> all efforts to successfully predict their behavior (in a way that
> is statistically significant).
>

Yes.

>>>> If you want to represent irrational numbers you need to use some
>>>> other encoding schema, e.g. "sqrt(2)" (8 bytes, voila!).
>>>
>>> pi? ;^D
>>
>> In mathematics, a notion like pi actually defines an irrational number
>> by fixing its properties. For some such irrational numbers it is
>> possible to give an infinite algorithm which produces the sequence of
>> its digits. The funny thing is that after fixing the number there is
>> no randomness any more, so e.g. a machine computing subsequent digits
>> of pi would make a pretty poor RNG ;-)
>
> What you're saying about pi is true of any deterministic random
> number generator. Yet there are lots of deterministic RNG's that do
> a fine job of generating random numbers. I know of no mathematical
> result that says consecutive digits of pi would fail any of the many
> statistical tests for RNG's that have been devised. As long as the
> starting point is not known, AFAIAA there is no reason to think
> consecutive digits of pi would be any worse as a random number
> generator than any other deterministic random number generator.
>
> (Note: what I am calling deterministic random number generators
> are sometimes referred to as PRNGs, for "pseudo random number
> generators".)

The key practical distinction, I think, is not whether or not the
sequence is deterministic - but whether or not anyone can determine it.
A "true" RNG is one where it is not physically feasible to determine
future values in the sequence, but it does not matter whether that is
because the values are inherently random (such as quantum noise), or if
it is simply impossible to determine them (perhaps because the
calculations take too much time and energy).

Regarding things like statistical random number tests, any test carried
out will be done over a finite sample of the data. A given RNG may pass
such tests and then later exhibit patterns such as repetition - there is
no way to determine this. (And a practical TRNG is a RNG that will pass
any such tests of any length that can be achieved.)

The digits of the decimal expansion of pi will (I believe) pass any
statistical test. But it will not pass a determinism test targeting pi.

>
> Incidentally, pi is not just irrational but transcendental. The
> property of being irrational is not by itself guarantee that a
> number would be a good source of random digits. Probably that
> statement is true for (arbitrary) transcendental numbers also.
> But pi is not just any transcendental number. :)

Neither the irrational nor transcendental properties are sufficient.
(They are not even necessary, in practical terms - it doesn't even
matter if a sequence repeats as long as it does not repeat during the
digits you are using.)

Liouville's constant, the sum of 10 ^ (-n!) for n = 1 to infinity, was
the first number proven to be transcendental. But since it is almost
entirely zeros, it would not make a useful random number source.

For theoretically infinite sequences, you want a normal number (or at
least, normal in a given base) - that is, one for which the probability
of seeing any given sequence of base b digits of length n gets
asymptotically closer to b ^ -n. I.e., all sequences are equally
likely, over time.