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From: Physfitfreak <Physfitfreak@gmail.com>
Newsgroups: comp.os.linux.advocacy
Subject: Re: Project Euclid Problem 26 SOLVED!!!
Date: Tue, 12 Mar 2024 18:02:22 -0500
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On 3/12/2024 4:00 PM, Farley Flud wrote:
> On Tue, 12 Mar 2024 15:16:21 -0500, Physfitfreak wrote:
> 
>>
>> It provides a rare peep into how things are for larger integers at the
>> denominator of rational numbers. Not derived, but actually computed.
>>
>> Then, somebody should derive such features by math alone, not computing.
>> At least the main features. It could be publishable too, if not derived
>> before. But I doubt it's something novel. Math people have done all
>> sorts of investigations, many of them even long ago.
>>
> 
> If you want, I can provide a file containing the data up to 100,000
> or perhaps 1 million.
> 
> The linear trend can be investigated.
> 
> I was thinking about doing this myself but I haven't got the time
> right now.  I am not a tenured professor and I do not have an abundance
> of free time (unfortunately).
> 
> 
>>
>> All the four features above should hold for any base, not just 10.
>> There's really nothing special about base 10. The base can be chosen as
>> any positive real number. In fact, it would be nice to have the same
>> plot, but for number system in base e :-)
>>
> 
> That is quite true -- for any integer base it will be the same.
> 
> For an irrational base, however, I cannot comment.
> 
> Correct me if I am wrong, but an irrational base has no practical
> or even theoretical utility in expressing INTEGER quantities, and
> INTEGER quantities are the basis for our counting system.
> 
> Sure, we can express PI as 1.0 in base PI but what does that
> improve?
> 


First, I wouldn't engage in creating a plot for a wider range of 
integers in denominator, just like you did, at this point. If it was for 
me, I'd first remove what's bothering me about how it is produced, and 
that's the fact that I don't know how that software works.

Again, if it was for me to do it, I'd first write a simple program 
myself, to find the length of repeating decimals for each number in the 
denominator, to make sure everything is the way it should be. If I would 
get identical results to what the software produced, only then, use the 
software for wider ranges of integers.

And about irrational bases and integer quantities, I remind you that the 
irrational e base system of numbers are full of integers :) You just 
have to let go of comparisons to base 10 situation, and only think in 
terms of base e as the only math you have at your disposal. Then 
everything works just like before, and for everything in math.

The two numbers in base e that I suggested in that challenge problem to 
add and get the result, were, in base e, two integers! It doesn't matter 
one bit that their base 10 number is not integer, because we're doing 
the math in base e now. Integers are perfectly described in the base e. 
You saw it for yourself that nothing came after the radix points in 
those numbers, so it wasn't even necessary to put the dots there :-)

The entire math, as it is now, can be based on a number system with base 
e. In this system, numbers 1 and 2 mean exactly what they do in base 10, 
but the rest of the numbers are of course different from their base 10 
representation. Let me give an example.

Say there are 5 Nazi cro-magnons standing on the deck of a U.S. navy war 
ship close to Yemen, preparing to get on F-35s to bomb the Houthis for 
no other reason than their dicks want so, and their dicks want so, 
because those Nazis are obedient servants of Satan. MIGHT AS WELL, and 
for no other reason that makes sense. Therefore, a Houthi in Yemen calls 
you and asks you this,

   "I want to know how many cro-magnons are standing there on the deck 
to adjust the anti-personnel warhead load on this missile accordingly, 
before they get on their F-35s. We want the lightest options we have, to 
make the missiles fastest as possible, but enough to wipe the Nazis 
clean off of that deck."

But you have a problem. Houthis, turns out, use the number system in 
base e only. They know nothing of a nonsensical "base 10" in use. So 
what would you do to tell them that there are "5" of them on the deck?

Fortunately, you followed Physfit's advice and wrote that little program 
that converts this unfamiliar number "5" into something Houthis 
understand as 5, and you relay the information to them as:

    "There are 12.02 ... number of cro-magnons standing on the deck."

With "..." representing more and more digits to any accuracy that you 
ever wish to have. By the way, 12.02 part is correct. I just calculated 
it. But it's not complete, of course. The numbers to the right of the 
radix point continue. But for Houthis, even 12.02 is enough to know 
exactly how many Nazis are on the deck.

So your argument that this conversion of 5 into 12.02.... introduces 
unnecessary difficulties, is really countered perfectly by countless 
cases of _integers_ in e base system, for which their 10 base 
representations have horribly looking never ending decimals. Why 
perfectly? Because life and business both, deals with much, much more 
cases of partial amounts to measure. Amounts of stuff that cannot be 
described with whole numbers no matter what base you're working in. 
That's why.

So e or 10 base or whatever base, is immaterial. Math works equivalently 
in all such cases, and as efficiently too.







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