Re: Project Euclid Problem 26 SOLVED!!!

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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