Programming is about mathematics and not coding.

Problem:

Any rational number a/b (in lowest terms) can be expressed either
as a terminating decimal or a repeating decimal.

We consider here only repeating decimals.

Given, a and b, which are intergers with any number of digits,
find the repeating portion in the decimal expansion.

That is, the decimal expansion will be:

a1, a2, a3, ... an, [r1, r2, r3 ... rn]

where [... ri ...] is the repeating portion.

Example:

1/28 = 0.03 571428 571428...

The non-repeating, leading portion can be very long and so too can
the repeating portion.

On 3/7/2024 12:11 PM, Nuxxie wrote:
> Programming is about mathematics and not coding.
>
> Problem:
>
> Any rational number a/b (in lowest terms) can be expressed either
> as a terminating decimal or a repeating decimal.
>
> We consider here only repeating decimals.
>
> Given, a and b, which are intergers with any number of digits,
> find the repeating portion in the decimal expansion.
>
> That is, the decimal expansion will be:
>
> a1, a2, a3, ... an, [r1, r2, r3 ... rn]
>
> where [... ri ...] is the repeating portion.
>
> Example:
>
> 1/28 = 0.03 571428 571428...
>
> The non-repeating, leading portion can be very long and so too can
> the repeating portion.
>

I don't think the dorks here know such stuff. And the good ones believe
it's waste of time but what they do (i.e. "fart") is not waste of time.

So I'll go ahead and ruin it for them all :)

(method of nines)

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On 3/7/2024 9:25 PM, Physfitfreak wrote:
> On 3/7/2024 12:11 PM, Nuxxie wrote:
>> Programming is about mathematics and not coding.
>>
>> Problem:
>>
>> Any rational number a/b (in lowest terms) can be expressed either
>> as a terminating decimal or a repeating decimal.
>>
>> We consider here only repeating decimals.
>>
>> Given, a and b, which are intergers with any number of digits,
>> find the repeating portion in the decimal expansion.
>>
>> That is, the decimal expansion will be:
>>
>> a1, a2, a3, ... an, [r1, r2, r3 ... rn]
>>
>> where [... ri ...] is the repeating portion.
>>
>> Example:
>>
>> 1/28 = 0.03 571428 571428...
>>
>> The non-repeating, leading portion can be very long and so too can
>> the repeating portion.
>>
>
>
> I don't think the dorks here know such stuff. And the good ones believe
> it's waste of time but what they do (i.e. "fart") is not waste of time.
>
> So I'll go ahead and ruin it for them all :)
>
> (method of nines)
>
>
>
>

Forgot to add that I suggest that when you post a challenge question,
you add sci.physics to target group as well. Reason for that is that
This newsgroup is pretty obscure and buried inside load of shit in
usenet. Sane healthy people who'd be interested in working on such
problems find sci.physics much more easier and quicker than a strange
named newsgroup like COLA. But unfortunately, so do the insane people :)

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On Thu, 07 Mar 2024 21:25:16 -0600, Physfitfreak wrote:

>
> I don't think the dorks here know such stuff. And the good ones believe
> it's waste of time but what they do (i.e. "fart") is not waste of time.
>

Project Euler, a well respected forum for programmers, doesn't think
that it's a waste of time:

https://projecteuler.net/problem=26

>
> So I'll go ahead and ruin it for them all :)
>
> (method of nines)
>

I don't know. Being a scientist/engineer, I have little background
in number theory.

But, after some research, it is related to the "multiplicative order"
of the denominator.

In Maxima CAS, there is a function that determines the multiplicative
order: zn_order(a, d), where a is the base and d the denominator.

Example:

zn_order(10,31);
15

Thus, 1/31 should have a repeating length of 15, and it does.

zn_order(10,79832137);
516846

1/79832137 should have a repeating length of 516846; I haven't checked.

But it fails for 1/28:

zn_order(10,28);
false

So the method is not totally general.

On Fri, 08 Mar 2024 14:20:12 +0000, Nuxxie wrote:

>
> But it fails for 1/28:
>
> zn_order(10,28);
> false
>

I found out the problem here.

The denominator, d, has to be factored as 2^i * 5^j * d', where
possibly i=1 and j=1.

In this case, 28 = 2^2 * 7 (i.e. i=2, j=0)

Now, zn_order(10,7) = 6

Thus 1/28 has a repeating portion of length 6 which is preceeded
by max(i,j) = 2 digits:

1/28 = 0.03571428571428...

This is fantastic stuff!

My main source, so far, is this link:

https://math.stackexchange.com/questions/377683/length-of-period-of-
decimal-expansion-of-a-fraction

Maxima uses multi-precision, based on GMP, and can handle extremely
large decimal expansions.

On Fri, 08 Mar 2024 14:54:29 +0000, Nuxxie wrote:

>
> 1/28 = 0.03571428571428...
>
> This is fantastic stuff!
>

Let's now do a big one.

1/37252902384619140625

First find the prime factorization:

ifactors(37252902384619140625);
[[5,11],[7,1],[47297,1],[2304403,1]]

Thus, we need to express this as:

5^11 * 762939440837

Now find the multiplicative order:

zn_order(10, 762939440837);

54494498496

The result:

The decimal expansion will begin with 11 digits followed
by a repeating porion of length 54,494,498,496.

This number would need 64 GiB of memory, but we could
write it out to a file using Maxima or bc or GMP.

I won't try but maybe someone else will.

This kind of stuff is what a computer is for. No fucking Netflix.
No fucking Gmail. Just this.

On Fri, 08 Mar 2024 15:43:54 +0000, Nuxxie wrote:

>
> 1/37252902384619140625
>
>
> The decimal expansion will begin with 11 digits followed by a repeating
> porion of length 54,494,498,496.
>
> This number would need 64 GiB of memory, but we could write it out to a
> file using Maxima or bc or GMP.
>

Someone please write out this number to a file. As 1 byte per character
it should be about 55 GiB in size.

Then verify that the decimals start with 11 digits and then repeat
after 54,494,498,496 + 11 digits. Just verify that the first 100 or
so digits are the same.

Y'all have big, fat mouths. Let's see if that can translate into
serious computing action.

The first to do it will receive a BIG PRIZE.

On 3/8/2024 10:04 AM, Nuxxie wrote:
> On Fri, 08 Mar 2024 15:43:54 +0000, Nuxxie wrote:
>
>>
>> 1/37252902384619140625
>>
>>
>> The decimal expansion will begin with 11 digits followed by a repeating
>> porion of length 54,494,498,496.
>>
>> This number would need 64 GiB of memory, but we could write it out to a
>> file using Maxima or bc or GMP.
>>
>
> Someone please write out this number to a file. As 1 byte per character
> it should be about 55 GiB in size.
>
> Then verify that the decimals start with 11 digits and then repeat
> after 54,494,498,496 + 11 digits. Just verify that the first 100 or
> so digits are the same.
>
> Y'all have big, fat mouths. Let's see if that can translate into
> serious computing action.
>
> The first to do it will receive a BIG PRIZE.
>

Again, you're turning the spoon twice around your head before placing it
in your mouth to eat your soup.

If you're after _proving_, mathematically, that a method will give the
repeating decimals, that's another matter. And I'm not good at that and
am not interested in stuff and degree of precise logic that
mathematicians are accustomed to and apply in their proofs. My
background is not math. That's why.

But if you're only interested in the answer, then method of nines is the
simplest and fastest way of getting it.

If you haven't seen it, then what kind of numerical methods class did
you attend? You must be way younger than me. I'm thinking the generation
after me with access to much better computers didn't need all that nitty
gritty to get by, and therefore profs simply omitted a bulk of such
material from the courses.

With an almost two lines of program, one can find a number consisting
only of digits of 9, for which the denominator of the fraction is a
divisor. Then with one more line of code, the fraction is rewritten with
that denominator consisting of 9's. The numerator thus obtained will be
the decimals that get repeated. As simple as that.

How many lines of code is that? 3 ? At most 4.

Example. Say we want to find the repeating decimals of 14/33 without
actually dividing the two numbers to visually find that out, but by
knowing in advance what those repeating decimals will be.

1- find the smallest number consisting of only digits of 9 that 33 is a
divisor for:

33 x 3 = 99

2. apply this factor of 3 to the fraction:

14/33 x 3/3 = 42/99

3. therefore the repeating decimals are 42.

That's how you have your soup :)

The cases involving other numbers appearing in the decimals before
repeating decimals begin are quite similar. You first convert the
fraction to different pieces to separate the repeating part of decimals
from the rest. This turns the term with repeating decimals to have a
bunch of zeros coming before the repeating part begins. Then you use a
slightly different form of method of nines (called method of nines with
zeros) to find the repeating decimals.

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On 3/8/2024 12:07 PM, Physfitfreak wrote:
> On 3/8/2024 10:04 AM, Nuxxie wrote:
>> On Fri, 08 Mar 2024 15:43:54 +0000, Nuxxie wrote:
>>
>>>
>>> 1/37252902384619140625
>>>
>>>
>>> The decimal expansion will begin with 11 digits followed by a repeating
>>> porion of length 54,494,498,496.
>>>
>>> This number would need 64 GiB of memory, but we could write it out to a
>>> file using Maxima or bc or GMP.
>>>
>>
>> Someone please write out this number to a file.  As 1 byte per character
>> it should be about 55 GiB in size.
>>
>> Then verify that the decimals start with 11 digits and then repeat
>> after 54,494,498,496 + 11 digits.  Just verify that the first 100 or
>> so digits are the same.
>>
>> Y'all have big, fat mouths.  Let's see if that can translate into
>> serious computing action.
>>
>> The first to do it will receive a BIG PRIZE.
>>
>
>
> Again, you're turning the spoon twice around your head before placing it
> in your mouth to eat your soup.
>
> If you're after _proving_, mathematically, that a method will give the
> repeating decimals, that's another matter. And I'm not good at that and
> am not interested in stuff and degree of precise logic that
> mathematicians are accustomed to and apply in their proofs. My
> background is not math. That's why.
>
> But if you're only interested in the answer, then method of nines is the
> simplest and fastest way of getting it.
>
> If you haven't seen it, then what kind of numerical methods class did
> you attend? You must be way younger than me. I'm thinking the generation
> after me with access to much better computers didn't need all that nitty
> gritty to get by, and therefore profs simply omitted a bulk of such
> material from the courses.
>
> With an almost two lines of program, one can find a number consisting
> only of digits of 9, for which the denominator of the fraction is a
> divisor. Then with one more line of code, the fraction is rewritten with
> that denominator consisting of 9's. The numerator thus obtained will be
> the decimals that get repeated. As simple as that.
>
> How many lines of code is that? 3 ? At most 4.
>
> Example. Say we want to find the repeating decimals of 14/33 without
> actually dividing the two numbers to visually find that out, but by
> knowing in advance what those repeating decimals will be.
>
> 1- find the smallest number consisting of only digits of 9 that 33 is a
> divisor for:
>
> 33 x 3 = 99
>
> 2. apply this factor of 3 to the fraction:
>
> 14/33 x 3/3 = 42/99
>
> 3. therefore the repeating decimals are 42.
>
> That's how you have your soup :)
>
> The cases involving other numbers appearing in the decimals before
> repeating decimals begin are quite similar. You first convert the
> fraction to different pieces to separate the repeating part of decimals
> from the rest. This turns the term with repeating decimals to have a
> bunch of zeros coming before the repeating part begins. Then you use a
> slightly different form of method of nines (called method of nines with
> zeros) to find the repeating decimals.
>
>
>
>

Method of nines have some more detail than just that. The example and
method I mentioned above applies only to cases where numerator and
denominator are of the same order of magnitude. If denominator is of a
magnitude of order n more than the numerator, then n zeros appear in the
repeating decimals before nonzeros in them begin. Example:

1/13 (i.e. there will be one repeating zero before non-zero repeating
decimals begin)

1. find the nines: (a two liner loop)

13 x 76923 = 999999

2. apply the factor:

1/13 x 76923/76923 = 76923/999999

3. therefore the repeating decimals are 076923 (not 76923).

Now let's take the example you brought earlier:

1/31

1. There will be one repeating zero before non-zero digits of repeating
decimals begin.

2. get the nines:

31 x 32258064516129 = 999999999999999

3. apply the factor:

1/31 x 32258064516129/32258064516129 = 32258064516129/999999999999999

4. therefore the repeating decimals will be 032258064516129

Note that the zeros coming before non-zero digits begin, are not a case
of zeros after radix that are _not_ part of the repeating decimals. The
latter cases are handled by method of nines with zeros which is slightly
different from method of nines, and I won't go into it. Those interested
can dig it themselves.

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On 3/8/2024 8:20 AM, Nuxxie wrote:
> Project Euler, a well respected forum for programmers, doesn't think
> that it's a waste of time:
>
> https://projecteuler.net/problem=26
>

It probably has moderators. So fuck that. I won't be there even if they
pay me.

But, sci.physics and sci.math doesn't think that it's a waste of time
either :) So as I said, I suggest that you include that newsgroup in
your challenge posts. Such posts would have a better chance to get
worked on than in a place like COLA.

I'm never in sci.math though cause that's not my background. But
sci.physics has room for math questions as well, and for good reasons
too. Especially the baby problems :)

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On Thu, 7 Mar 2024 21:25:16 -0600, Physfitfreak wrote:

> (method of nines)

Ah, nines. When I was in elementary school I got pretty good with the
Trachtenberg system. I would check the answer by casting out the nines.

https://en.wikipedia.org/wiki/Trachtenberg_system
https://en.wikipedia.org/wiki/Casting_out_nines

That really pissed off my teachers. "Show your work!"

On 3/8/2024 2:56 PM, rbowman wrote:
> On Thu, 7 Mar 2024 21:25:16 -0600, Physfitfreak wrote:
>
>> (method of nines)
>
> Ah, nines. When I was in elementary school I got pretty good with the
> Trachtenberg system. I would check the answer by casting out the nines.
>
> https://en.wikipedia.org/wiki/Trachtenberg_system
> https://en.wikipedia.org/wiki/Casting_out_nines
>
> That really pissed off my teachers. "Show your work!"

elementary school? wtf?

I figured you were some kind of Young Sheldon-ish savant.

I got very close to perfect scores in high school geometry. It's all
downhill since.

On 3/7/2024 10:25 PM, Physfitfreak wrote:
> On 3/7/2024 12:11 PM, Nuxxie wrote:
>> Programming is about mathematics and not coding.
>>
>> Problem:
>>
>> Any rational number a/b (in lowest terms) can be expressed either
>> as a terminating decimal or a repeating decimal.
>>
>> We consider here only repeating decimals.
>>
>> Given, a and b, which are intergers with any number of digits,
>> find the repeating portion in the decimal expansion.
>>
>> That is, the decimal expansion will be:
>>
>> a1, a2, a3, ... an, [r1, r2, r3 ... rn]
>>
>> where [... ri ...] is the repeating portion.
>>
>> Example:
>>
>> 1/28 = 0.03 571428 571428...
>>
>> The non-repeating, leading portion can be very long and so too can
>> the repeating portion.
>>
>
>
> I don't think the dorks here know such stuff. And the good ones believe
> it's waste of time but what they do (i.e. "fart") is not waste of time.
>
> So I'll go ahead and ruin it for them all :)
>
> (method of nines)

Ruin away, Sandy. You 2 neurotics are the only ones who care about such
nonsense anyway.

On 3/8/2024 1:56 PM, rbowman wrote:
> On Thu, 7 Mar 2024 21:25:16 -0600, Physfitfreak wrote:
>
>> (method of nines)
>
> Ah, nines. When I was in elementary school I got pretty good with the
> Trachtenberg system. I would check the answer by casting out the nines.
>
> https://en.wikipedia.org/wiki/Trachtenberg_system
> https://en.wikipedia.org/wiki/Casting_out_nines
>
> That really pissed off my teachers. "Show your work!"

Hehe :)

Very interesting. So much stuff that can be done with 9s :) All quite
useful. Most of them I have had no encounter with.

I didn't encounter the above "method of nines" until the need for it
came up in grad school.

At one of my jobs, a printing clerk astonished me with calculating large
multiplications in his mind, very fast. I kept asking him how he did it,
and he always only smiled at me :) Every time we were bored we would
begin a race getting the result of some multiplication in mind. Every
single time he won of course. It was fun.

Strange that I remembered him after decades. He was a diehard
southerner, still in war with "Yankees" and craving for lynching Blacks
and jews :) His number one worry in the world was the loss of the pure
White race. It was really on his mind every day. I didn't help that
either, cause I always joked with him, "You're an endangered species",
and he felt the truth of it very deeply :-)

I don't know about right now, but Texas back then had a good portion of
its people just about like how he was. Same world view, and very
concerned and worried. And fiercely anti-Yankee. I mean just as if it
was still 1865. The moment news of Twin Towers coming down hit the TV,
you could see all these pickup trucks driving in the streets carrying
large flags of the Confederates, for days. They were so much hoping that
their militia had done that. A few days later they all got so depressed
and silent, finding out it was a Palestinian-related issue, having
nothing to do with the South. I sensed and saw all that very closely!
South got very depressed by the news of it, and the rest of the USA got
very relieved! Cause they, too, first thought it was the militia.

No idea how things are now, though.

Bad diversion here, but was worth it, cause he was actually a heck of a
guy and deserved to be mentioned at least this much. Very patriotic in a
Southern way. You guys up there don't know what I'm talking about.

Indians have many ways of calculating arithmetic operations. It must be
something that's stressed in their schools. I even have a little book by
an Indian woman that teaches many methods of calculating fast. I used to
know the most useful ones among it but I've forgotten it all now.

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On 3/8/2024 9:20 AM, Lameass Larry Piet wrote:
> On Thu, 07 Mar 2024 21:25:16 -0600, Physfitfreak wrote:
>
>>
>> I don't think the dorks here know such stuff. And the good ones believe
>> it's waste of time but what they do (i.e. "fart") is not waste of time.
>>
>
> Project Euler, a well respected forum for programmers, doesn't think
> that it's a waste of time:
>
> https://projecteuler.net/problem=26

Remember when you said the exact opposite?

"That entire "Project Euler" web site is pure baloney." Feeb July 2021

"What a totally lame project [Project Euler]. It's where all the string
manipulators congregate." Feeb July 2021

phony twat.

On Fri, 8 Mar 2024 15:12:18 -0500, DFS wrote:

> I got very close to perfect scores in high school geometry. It's all
> downhill since.

I never got into that QED stuff. Being a code monkey I'll take some old
Greek's word for it an cut to the chase.

On Fri, 8 Mar 2024 15:16:29 -0600, Physfitfreak wrote:

> At one of my jobs, a printing clerk astonished me with calculating large
> multiplications in his mind, very fast. I kept asking him how he did it,
> and he always only smiled at me Every time we were bored we would begin
> a race getting the result of some multiplication in mind. Every single
> time he won of course. It was fun.

I used to be good at it but it's a definite use it or lose it skill.

I'm a Yankee by birth and even had a great something grandfather that
fought in that war under Miles but I think it was a raw deal. Leaving the
whole slavery red herring aside I am not comfortable with the Roach Motel
theory of government.

https://www.youtube.com/watch?v=jKhGHxO-woc

On 3/8/2024 1:06 PM, Physfitfreak wrote:
> On 3/8/2024 12:07 PM, Physfitfreak wrote:
>>
>
> 4. therefore the repeating decimals will be 032258064516129
>
>
> Note that the zeros coming before non-zero digits begin, are not a case
> of zeros after radix that are _not_ part of the repeating decimals. The
> latter cases are handled by method of nines with zeros which is slightly
> different from method of nines, and I won't go into it. Those interested
> can dig it themselves.
>
>
>
>
>
>
>
>

And that's why your 1/28 couldn't be handled by your software, because I
think it requires method of nines with zeros, not the method of nines.
Whoever created that function, perhaps didn't go as far as you expected.

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On 3/8/2024 10:43 AM, Larry 'Nuxxie' Piet wrote:

> The result:
>
> The decimal expansion will begin with 11 digits followed
> by a repeating porion of length 54,494,498,496.
>
> This number would need 64 GiB of memory, but we could
> write it out to a file using Maxima or bc or GMP.
>
> I won't try but maybe someone else will.
>
> This kind of stuff is what a computer is for. No fucking Netflix.
> No fucking Gmail. Just this.

no gmail? idiot.

no online sports or videos? idiot.

Screw that useless, boring math twaddle. The popular trend today is
Usenet databases:

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