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Re: A Problem To Solve :-)

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On 3/2/2024 6:55 PM, Physfitfreak wrote:
> Note that in general you can write the decimals as sum of fractions with 
> powers of 10 in the denominators of them. If you replace those 10s by 2 
> x 5 then you can convert that decimal number to a fraction with nothing 
> in its denominator but factors of 2 and 5 _only_. Since this holds for 
> any fraction with terminating decimal, then all you need to show to 
> prove the fraction does not have terminating decimals is to show that a 
> prime other than 2 or 5 is also a factor in the denominator.


I missed one logical step. Here is a better explanation.

Any fraction can be written as a number with decimals that either 
terminate, or continue for ever.

But _if_ the fraction has terminating decimals, then the number can be 
written as a _finite_ sum of fractions with various powers of 10 in 
their denominators. Like for instance:

11/8 = 1.375 = 1/[(10)^0] + 3/[(10)^1] + 7/[(10)^2] + 5/[(10)^3]

      = 1 + 3/(2 x 5) + 7/(2^2 x 5^2) + 5/(2^3 x 5^3)

Now add these fractions. You place the greatest common divisor which is 
(2^3 x 5^3) in the denominator of the resultant fraction, and use the 
elementary method of adding the fractions to get:

     denominator of the result = (2^3 x 5^3)
     numerator of the result = (2^3 x 5^3) + 3(2^2 x 5^2) + 7(2 x 5) + 5

So, as you see, for a fraction with terminating decimals you would 
_only_ have powers of 2 and/or 5 in the denominator, plus the fraction 
will be exactly equal to what it first was (in this case reducing to 11/8).

But with a fraction with non-terminating decimals, say 11/6, you would have:

11/6 = 1.8333333333... = 1/[(10)^0] + 8/[(10)^1] + 3/[(10)^2] + 
3/[(10)^3] + ...

So you can't trivially get the greatest common divisor for it so you 
could sum the fractions. To be able to make the sum, you can do this:

11/6 = 1/[(10)^0] + 8/[(10)^1] + 3/(90)

Now you'd be able to perform the sum:

      denominator of the result = 90 = 2 x 5 x 3 x 3
      numerator of the result  = 90 + 72 + 3 = 165

and as you see, the denominator now has had to have prime number(s) 
other than 2 or 5.

So this check can be used to easily find whether 11/6 has terminating or 
non-terminating decimals:

denominator = 2 x 3

and since the prime 3 appears there, the fraction will have 
non-terminating decimals.



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Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-12 01:39 -0600
  Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-14 12:50 -0600
    Re: A Problem To Solve :-) Lester Thorpe <lt@gnu.rocks> - 2024-02-14 20:33 +0000
      Re: A Problem To Solve :-) Athel Cornish-Bowden <me@yahoo.com> - 2024-02-15 11:11 +0100
      Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-15 13:22 -0600
    Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-18 01:51 -0600
      Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-21 00:38 -0600
        Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-27 16:13 -0600
          Re: A Problem To Solve :-) Kuang Tze Kui <ugdiun@kgknudu.cn> - 2024-02-27 22:46 +0000
          Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-01 20:50 -0600
            Re: A Problem To Solve :-) Farley Flud <ff@linux.rocks> - 2024-03-02 12:10 +0000
              Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-02 12:06 -0600
                Re: A Problem To Solve :-) Farley Flud <ff@linux.rocks> - 2024-03-02 19:44 +0000
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-02 21:55 -0600
                Re: A Problem To Solve :-) Farley Flud <ff@linux.rocks> - 2024-03-03 10:24 +0000
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-03 10:03 -0600
                Re: A Problem To Solve :-) rbowman <bowman@montana.com> - 2024-03-03 20:21 +0000
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-03 14:46 -0600
                Re: A Problem To Solve :-) rbowman <bowman@montana.com> - 2024-03-04 01:27 +0000
                Re: A Problem To Solve :-) Farley Flud <ff@linux.rocks> - 2024-03-03 19:28 +0000
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-03 14:34 -0600
                Re: A Problem To Solve :-) Stéphane CARPENTIER <sc@fiat-linux.fr> - 2024-03-03 10:33 +0000
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-03 14:03 -0600
                Re: A Problem To Solve :-) Dowell Forgács Barabás <keelaa@dsb.hu> - 2024-03-04 16:59 +0000
                Re: A Problem To Solve :-) Stéphane CARPENTIER <sc@fiat-linux.fr> - 2024-03-03 10:26 +0000
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-03 10:08 -0600
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-03 14:04 -0600
                Re: A Problem To Solve :-) Stéphane CARPENTIER <sc@fiat-linux.fr> - 2024-03-03 20:42 +0000
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-03 14:48 -0600
                Re: A Problem To Solve :-) Stéphane CARPENTIER <sc@fiat-linux.fr> - 2024-03-03 21:20 +0000
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-03 15:30 -0600
            Re: A Problem To Solve :-) Farley Flud <ff@linux.rocks> - 2024-03-02 14:09 +0000
              Re: A Problem To Solve :-) Farley Flud <fflud@gnu.rocks> - 2024-03-02 19:18 +0000
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-02 19:01 -0600
              Re: A Problem To Solve :-) Chris Ahlstrom <OFeem1987@teleworm.us> - 2024-03-02 16:22 -0500
                Re: A Problem To Solve :-) Farley Flud <ff@linux.rocks> - 2024-03-02 22:27 +0000
                Re: A Problem To Solve :-) rbowman <bowman@montana.com> - 2024-03-03 00:09 +0000
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-02 19:17 -0600
                Re: A Problem To Solve :-) rbowman <bowman@montana.com> - 2024-03-03 04:18 +0000
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-03 10:17 -0600
              Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-02 18:55 -0600
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-03 12:41 -0600
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-03 13:02 -0600
                Re: A Problem To Solve :-) Farley Flud <ff@linux.rocks> - 2024-03-03 19:25 +0000
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-03 14:22 -0600
                Re: A Problem To Solve :-) rbowman <bowman@montana.com> - 2024-03-04 03:01 +0000
            Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-05 14:42 -0600
              the answer is “Pwnage” -hh <recscuba_google@huntzinger.com> - 2024-03-05 18:20 -0500
                Re: the answer is “Pwnage” Farley Flud <ff@linux.rocks> - 2024-03-05 23:47 +0000
                Re: the answer is “Pwnage” Physfitfreak <Physfitfreak@gmail.com> - 2024-03-05 22:52 -0600
                Re: the answer is “Pwnage” Physfitfreak <Physfitfreak@gmail.com> - 2024-03-05 22:41 -0600
              Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-07 17:29 -0600
                Re: A Problem To Solve :-) Chris Ahlstrom <OFeem1987@teleworm.us> - 2024-03-07 19:18 -0500
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-07 19:02 -0600
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-07 19:27 -0600
                Re: A Problem To Solve :-) Chris Ahlstrom <OFeem1987@teleworm.us> - 2024-03-08 07:38 -0500
                Nice. </Sarcasm> "Relf"  <Usenet@Jeff-Relf.Me> - 2024-03-07 16:46 -0800
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-10 16:11 -0500
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-16 18:48 -0500
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-17 01:04 -0500
                Re: A Problem To Solve :-) Farley Flud <ff@linux.rocks> - 2024-03-18 19:46 +0000
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-18 16:18 -0500
                Re: A Problem To Solve :-) Farley Flud <ff@linux.rocks> - 2024-03-18 21:39 +0000
                Re: A Problem To Solve :-) rbowman <bowman@montana.com> - 2024-03-18 23:15 +0000
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-18 19:02 -0500
                Re: A Problem To Solve :-) rbowman <bowman@montana.com> - 2024-03-19 04:56 +0000
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-19 12:14 -0500
                Re: A Problem To Solve :-) Chris Ahlstrom <OFeem1987@teleworm.us> - 2024-03-19 07:51 -0400
                Re: A Problem To Solve :-) Chris Ahlstrom <OFeem1987@teleworm.us> - 2024-03-18 17:58 -0400
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-22 19:07 -0500

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