(07-31-2022, 08:57 PM)Pete Wrote: For you math folks, if you have a more elegant way to do this, I'd love to see it. Also, please excuse the message for "Eventual infinite repetend." I used this made up term to describe fractions like 1 / 6 where the digits repeat, but not immediately following the decimal point like 1 / 3 does. If you know the correct term for this type of repetend, please let me know.
Pete
There is no official name for the digits that precede the repetend.
I have seen many different terms used on various web pages about repetends such as "non-repeating digits" of the repetend and "prefix" of the repetend.
Personally, I prefer to use "unique digit(s)" of the repetend because that's the term I thought of to best describe the non-repeating part of a repeating decimal number when I started to update a "divide" program I wrote for the TRS-80 Color Computer back in 1989 so that it could find repetends using QB4.5 a few years ago.
So, I guess the short story is that you can call the non-repeating part of a repeating decimal number anything you want to as long as it's within reason / makes sense [it properly describes the digit(s)].
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I guess you have already found that for (1/x), the repetend can be almost any length BUT it will never be greater than (x-1).
(For example, 1 / 7 has a repetend of 6 digits and 1 / 65537 has a repetend of 65536 digits.)
So far as "unique digits", the most I've ever found using 1 divided by a 15 digit number is 48 :
1 / 844,424,930,131,968 = .000000000000001184237892933500309785207112630208333333
(This answer has forty-eight unique digits, and then one digit repeats infinitely.)
There can also be times when the "unique digits" might APPEAR to be the repetend but when the calculation is carried out further the true repetend appears :
10,976.88888888888888877912 / 6 = 1,829.48148148148148146318666666666666666666666
(It looks like "481" will be the repeating pattern at first, but only one digit repeats infinitely.)
The longest TERMINATING decimal number I've found using 1 divided by 15 digits is 49 digits long :
1 / 562,949,953,421,312 = .0000000000000017763568394002504646778106689453125
(This is a 49 digit terminating decimal number.)
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As soon as I can get the program cleaned up a little, I'll upload it. I also need to edit or remove a few references to the .net and .org sites, so it will be at least a few days or more before I can get everything ready.
I'm sure you can easily make use of it to explore repetends and the number of digits required to be checked in order to find them (at least the way I do it).
NOTE : I am not a "professional" mathematician by any means, just someone who became facinated by repeating decimal numbers way back in the 8th grade....