If you examine the decimal equivalents of the reciprocals of the powers of three you come upon a curious result.
period : 1 (30)
period : 1 (30)
period : 3 (31)
period : 9 (32)
period : 27 (33)
You are welcome to continue – but, if the pattern holds, the period of the next fraction will be 81 !! But I want to look at the pattern within the last decimal equivalent, the reciprocal of 35.
At first it looks very predictable, a set of three digits, 004, with each digit increasing by 1 as you move to the next triplet, 115, then 226 and so on.
The pattern seems to break down after 559 – but does it ?
If the pattern were to continue, the next triplet should be 66(10).
Well, the zero is there – what happened to the 1 ?
It seems it got ‘carried’ back and added to the middle 6, creating the 670 set. In a similar fashion, the next set, 77(11) becomes 781. Then 88(12) becomes 892 – but it isn’t- look at the number, it is 893. What happened ?
We need to go further. The next set after 893 should be 99(13). The 1 is carried back to the middle 9, making it a 10 – that 1 is carried back to the first 9, it becomes a 10. Finally, that 1 is carried back to the last digit of the previous triplet. That creates the 893 when we had expected 892. So the 99(13) triplet is now 003. The 003 becomes 004 for the same reason the 892 became 893.
Why is this curious ? Take the 670 triplet for example. How does the middle digit know to change to 7 from a 6 when we haven’t yet calculated the last digit of the triplet, 10, which caused the 6 to change to a 7 ? Go figure !!
The division algorithm is on the next page.
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